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+{"text":"\\section{Introduction}\n\\noindent Detecting anomalies in real-time data sources is critical thanks to the steady rise in the complexity of modern systems, ranging from satellite system monitoring to cyber-security. Such systems often produce multi-channel time series data that automatically detecting anomalous moments can be quite challenging to any anomaly detection (AD) system due to its intrinsic inter-correlation, seasonality, trendiness, and irregularity traits. Speedy detection, along with timely corrective measures before any catastrophic failure, are also key considerations for time-series AD systems.\n\n Multivariate time-series (MTS) AD on seasonality-heavy data can be challenging to most techniques proposed in the literature. Classical time-series forecasting techniques, such as Autoregressive Integrated Moving Average (ARIMA) \\cite{arima} and Statistical Process Control (SPC) \\cite{spc}, in general cannot adequately capture the inter-dependencies among MTS. Also, classical density or distance-based models, such as K-Nearest Neighbors (KNN) \\cite{knn}, usually ignore the effect of temporal dependencies and\/or seasonality in time series. In recent years, deep learning architectures have achieved great success due to their ability to learn the latent representation of normal samples, such as Auto-encoders \\cite{deep-ae} and Generative Adversarial Networks (GAN) \\cite{gan2}. However, such advanced AD methods suffer from high false positive rate (FPR) when applied to seasonal MTS \\cite{season-fpr}. Furthermore, majority of the existing AD methods are built on an unrealistic assumption that the training data is contamination free, which is rarely the case in real-world applications.\n\n\\begin{figure}[t!]\n  \\centering \n  \\includegraphics[width=0.83\\columnwidth]{Figures\/GAN_architecture2.png}\n  \\caption{RSM-GAN architecture with loss definitions}\n  \\label{fig:GAN} \n\\end{figure}\n\n This paper explores some of the challenges in real-world MTS, namely multi-period seasonality and training data contamination, by proposing a GAN-based architecture, named Robust Seasonal Multivariate GAN (RSM-GAN), that has an encoder-decoder-encoder structure as shown in Figure \\ref{fig:GAN}. Co-training of an additional encoder enables this model to be robust against noise and contamination in training data. A novel smoothed attention mechanism is employed in recurrent layers of the encoders to account for multiple seasonality patterns in a data-driven manner. Also, we propose a causal inference framework for root cause identification. We conduct extensive empirical studies on synthetic data with various levels of seasonality and contamination, along with a real-world encryption key dataset. The results show superiority of RSM-GAN for timely and precise detection of anomalies and root causes as compared to state-of-the-art baseline models.\n\n\n Contributions of our work can be summarized as follows: (1) we propose a convolutional recurrent Wasserstein GAN architecture (RSM-GAN) that detects anomalies in MTS data precisely\n; (2) we explicitly model seasonality as part of the RSM-GAN architecture through a novel smoothed attention mechanism; (3) we apply an additional encoder to handle the contaminated training data; (4) we propose a scoring and causal inference framework to accurately and timely identify anomalies and to pinpoint unspecified number of root cause(s). The RSM-GAN framework enables system operators to react to abnormalities swiftly and in real-time manner, while giving them critical information about the root cause(s) and severity of the anomalies.\n\n\n\n\\section{Related Work}\nMTS anomaly detection has long been an active research area because of its critical importance in monitoring high risk tasks. \nClassical time series analysis models such as Vector Auto-regression (VAR) \\cite{var}, and latent state based models such as Kalman Filters \\cite{kalman} have been applied to MTS, but they are sensitive to noise and prone to misspecification. Classical machine learning methods are also widely used that can be categorized into distance-based methods such as the KNN \\cite{knn}, classification-based methods such as One-Class SVM \\cite{one-svm}, and ensemble methods such as Isolation Forest \\cite{iforest}. These general purpose AD methods do not account for temporal dependencies nor the seasonality patterns that are ubiquitous in MTS, which lead to non-satisfactory performance in real applications. Recently, deep neural networks with architectures such as auto-encoder and GAN-based, have shown great promise for AD in various domains. Autoencoder-based models learn low-dimensional latent representations and utilize reconstruction errors as the score to detect anomalies \\cite{autoencoder1,autoencoder2,autoencoder3}. GAN-based models leverage adversarial learning for mapping high-dimensional training data to the latent space and later use latent space to calculate reconstruction loss as the anomaly score \\cite{ganomaly,gan3image,gan4image}.\n\nRecurrent neural network (RNN)-based approaches have been employed for MTS AD \\cite{lstmed,rnn-ad}. \\cite{madgan} proposed GAN-AD, which is the first work to apply recurrent GAN-based approach to MTS anomaly detection. However, the GAN-AD architecture is not efficient for real-time anomaly detection due to costly invert mapping step while testing. Multi-Scale Convolutional Recurrent Encoder-Decoder (MSCRED) is a deep autoencoder-based AD framework applied to MTS data \\cite{mscred}. MSCRED captures inter-correlation and temporal dependency between time-series by convolutional-LSTM networks and therefore, achieves state-of-the-art performance. However, non of these models account for seasonal and contaminated training data.\nA few studies have addressed seasonality by applying Fourier transform, such as Seasonal ARIMA \\cite{sarima}, or time-series decomposition methods \\cite{fourier-season}. Such treatments are inefficient when applied to high-dimensional MTS data while they do not account for multi-period seasonality. RSM-GAN is designed to address heavy seasonality using attention mechanism, and to improve robustness to severe levels of contamination by co-training of an encoder.\n \n\n\n\\section{Methodology}\\label{method}\nWe define an MTS as $X=(X_1,...,X_n)\\in\\mathbb{R}^{n\\times T}$, where $n$ is the number of time series, and $T$ is the length of the training data. We aim to predict two AD outcomes: 1) the time points $t$ after $T$ that are associated with anomalies, and 2) time series $ i \\in \\{1,..,n\\}$ causing the anomalies.\nIn the following, we first describe how we transform the raw MTS to be consumed by a convolutional recurrent GAN. Then we introduce the RSM-GAN architecture and the seasonal attention mechanism. Finally, we describe anomaly scoring and causal inference procedure to identify anomalies and the root causes in the prediction phase.\n\n\\subsection{RSM-GAN Framework}\n\\subsubsection{MTS to Image Conversion} To extend GAN to MTS and to capture inter-correlation between multiple time series, we convert the MTS into an image-like structure through construction of the so-called multi-channel correlation matrix (MCM), inspired by \\cite{song2018deep,mscred}.\nSpecifically, we define multiple windows of different sizes $W=(w_1,...,w_C)$, and calculate the pairwise inner product (correlation) of time series within each window. At a specific time point $t$, we generate $C$ matrices (channels) of shape $n\\times n$, where each element of matrix $S_t^c$ for a window of size $w_c$ is calculated by this formula:\n\\begin{equation}\n    s_{ij}=\\frac{\\sum_{\\delta=0}^{w_c}x_i^{t-\\delta}\\cdot x_j^{t-\\delta}}{w_c}\n\\end{equation}\n In this work, we select windows $W=(5, 10, 30)$. This results in $3$ channels of $n\\times n$ correlation matrices for time point $t$ noted as $S_t$. To convert the span of MTS into this shape, we consider a step size $p=5$. Therefore, $X$ is transformed to $S=(S_1,...,S_M)\\in\\mathbb{R}^{M \\times n\\times n\\times C}$, where  $M=\\lfloor\\frac{T}{p}\\rfloor$ steps represented by MCMs. Finally, to capture the temporal dependency between consecutive steps, we stack $h=4$ previous steps to the current step $t$ to prepare the input to the GAN-based model. Later, we extend MCM to also capture seasonality unique to MTS.\n\n\\subsubsection{RSM-GAN Architecture} \nThe idea behind using a GAN to detect anomalies is intuitive. During training, a GAN utilize adversarial learning to capture the distribution of the input data. Then, if anomalies are present during prediction, the networks would fail to reconstruct the input, thus produce large losses. In most deep AD literature, the training data is explicitly assumed to be normal with no contamination. In a study, \\cite{encoder} have shown that simultaneous training of an encoder with GAN improves the robustness of the model towards contamination. This is mainly because the joint encoder forces similar inputs to lie close to each other by optimizing the latent loss, and thus account for the contamination while training. To this end, we adopt an encoder-decoder-encoder structure \\cite{ganomaly}, with the additional encoder, to optimize input reconstruction in both original and latent space. Specifically, in Figure \\ref{fig:GAN}, the generator $G$ has autoencoder structure that the encoder ($G_E$) and decoder ($G_D$) interact with each other to minimize the contextual loss: the $l_2$ distance between input $x$ and reconstructed input $G(x)=x'$. An additional encoder $E$ is trained jointly with the generator to minimize the latent loss: the $l_2$ distance between latent vector $z$ and reconstructed latent vector $z'$. Finally, the discriminator $D$ is tasked to distinguish between the original input $x$ and the generated input $x'$. Following the recent advancements on GAN, we employ the Wasserstein GAN with gradient penalty (WGAN-GP) \\cite{wgan-gp} to ensure stable gradients, avoid the collapsing mode, and thus improve the training. \nTherefore, the final objective functions for the generator and discriminator are as following:\n\\begin{equation}\n\\begin{aligned}\n    L_G =  \\min_{G}\\min_{E} & \\Big(  w_1\\mathbb{E}_{x\\sim p_x} \\| x-x' \\|_2 +  w_2 \\mathbb{E}_{x\\sim p_x} \\| G_E(x)-E(x') \\|_2 \\\\ & +  w_3  \\mathbb{E}_{x\\sim p_x}[f_\\theta(x')]\\Big)\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n    L_D = \\max_{\\theta \\in \\Theta} \\mathbb{E}_{x\\sim p_x}[f_\\theta(x)] - \\mathbb{E}_{x\\sim p_x} [f_\\theta(x')]\n\\end{equation}\n\\noindent where $\\theta$ is the discriminator's parameter and ($w_1$, $w_2$, $w_3$) are weights controlling the effect of each loss. The choice of contextual loss weight, has the largest effect on training convergence and we chose (50, 1, 1) weights for optimal training. We employ Adam optimizer to optimize the above losses for $G$ and $D$ alternatively. Each encoder in Figure \\ref{fig:GAN} is composed of multiple convolutional layers, each followed by convolutional-LSTM layers to capture both spatial and temporal dependencies in input. The detailed inner structure of each component is described in Appendix \\ref{sec:apx_inner}.\n\n\n\\subsubsection{Seasonality Adjustment via Attention Mechanism} \nIn order to adjust the seasonality in MTS data, we stack previous seasonal steps to the input data, and allow the convolutional-LSTM cells in the encoder to capture temporal dependencies through an attention mechanism. Specifically, in addition to $h$ previous immediate steps, we append $m_i$ previous seasonal steps per seasonal pattern $i$. To illustrate, assume the input has both the daily and weekly seasonality. To prepare input for time step $t$, we stack MCMs of up to $m_1$ days ago at the same time, and up to $m_2$ weeks ago at the same time.  \nAdditionally, to account for the fact that seasonal patterns are often not exact, we smooth the seasonal steps by averaging over steps in a neighboring window of 6 steps.\\\\\nMoreover, even though the $h$ previous steps are closer to the current time step, but the previous seasonal steps might be a better indicator to reconstruct the current step. Therefore, an attention mechanism is employed to assign weights to each step based on the similarity of the hidden state representations in the last layer using:\n\\begin{equation}\n    \\mathcal{H'}_t = \\sum_{i\\in (t-N,t)} \\alpha_i \\mathcal{H}_i  \\text{, where } \n    \\alpha_i=\\mathrm{softmax}\\Big(\\frac{Vec(\\mathcal{H}_t)^T Vec(\\mathcal{H}_i)}{\\mathcal{X}}\\Big)\n\\end{equation}\nwhere $N=h+\\Sigma m_i$, $Vec(\\cdot)$ denotes the vector, and $\\mathcal{X}=5$ is the rescaling factor. Figure \\ref{fig:attention} presents the structure of the described smoothed attention mechanism.\nFinally, to make our model even more adaptable to real-world datasets that often exhibit holiday effects, we multiply the attention weight $\\alpha_i$ by a binary bit $b_i \\in \\{0,1\\}$, where $b_i=0$ in case of holidays or other exceptional behavior in previous steps. This way, we eliminate the effect of undesired steps from modeling the current step.\n\n\\begin{figure}[t]\n  \\centering \n  \\includegraphics[width=0.88\\columnwidth]{Figures\/attn.png} \n  \\caption{Smoothed attention mechanism}\n  \\label{fig:attention} \n\\end{figure}\n\n\\subsection{Prediction Phase}\n\\subsubsection{Anomaly Score Assignment} \\label{sec:AS_assign}\nThe residual MCM matrix from the first channel, $R_x=x_{:,:,0}-x_{:,:,0}'$, are indicative of anomalies while predicting. \nWe define broken tiles as the elements of $R_x$ that have error value of greater than $\\theta_b$. Previous studies have defined a scoring method based on the number of broken tiles in $R_x$ that we call context$_{b}$ \\cite{mscred}. However, this score is insensitive to non-severe anomalies, and lowering the threshold would result in high FPR. Since each row\/column in $R_x$ is associated with a time series, the ones with the largest number of broken tiles are contributing the most to the anomalies. Therefore, by defining a threshold $\\theta_h \\leq \\theta_b$, we propose to only count the number of broken tiles in rows\/columns with more than half broken and name this method context$_{h}$. The above thresholds $\\theta = \\beta \\times \\eta_{.996}(E_\\mathrm{train})$, which is calculated based on $99.6^{th}$ percentile of error in the training residual matrices, and the best $\\beta$ is chosen by a grid search on validation set.\n\n\n\\subsubsection{Root Cause Framework} Large errors in rows\/columns of $R_x$ are indicative of anomalous behavior in those time series. To identify which are contributing the most to anomalies, we need a root-cause scoring system to assign a score to each time series based on the severity of its errors. We present two different methods: 1) number of broken tiles (NB) (using the optimized $\\theta_b$), and 2) sum of absolute errors (AE). \n\n Furthermore, the number of root causes, $k$, is unknown in real-world applications. Despite previous studies, we allow the elbow method \\cite{elbow} to find the optimal $k$ number of time series from the root cause scores. In this approach, by sorting the scores and plotting the curve, we aim to find the point where the scores become very small and close to each other (elbow point). Time series associated with the scores greater than elbow point are identified as root causes. Thus, we define the elbow point as the point with maximum distance from a vector that connects the first and last scores. \n\n\n\\section{Experimental Setup}\n\\subsection{Data}\n\\textbf{Synthetic Data:} We generate synthetic sinusoidal-based multivariate time-series with different seasonal period and contamination levels to evaluate our model comprehensively. Ten time series with 2 months worth of data by minute sampling frequency are generated with length $T=80,640$. Also, anomalies with varying duration and intensity are injected to the training and test sets. The detailed data generation process in discussed in Appendix \\ref{sec:apx_synth}.\n\n\\textbf{Encryption Key Data:} Our encryption-key dataset contains $7$ time series generated from a project's encryption process. Each time series represents the number of requests for a specific encryption key per minute. The dataset contains $4$ months of data with length $T=156,465$. Four anomalies with various length and scales are identified in the test sequence by a security expert, and we randomly injected $5$ additional anomalies into both the training and test sets.\n\n\n\\subsection{Evaluation}\nRSM-GAN is compared against two classical machine learning models, i.e., One-class SVM (OC-SVM) \\cite{one-svm} and Isolation Forest \\cite{iforest}. and a deep autoencoder-based AD model called MSCRED \\cite{mscred}. MSCRED is run in a sufficient number of epochs and its best performance is reported. We also have tried GAN-AD \\cite{madgan} in multiple settings, but the results are not reported here due to its inefficient and faulty performance. For evaluation, in addition to precision, recall, FPR, and F1 score, we include the \\textbf{Numenta Anomaly Benchmark (NAB)} score \\cite{numenta}. NAB is a standard open source framework for evaluating real-time AD algorithms. The NAB assigns score to each true positive detection based on their relative position to the anomaly window (by a scaled \\textit{sigmoid} function), and negative score to false positives and false negatives.\n\n In all experiments, the first half of the time series are used for training and the remainder for validation\/test by 1\/5 ratio. RSM-GAN is implemented in Tensorflow and trained in 300 epochs, in batches of size $32$, on an AWS Sagemaker instance with four $16$GB GPUs.\nAll the results are produced by an average over five independent runs.\n\n\\section{Results and Discussion}\n\\subsection{Anomaly Score and Root Cause Assessment}\nIn this section, we first compare the performance of our RSM-GAN using the two context$_{b}$ and context$_{h}$ anomaly score assignment methods described in Section \\ref{sec:AS_assign}. Table \\ref{tab:scores} reports the performance on synthetic MTS with no contamination and seasonality with the optimized threshold as illustrated. As we can see, our proposed context$_{h}$ method outperforms context$_{b}$ for all metrics except of recall. Specifically, context$_{h}$ improves the precision and FPR by $6.2\\%$ and $0.08\\%$, respectively. Since the same result holds for other settings, we report the results based on context$_{h}$ scoring in the following experiments.\n\n\\begin{table}[tb]\n\\caption{Different anomaly score assignment performances}\n\\begin{adjustbox}{width=0.97\\columnwidth,center}\n\\begin{tabular}{c|c|ccccc}\n\\textbf{Score} &\\textbf{Threshold}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} &\\textbf{FPR} & \\textbf{NAB Score} \\\\ \\hline\ncontext$_{b}$ & 0.0019 & 0.784 & \\textbf{0.958} &\t0.862 &\t0.0023 &\t0.813 \\\\\ncontext$_{h}$ & 0.00026 &\\textbf{0.846} &\t0.916 &\t\\textbf{0.880} &\t\\textbf{0.0015} &\t\\textbf{0.859} \\\\\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:scores}\n\\end{table}\n\n Next, we compare the two root cause scoring methods for the baseline MSCRED and our RSM-GAN. Root causes are identified based on the average of $R_x$'s in an anomaly window. \nSynthetic data used in this experiment has two combined seasonal patterns and ten anomalies in training sequence. Overall, RSM-GAN outperforms MSCRED (marked by *). As the results suggest, the NB method performs the best for MSCRED. However, for RSM-GAN the AE leads to the best performance. Since the same results hold for other settings, we report NB for MSCRED and AE for RSM-GAN in subsequent sections.\n\n\\begin{table}[tb]\n\\caption{Different root cause identification performances}\n\\begin{adjustbox}{width=0.9\\columnwidth,center}\n\\begin{tabular}{c||c|ccc}\n\\textbf{Model} &\\textbf{Scoring}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} \\\\ \\hline\n\\multirowcell{2}{MSCRED} \n& Number of broken (NB) & 0.5154 &\t\\textbf{0.7933} &\t\\textbf{0.6249} \\\\\n& Absolute error (AE) & \\textbf{0.5504} &\t0.7066 &\t0.6188 \\\\ \\hline\n\n\\multirowcell{2}{RSM-GAN} \n& Number of broken (NB) & 0.4960 &\t0.8500 &\t0.6264 \\\\\n& Absolute error (AE) & \\textbf{0.6883*} &\t\\textbf{0.8666*} &\t\\textbf{0.7672*} \\\\ \n\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:rootcause}\n\\end{table}\n\n\n\n\\subsection{Contamination Resistance Assessment}\nWe assess the robustness of RSM-GAN towards different levels of contamination in training data. In this experiment, we start with no contamination and at each subsequent level, we add $5$ more random anomalies with varying duration to the training data. The percentages presented in the first column in Table \\ref{tab:contamination} shows the proportions of the anomalous time points in train\/test time span.  Results in Table \\ref{tab:contamination} suggest that our proposed model outperforms all baseline models at all contamination levels for all metrics except of the recall. Note that the 100\\% recall for classic baselines  are at the expense of FPR as high as $26.4\\%$. Furthermore, comparison of the NAB score shows that our model has more timely detection and less irrelevant false positives. Lastly, as we can see, the MSCRED performance drops drastically as the contamination level increases, due to the normal training data assumption and the encoder-decoder architecture.\n\n\\begin{table*}[hbt]\n\\caption{Model Performance on synthetic data with different levels of training contamination and random seasonality}\n\\begin{adjustbox}{width=0.69\\textwidth,center}\n\\begin{tabular}{c||c|ccccc|c}\n\\textbf{Contamination} &\\textbf{Model}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} &\\textbf{FPR} & \\textbf{NAB Score} & \\textbf{Root Cause Recall} \\\\ \\hline\n\\multirowcell{4}{No contamination \\\\ train: 0 (0) \\\\ test: 10 (\\%0.82)} & OC-SVM & 0.1581 &\t\\textbf{1.0000} &\t0.2730 &\t0.0473 &\t-8.4370 & - \\\\\n& Isolation Forest & 0.0326 &\t\\textbf{1.0000} &\t0.0631 &\t0.2640 &\t-51.4998 & - \\\\\n& MSCRED & 0.8000 &\t0.8450 &\t0.8219 &\t0.0018 &\t0.7495 & \\textbf{0.7533} \\\\\n& RSM-GAN & \\textbf{0.8461} &\t0.9166 &\t\\textbf{0.8800} &\t\\textbf{0.0015} &\t\\textbf{0.8598} &\t0.6333 \\\\ \\hline\n\n\\multirowcell{4}{Mild contamination \\\\ train: 5 (\\%0.43) \\\\ test: 10 (\\%0.76)} \n& OC-SVM & 0.2810 &\t\\textbf{1.0000} &\t0.4387 &\t0.0218 &\t-3.3411 & - \\\\\n& Isolation Forest & 0.3134 &\t\\textbf{1.0000} &\t0.4772 &\t0.0187 &\t-2.7199 & - \\\\\n& MSCRED & 0.6949 &\t0.6029 &\t0.6457 &\t0.0023 &\t0.2721 &\t0.5483 \\\\\n& RSM-GAN & \\textbf{0.8906} &\t0.7500 &\t\\textbf{0.8143} &\t\\textbf{0.0009} &\t\\textbf{0.8865} &\t\\textbf{0.7700} \\\\ \\hline\n\n\\multirowcell{4}{Medium contamination \\\\ train: 10 (\\%0.82) \\\\ test: 10 (\\%0.85)} \n& OC-SVM & 0.4611 &\t\\textbf{1.0000} &\t0.6311 &\t0.0113 &\t-1.2351 & - \\\\\n& Isolation Forest & 0.6311 &\t\\textbf{1.0000} &\t0.7739 &\t0.0056 &\t-0.1250 & - \\\\\n& MSCRED & 0.6548 &\t0.7143 &\t0.6832 &\t0.0036 &\t0.2712 &\t0.6217 \\\\\n& RSM-GAN & \\textbf{0.8553} &\t0.8442 &\t\\textbf{0.8497} &\t\\textbf{0.0014} &\t\\textbf{0.8511} &\t\\textbf{0.8083} \\\\ \\hline\n\n\\multirowcell{4}{Severe contamination \\\\ train: 15 (\\%1.19) \\\\ test: 15 (\\%1.18)} \n& OC-SVM & 0.5691 &\\textbf{\t1.0000} &\t0.7254 &\t0.0102 &\t-0.3365 & - \\\\\n& Isolation Forest & 0.8425 &\t\\textbf{1.0000} &\t\\textbf{0.9145} &\t0.0025 &\t0.6667 & - \\\\\n& MSCRED & 0.5493 &\t0.7290 &\t0.6265 &\t0.0080 &\t0.0202 &\t0.6611 \\\\\n& RSM-GAN & \\textbf{0.8692} &\t0.8774 &\t0.8732 &\t\\textbf{0.0018} &\t\\textbf{0.8872} &\t\\textbf{0.8133} \\\\ \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:contamination}\n\\end{table*}\n\n\n\n\n\n\n\n\\begin{table*}[htb!]\n\\caption{Model performance on synthetic data with different seasonal patterns and no training contamination}\n\\begin{adjustbox}{width=0.69\\textwidth,center}\n\\begin{tabular}{c||c|ccccc|c}\n\\textbf{Seasonality} &\\textbf{Model}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} &\\textbf{FPR} & \\textbf{NAB Score} & \\textbf{Root Cause Recall} \\\\ \\hline\n\n\\multirowcell{4}{Random seasonality} \n& OC-SVM & 0.4579 &\t0.9819 &\t0.6245 &\t0.0097 &\t-8.6320 & - \\\\\n& Isolation Forest & 0.0325 &\t\\textbf{1.0000} &\t0.0630 &\t0.2646 &\t-51.606 &- \\\\\n& MSCRED & 0.8000 &\t0.8451 &\t0.8219 &\t0.0019 &\t0.7495 & \\textbf{0.7533} \\\\\n& RSM-GAN & \\textbf{0.8462} &\t0.9167 &\t\\textbf{0.8800} &\t\\textbf{0.0015} &\t\\textbf{0.8598} & 0.6333 \\\\ \\hline\n\n\\multirowcell{4}{Daily seasonality} \n& OC-SVM & 0.1770 &\t\\textbf{1.0000} &\t0.3008 &\t0.0532 &\t-9.5465 &  - \\\\\n& Isolation Forest & 0.1387 &\t\\textbf{1.0000} &\t0.2436 &\t0.0710 &\t-13.107 & - \\\\\n& MSCRED & 0.7347 &\t0.7912 &\t0.7619 &\t0.0033 &\t0.3775 & \\textbf{0.7467} \\\\\n& RSM-GAN & \\textbf{0.9012} &\t0.7935 &\t\\textbf{0.8439} &\t\\textbf{0.0010} &\t\\textbf{0.5175} & 0.6717 \\\\ \\hline\n\n\\multirowcell{4}{Daily and weekly \\\\ seasonality} \n& OC-SVM & 0.1883 &\t\\textbf{0.9487} &\t0.3142 &\t0.0400 &\t-6.9745 &  - \\\\\n& Isolation Forest & 0.1783 &\t\\textbf{0.9487} &\t0.3002 &\t0.0428 &\t-7.5278 &  - \\\\\n& MSCRED & 0.6548 &\t0.7143 &\t0.6832 &\t0.0036 &\t0.2712 &\t\\textbf{0.6217} \\\\\n& RSM-GAN & \\textbf{0.9000} &\t0.6750 &\t\\textbf{0.7714} &\t\\textbf{0.0008} &\t\\textbf{0.5461} & 0.4650 \\\\ \\hline\n\n\\multirowcell{4}{Weekly and monthly \\\\ seasonality \\\\ with holidays} \n& OC-SVM & 0.2361 &\t\\textbf{0.9444} &\t0.3778 &\t0.0425 &\t-1.7362 & - \\\\\n& Isolation Forest & 0.2783 &\t0.8889 &\t0.4238 &\t0.0321 &\t-1.0773 &  - \\\\\n& MSCRED & 0.0860 &\t0.7059 &\t0.1534 &\t0.0983 &\t-5.1340 & 0.6067 \\\\\n& RSM-GAN & \\textbf{0.6522} &\t0.8108 &\t\\textbf{0.7229} &\t\\textbf{0.0063} &\t\\textbf{0.5617} & \\textbf{0.8667} \\\\ \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:seasonality}\n\\end{table*}\n\n\n\n\\begin{table*}[htb!]\n\\caption{Model performance on encryption key and synthetic two-period seasonal MTS with medium contamination}\n\\begin{adjustbox}{width=0.64\\textwidth,center}\n\\begin{tabular}{c||c|ccccc|c}\n\\textbf{Dataset} &\\textbf{Model}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} &\\textbf{FPR} & \\textbf{NAB Score} & \\textbf{Root Cause Recall} \\\\ \\hline\n\n\\multirowcell{4}{Encryption \\\\ key} \n& OC-SVM & 0.1532 &\t0.2977 &\t0.2023 &\t0.0063 &\t-17.4715& - \\\\\n& Isolation Forest & 0.3861 &\t\\textbf{0.4649} &\t0.4219 &\t0.0028 &\t-6.9343\t& - \\\\\n& MSCRED & 0.1963 &\t0.2442 &\t0.2176 &\t0.0055 &\t-1.1047\t& 0.4709 \\\\\n& RSM-GAN & \\textbf{0.6852} &\t0.4405 &\t\\textbf{0.5362} &\t\\textbf{0.0011} &\t\\textbf{0.2992}\t& \\textbf{0.5093} \\\\ \\hline\n\n\\multirowcell{4}{Synthetic} \n& OC-SVM & 0.6772 &\t0.9185 &\t0.7772 &\t0.0038 &\t-2.7621 & - \\\\\n& Isolation Forest & 0.7293 &\t\\textbf{0.9610} &\t0.8221 &\t0.0033 &\t-2.2490 & - \\\\\n& MSCRED & 0.6228 &\t0.7403 &\t0.6746 &\t0.0043 &\t0.2753 &\t0.6600 \\\\\n& RSM-GAN & \\textbf{0.8884} &\t0.8438 &\t\\textbf{0.8649} &\t\\textbf{0.0010} &\t\\textbf{0.8986} &\t\\textbf{0.7870} \\\\ \\hline\n\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:realworld}\n\\end{table*}\n\n\\subsection{Seasonality Adjustment Assessment}\n Next, we assess the performance of our proposed attention mechanism, assuming no training contamination exists. In the first experiment, synthetic MTS includes 2 months of data, sampled per minute, with only random seasonality. Daily and weekly seasonality patterns are added at each further step. In the last experiment, we simulate $3$ years of hourly data, and add special patterns to illustrate holiday effect in both training and test data. The test set of each experiment is contaminated with $10$ random anomalies. Comparing the results in Table \\ref{tab:seasonality}, RSM-GAN shows consistent performance due to the attention adjustment strategy. All the other baseline models, especially MSCRED's performance deteriorate with increased complexity of seasonal patterns.\nIn the last experiment in Table \\ref{tab:seasonality}, all of the abnormalities injected to holidays are misidentified by the baseline models as anomalies, since no holiday adjustment is incorporated in those models and thus, low precision and high FPR has emerged. In RSM-GAN, multiplying the binary vectors of holidays with the attention weights enables accountability for extreme events and leads to the best performance in almost all metrics. \n\n\n\n\n\n\n\n\n\\subsection{Performance on Real-world Dataset}\n\\noindent This section evaluates our model on a real-world encryption key dataset that has both daily and weekly seasonality. To be comprehensive, we also create a synthetic dataset with similar seasonality patterns and 10 anomalies in the training set. \nFrom Table \\ref{tab:realworld}, we make the following observations: 1) RSM-GAN consistently outperforms all the baseline models for anomaly detection and root cause identification recall in both datasets. 2) Not surprisingly, for all the models, performance on the synthetic data is better than that of encryption key data. It is due to the excessive irregularities and noise in the encryption key data. \n\\begin{figure}[bt]\n  \\centering \n  \\includegraphics[width=\\columnwidth]{Figures\/idps_final.png}\n   \\caption{Anomaly score assignment on encryption key data}\n   \\label{fig:final_plot}\n\\end{figure}\n3) Figure \\ref{fig:final_plot} illustrates the anomaly scores assigned to each time point in test dataset by each algorithm, with the bottom plot presenting the ground truth. As we can see, even though isolation forest has the highest recall rate, it also detects many false positives not related to the actual anomaly windows, leading to negative NAB scores. 4) By comparing our model to MSCRED in Figure \\ref{fig:final_plot}, we can see that MSCRED not only has much higher FPR, but it also fails to capture some anomalies. \n\n\n\n\n\n\n\\section{Conclusion}\nIn this work, we proposed a GAN-based AD framework to handle contaminated and seasonality-heavy multivariate time-series. RSM-GAN leverages adversarial learning to accurately capture temporal and spatial dependencies in the data, while simultaneously training an additional encoder to handle training data contamination. The novel attention mechanism in the recurrent layers of RSM-GAN enables the model to adjust complex seasonal patterns often found in the real-world data. We conducted extensive empirical studies and results show that our architecture together with a new score assignment and causal inference lead to an exceptional performance over advanced baseline models on both synthetic and real-world datasets.\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\section{Introduction}\n\\noindent Detecting anomalies in real-time data sources is critical thanks to the steady rise in the complexity of modern systems, ranging from satellite system monitoring to cyber-security. Such systems often produce multi-channel time series data that automatically detecting anomalous moments can be quite challenging to any anomaly detection (AD) system due to its intrinsic inter-correlation, seasonality, trendiness, and irregularity traits. Speedy detection, along with timely corrective measures before any catastrophic failure, are also key considerations for time-series AD systems.\n\n Multivariate time-series (MTS) AD on seasonality-heavy data can be challenging to most techniques proposed in the literature. Classical time-series forecasting techniques, such as Autoregressive Integrated Moving Average (ARIMA) \\cite{arima} and Statistical Process Control (SPC) \\cite{spc}, in general cannot adequately capture the inter-dependencies among MTS. Also, classical density or distance-based models, such as K-Nearest Neighbors (KNN) \\cite{knn}, usually ignore the effect of temporal dependencies and\/or seasonality in time series. In recent years, deep learning architectures have achieved great success due to their ability to learn the latent representation of normal samples, such as Auto-encoders \\cite{deep-ae} and Generative Adversarial Networks (GAN) \\cite{gan2}. However, such advanced AD methods suffer from high false positive rate (FPR) when applied to seasonal MTS \\cite{season-fpr}. Furthermore, majority of the existing AD methods are built on an unrealistic assumption that the training data is contamination free, which is rarely the case in real-world applications.\n\n\\begin{figure}[t!]\n  \\centering \n  \\includegraphics[width=0.83\\columnwidth]{Figures\/GAN_architecture2.png}\n  \\caption{RSM-GAN architecture with loss definitions}\n  \\label{fig:GAN} \n\\end{figure}\n\n This paper explores some of the challenges in real-world MTS, namely multi-period seasonality and training data contamination, by proposing a GAN-based architecture, named Robust Seasonal Multivariate GAN (RSM-GAN), that has an encoder-decoder-encoder structure as shown in Figure \\ref{fig:GAN}. Co-training of an additional encoder enables this model to be robust against noise and contamination in training data. A novel smoothed attention mechanism is employed in recurrent layers of the encoders to account for multiple seasonality patterns in a data-driven manner. Also, we propose a causal inference framework for root cause identification. We conduct extensive empirical studies on synthetic data with various levels of seasonality and contamination, along with a real-world encryption key dataset. The results show superiority of RSM-GAN for timely and precise detection of anomalies and root causes as compared to state-of-the-art baseline models.\n\n\n Contributions of our work can be summarized as follows: (1) we propose a convolutional recurrent Wasserstein GAN architecture (RSM-GAN) that detects anomalies in MTS data precisely\n; (2) we explicitly model seasonality as part of the RSM-GAN architecture through a novel smoothed attention mechanism; (3) we apply an additional encoder to handle the contaminated training data; (4) we propose a scoring and causal inference framework to accurately and timely identify anomalies and to pinpoint unspecified number of root cause(s). The RSM-GAN framework enables system operators to react to abnormalities swiftly and in real-time manner, while giving them critical information about the root cause(s) and severity of the anomalies.\n\n\n\n\\section{Related Work}\nMTS anomaly detection has long been an active research area because of its critical importance in monitoring high risk tasks. \nClassical time series analysis models such as Vector Auto-regression (VAR) \\cite{var}, and latent state based models such as Kalman Filters \\cite{kalman} have been applied to MTS, but they are sensitive to noise and prone to misspecification. Classical machine learning methods are also widely used that can be categorized into distance-based methods such as the KNN \\cite{knn}, classification-based methods such as One-Class SVM \\cite{one-svm}, and ensemble methods such as Isolation Forest \\cite{iforest}. These general purpose AD methods do not account for temporal dependencies nor the seasonality patterns that are ubiquitous in MTS, which lead to non-satisfactory performance in real applications. Recently, deep neural networks with architectures such as auto-encoder and GAN-based, have shown great promise for AD in various domains. Autoencoder-based models learn low-dimensional latent representations and utilize reconstruction errors as the score to detect anomalies \\cite{autoencoder1,autoencoder2,autoencoder3}. GAN-based models leverage adversarial learning for mapping high-dimensional training data to the latent space and later use latent space to calculate reconstruction loss as the anomaly score \\cite{ganomaly,gan3image,gan4image}.\n\nRecurrent neural network (RNN)-based approaches have been employed for MTS AD \\cite{lstmed,rnn-ad}. \\cite{madgan} proposed GAN-AD, which is the first work to apply recurrent GAN-based approach to MTS anomaly detection. However, the GAN-AD architecture is not efficient for real-time anomaly detection due to costly invert mapping step while testing. Multi-Scale Convolutional Recurrent Encoder-Decoder (MSCRED) is a deep autoencoder-based AD framework applied to MTS data \\cite{mscred}. MSCRED captures inter-correlation and temporal dependency between time-series by convolutional-LSTM networks and therefore, achieves state-of-the-art performance. However, non of these models account for seasonal and contaminated training data.\nA few studies have addressed seasonality by applying Fourier transform, such as Seasonal ARIMA \\cite{sarima}, or time-series decomposition methods \\cite{fourier-season}. Such treatments are inefficient when applied to high-dimensional MTS data while they do not account for multi-period seasonality. RSM-GAN is designed to address heavy seasonality using attention mechanism, and to improve robustness to severe levels of contamination by co-training of an encoder.\n \n\n\n\\section{Methodology}\\label{method}\nWe define an MTS as $X=(X_1,...,X_n)\\in\\mathbb{R}^{n\\times T}$, where $n$ is the number of time series, and $T$ is the length of the training data. We aim to predict two AD outcomes: 1) the time points $t$ after $T$ that are associated with anomalies, and 2) time series $ i \\in \\{1,..,n\\}$ causing the anomalies.\nIn the following, we first describe how we transform the raw MTS to be consumed by a convolutional recurrent GAN. Then we introduce the RSM-GAN architecture and the seasonal attention mechanism. Finally, we describe anomaly scoring and causal inference procedure to identify anomalies and the root causes in the prediction phase.\n\n\\subsection{RSM-GAN Framework}\n\\subsubsection{MTS to Image Conversion} To extend GAN to MTS and to capture inter-correlation between multiple time series, we convert the MTS into an image-like structure through construction of the so-called multi-channel correlation matrix (MCM), inspired by \\cite{song2018deep,mscred}.\nSpecifically, we define multiple windows of different sizes $W=(w_1,...,w_C)$, and calculate the pairwise inner product (correlation) of time series within each window. At a specific time point $t$, we generate $C$ matrices (channels) of shape $n\\times n$, where each element of matrix $S_t^c$ for a window of size $w_c$ is calculated by this formula:\n\\begin{equation}\n    s_{ij}=\\frac{\\sum_{\\delta=0}^{w_c}x_i^{t-\\delta}\\cdot x_j^{t-\\delta}}{w_c}\n\\end{equation}\n In this work, we select windows $W=(5, 10, 30)$. This results in $3$ channels of $n\\times n$ correlation matrices for time point $t$ noted as $S_t$. To convert the span of MTS into this shape, we consider a step size $p=5$. Therefore, $X$ is transformed to $S=(S_1,...,S_M)\\in\\mathbb{R}^{M \\times n\\times n\\times C}$, where  $M=\\lfloor\\frac{T}{p}\\rfloor$ steps represented by MCMs. Finally, to capture the temporal dependency between consecutive steps, we stack $h=4$ previous steps to the current step $t$ to prepare the input to the GAN-based model. Later, we extend MCM to also capture seasonality unique to MTS.\n\n\\subsubsection{RSM-GAN Architecture} \nThe idea behind using a GAN to detect anomalies is intuitive. During training, a GAN utilize adversarial learning to capture the distribution of the input data. Then, if anomalies are present during prediction, the networks would fail to reconstruct the input, thus produce large losses. In most deep AD literature, the training data is explicitly assumed to be normal with no contamination. In a study, \\cite{encoder} have shown that simultaneous training of an encoder with GAN improves the robustness of the model towards contamination. This is mainly because the joint encoder forces similar inputs to lie close to each other by optimizing the latent loss, and thus account for the contamination while training. To this end, we adopt an encoder-decoder-encoder structure \\cite{ganomaly}, with the additional encoder, to optimize input reconstruction in both original and latent space. Specifically, in Figure \\ref{fig:GAN}, the generator $G$ has autoencoder structure that the encoder ($G_E$) and decoder ($G_D$) interact with each other to minimize the contextual loss: the $l_2$ distance between input $x$ and reconstructed input $G(x)=x'$. An additional encoder $E$ is trained jointly with the generator to minimize the latent loss: the $l_2$ distance between latent vector $z$ and reconstructed latent vector $z'$. Finally, the discriminator $D$ is tasked to distinguish between the original input $x$ and the generated input $x'$. Following the recent advancements on GAN, we employ the Wasserstein GAN with gradient penalty (WGAN-GP) \\cite{wgan-gp} to ensure stable gradients, avoid the collapsing mode, and thus improve the training. \nTherefore, the final objective functions for the generator and discriminator are as following:\n\\begin{equation}\n\\begin{aligned}\n    L_G =  \\min_{G}\\min_{E} & \\Big(  w_1\\mathbb{E}_{x\\sim p_x} \\| x-x' \\|_2 +  w_2 \\mathbb{E}_{x\\sim p_x} \\| G_E(x)-E(x') \\|_2 \\\\ & +  w_3  \\mathbb{E}_{x\\sim p_x}[f_\\theta(x')]\\Big)\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n    L_D = \\max_{\\theta \\in \\Theta} \\mathbb{E}_{x\\sim p_x}[f_\\theta(x)] - \\mathbb{E}_{x\\sim p_x} [f_\\theta(x')]\n\\end{equation}\n\\noindent where $\\theta$ is the discriminator's parameter and ($w_1$, $w_2$, $w_3$) are weights controlling the effect of each loss. The choice of contextual loss weight, has the largest effect on training convergence and we chose (50, 1, 1) weights for optimal training. We employ Adam optimizer to optimize the above losses for $G$ and $D$ alternatively. Each encoder in Figure \\ref{fig:GAN} is composed of multiple convolutional layers, each followed by convolutional-LSTM layers to capture both spatial and temporal dependencies in input. The detailed inner structure of each component is described in Appendix \\ref{sec:apx_inner}.\n\n\n\\subsubsection{Seasonality Adjustment via Attention Mechanism} \nIn order to adjust the seasonality in MTS data, we stack previous seasonal steps to the input data, and allow the convolutional-LSTM cells in the encoder to capture temporal dependencies through an attention mechanism. Specifically, in addition to $h$ previous immediate steps, we append $m_i$ previous seasonal steps per seasonal pattern $i$. To illustrate, assume the input has both the daily and weekly seasonality. To prepare input for time step $t$, we stack MCMs of up to $m_1$ days ago at the same time, and up to $m_2$ weeks ago at the same time.  \nAdditionally, to account for the fact that seasonal patterns are often not exact, we smooth the seasonal steps by averaging over steps in a neighboring window of 6 steps.\\\\\nMoreover, even though the $h$ previous steps are closer to the current time step, but the previous seasonal steps might be a better indicator to reconstruct the current step. Therefore, an attention mechanism is employed to assign weights to each step based on the similarity of the hidden state representations in the last layer using:\n\\begin{equation}\n    \\mathcal{H'}_t = \\sum_{i\\in (t-N,t)} \\alpha_i \\mathcal{H}_i  \\text{, where } \n    \\alpha_i=\\mathrm{softmax}\\Big(\\frac{Vec(\\mathcal{H}_t)^T Vec(\\mathcal{H}_i)}{\\mathcal{X}}\\Big)\n\\end{equation}\nwhere $N=h+\\Sigma m_i$, $Vec(\\cdot)$ denotes the vector, and $\\mathcal{X}=5$ is the rescaling factor. Figure \\ref{fig:attention} presents the structure of the described smoothed attention mechanism.\nFinally, to make our model even more adaptable to real-world datasets that often exhibit holiday effects, we multiply the attention weight $\\alpha_i$ by a binary bit $b_i \\in \\{0,1\\}$, where $b_i=0$ in case of holidays or other exceptional behavior in previous steps. This way, we eliminate the effect of undesired steps from modeling the current step.\n\n\\begin{figure}[t]\n  \\centering \n  \\includegraphics[width=0.88\\columnwidth]{Figures\/attn.png} \n  \\caption{Smoothed attention mechanism}\n  \\label{fig:attention} \n\\end{figure}\n\n\\subsection{Prediction Phase}\n\\subsubsection{Anomaly Score Assignment} \\label{sec:AS_assign}\nThe residual MCM matrix from the first channel, $R_x=x_{:,:,0}-x_{:,:,0}'$, are indicative of anomalies while predicting. \nWe define broken tiles as the elements of $R_x$ that have error value of greater than $\\theta_b$. Previous studies have defined a scoring method based on the number of broken tiles in $R_x$ that we call context$_{b}$ \\cite{mscred}. However, this score is insensitive to non-severe anomalies, and lowering the threshold would result in high FPR. Since each row\/column in $R_x$ is associated with a time series, the ones with the largest number of broken tiles are contributing the most to the anomalies. Therefore, by defining a threshold $\\theta_h \\leq \\theta_b$, we propose to only count the number of broken tiles in rows\/columns with more than half broken and name this method context$_{h}$. The above thresholds $\\theta = \\beta \\times \\eta_{.996}(E_\\mathrm{train})$, which is calculated based on $99.6^{th}$ percentile of error in the training residual matrices, and the best $\\beta$ is chosen by a grid search on validation set.\n\n\n\\subsubsection{Root Cause Framework} Large errors in rows\/columns of $R_x$ are indicative of anomalous behavior in those time series. To identify which are contributing the most to anomalies, we need a root-cause scoring system to assign a score to each time series based on the severity of its errors. We present two different methods: 1) number of broken tiles (NB) (using the optimized $\\theta_b$), and 2) sum of absolute errors (AE). \n\n Furthermore, the number of root causes, $k$, is unknown in real-world applications. Despite previous studies, we allow the elbow method \\cite{elbow} to find the optimal $k$ number of time series from the root cause scores. In this approach, by sorting the scores and plotting the curve, we aim to find the point where the scores become very small and close to each other (elbow point). Time series associated with the scores greater than elbow point are identified as root causes. Thus, we define the elbow point as the point with maximum distance from a vector that connects the first and last scores. \n\n\n\\section{Experimental Setup}\n\\subsection{Data}\n\\textbf{Synthetic Data:} We generate synthetic sinusoidal-based multivariate time-series with different seasonal period and contamination levels to evaluate our model comprehensively. Ten time series with 2 months worth of data by minute sampling frequency are generated with length $T=80,640$. Also, anomalies with varying duration and intensity are injected to the training and test sets. The detailed data generation process in discussed in Appendix \\ref{sec:apx_synth}.\n\n\\textbf{Encryption Key Data:} Our encryption-key dataset contains $7$ time series generated from a project's encryption process. Each time series represents the number of requests for a specific encryption key per minute. The dataset contains $4$ months of data with length $T=156,465$. Four anomalies with various length and scales are identified in the test sequence by a security expert, and we randomly injected $5$ additional anomalies into both the training and test sets.\n\n\n\\subsection{Evaluation}\nRSM-GAN is compared against two classical machine learning models, i.e., One-class SVM (OC-SVM) \\cite{one-svm} and Isolation Forest \\cite{iforest}. and a deep autoencoder-based AD model called MSCRED \\cite{mscred}. MSCRED is run in a sufficient number of epochs and its best performance is reported. We also have tried GAN-AD \\cite{madgan} in multiple settings, but the results are not reported here due to its inefficient and faulty performance. For evaluation, in addition to precision, recall, FPR, and F1 score, we include the \\textbf{Numenta Anomaly Benchmark (NAB)} score \\cite{numenta}. NAB is a standard open source framework for evaluating real-time AD algorithms. The NAB assigns score to each true positive detection based on their relative position to the anomaly window (by a scaled \\textit{sigmoid} function), and negative score to false positives and false negatives.\n\n In all experiments, the first half of the time series are used for training and the remainder for validation\/test by 1\/5 ratio. RSM-GAN is implemented in Tensorflow and trained in 300 epochs, in batches of size $32$, on an AWS Sagemaker instance with four $16$GB GPUs.\nAll the results are produced by an average over five independent runs.\n\n\\section{Results and Discussion}\n\\subsection{Anomaly Score and Root Cause Assessment}\nIn this section, we first compare the performance of our RSM-GAN using the two context$_{b}$ and context$_{h}$ anomaly score assignment methods described in Section \\ref{sec:AS_assign}. Table \\ref{tab:scores} reports the performance on synthetic MTS with no contamination and seasonality with the optimized threshold as illustrated. As we can see, our proposed context$_{h}$ method outperforms context$_{b}$ for all metrics except of recall. Specifically, context$_{h}$ improves the precision and FPR by $6.2\\%$ and $0.08\\%$, respectively. Since the same result holds for other settings, we report the results based on context$_{h}$ scoring in the following experiments.\n\n\\begin{table}[tb]\n\\caption{Different anomaly score assignment performances}\n\\begin{adjustbox}{width=0.97\\columnwidth,center}\n\\begin{tabular}{c|c|ccccc}\n\\textbf{Score} &\\textbf{Threshold}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} &\\textbf{FPR} & \\textbf{NAB Score} \\\\ \\hline\ncontext$_{b}$ & 0.0019 & 0.784 & \\textbf{0.958} &\t0.862 &\t0.0023 &\t0.813 \\\\\ncontext$_{h}$ & 0.00026 &\\textbf{0.846} &\t0.916 &\t\\textbf{0.880} &\t\\textbf{0.0015} &\t\\textbf{0.859} \\\\\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:scores}\n\\end{table}\n\n Next, we compare the two root cause scoring methods for the baseline MSCRED and our RSM-GAN. Root causes are identified based on the average of $R_x$'s in an anomaly window. \nSynthetic data used in this experiment has two combined seasonal patterns and ten anomalies in training sequence. Overall, RSM-GAN outperforms MSCRED (marked by *). As the results suggest, the NB method performs the best for MSCRED. However, for RSM-GAN the AE leads to the best performance. Since the same results hold for other settings, we report NB for MSCRED and AE for RSM-GAN in subsequent sections.\n\n\\begin{table}[tb]\n\\caption{Different root cause identification performances}\n\\begin{adjustbox}{width=0.9\\columnwidth,center}\n\\begin{tabular}{c||c|ccc}\n\\textbf{Model} &\\textbf{Scoring}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} \\\\ \\hline\n\\multirowcell{2}{MSCRED} \n& Number of broken (NB) & 0.5154 &\t\\textbf{0.7933} &\t\\textbf{0.6249} \\\\\n& Absolute error (AE) & \\textbf{0.5504} &\t0.7066 &\t0.6188 \\\\ \\hline\n\n\\multirowcell{2}{RSM-GAN} \n& Number of broken (NB) & 0.4960 &\t0.8500 &\t0.6264 \\\\\n& Absolute error (AE) & \\textbf{0.6883*} &\t\\textbf{0.8666*} &\t\\textbf{0.7672*} \\\\ \n\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:rootcause}\n\\end{table}\n\n\n\n\\subsection{Contamination Resistance Assessment}\nWe assess the robustness of RSM-GAN towards different levels of contamination in training data. In this experiment, we start with no contamination and at each subsequent level, we add $5$ more random anomalies with varying duration to the training data. The percentages presented in the first column in Table \\ref{tab:contamination} shows the proportions of the anomalous time points in train\/test time span.  Results in Table \\ref{tab:contamination} suggest that our proposed model outperforms all baseline models at all contamination levels for all metrics except of the recall. Note that the 100\\% recall for classic baselines  are at the expense of FPR as high as $26.4\\%$. Furthermore, comparison of the NAB score shows that our model has more timely detection and less irrelevant false positives. Lastly, as we can see, the MSCRED performance drops drastically as the contamination level increases, due to the normal training data assumption and the encoder-decoder architecture.\n\n\\begin{table*}[hbt]\n\\caption{Model Performance on synthetic data with different levels of training contamination and random seasonality}\n\\begin{adjustbox}{width=0.69\\textwidth,center}\n\\begin{tabular}{c||c|ccccc|c}\n\\textbf{Contamination} &\\textbf{Model}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} &\\textbf{FPR} & \\textbf{NAB Score} & \\textbf{Root Cause Recall} \\\\ \\hline\n\\multirowcell{4}{No contamination \\\\ train: 0 (0) \\\\ test: 10 (\\%0.82)} & OC-SVM & 0.1581 &\t\\textbf{1.0000} &\t0.2730 &\t0.0473 &\t-8.4370 & - \\\\\n& Isolation Forest & 0.0326 &\t\\textbf{1.0000} &\t0.0631 &\t0.2640 &\t-51.4998 & - \\\\\n& MSCRED & 0.8000 &\t0.8450 &\t0.8219 &\t0.0018 &\t0.7495 & \\textbf{0.7533} \\\\\n& RSM-GAN & \\textbf{0.8461} &\t0.9166 &\t\\textbf{0.8800} &\t\\textbf{0.0015} &\t\\textbf{0.8598} &\t0.6333 \\\\ \\hline\n\n\\multirowcell{4}{Mild contamination \\\\ train: 5 (\\%0.43) \\\\ test: 10 (\\%0.76)} \n& OC-SVM & 0.2810 &\t\\textbf{1.0000} &\t0.4387 &\t0.0218 &\t-3.3411 & - \\\\\n& Isolation Forest & 0.3134 &\t\\textbf{1.0000} &\t0.4772 &\t0.0187 &\t-2.7199 & - \\\\\n& MSCRED & 0.6949 &\t0.6029 &\t0.6457 &\t0.0023 &\t0.2721 &\t0.5483 \\\\\n& RSM-GAN & \\textbf{0.8906} &\t0.7500 &\t\\textbf{0.8143} &\t\\textbf{0.0009} &\t\\textbf{0.8865} &\t\\textbf{0.7700} \\\\ \\hline\n\n\\multirowcell{4}{Medium contamination \\\\ train: 10 (\\%0.82) \\\\ test: 10 (\\%0.85)} \n& OC-SVM & 0.4611 &\t\\textbf{1.0000} &\t0.6311 &\t0.0113 &\t-1.2351 & - \\\\\n& Isolation Forest & 0.6311 &\t\\textbf{1.0000} &\t0.7739 &\t0.0056 &\t-0.1250 & - \\\\\n& MSCRED & 0.6548 &\t0.7143 &\t0.6832 &\t0.0036 &\t0.2712 &\t0.6217 \\\\\n& RSM-GAN & \\textbf{0.8553} &\t0.8442 &\t\\textbf{0.8497} &\t\\textbf{0.0014} &\t\\textbf{0.8511} &\t\\textbf{0.8083} \\\\ \\hline\n\n\\multirowcell{4}{Severe contamination \\\\ train: 15 (\\%1.19) \\\\ test: 15 (\\%1.18)} \n& OC-SVM & 0.5691 &\\textbf{\t1.0000} &\t0.7254 &\t0.0102 &\t-0.3365 & - \\\\\n& Isolation Forest & 0.8425 &\t\\textbf{1.0000} &\t\\textbf{0.9145} &\t0.0025 &\t0.6667 & - \\\\\n& MSCRED & 0.5493 &\t0.7290 &\t0.6265 &\t0.0080 &\t0.0202 &\t0.6611 \\\\\n& RSM-GAN & \\textbf{0.8692} &\t0.8774 &\t0.8732 &\t\\textbf{0.0018} &\t\\textbf{0.8872} &\t\\textbf{0.8133} \\\\ \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:contamination}\n\\end{table*}\n\n\n\n\n\n\n\n\\begin{table*}[htb!]\n\\caption{Model performance on synthetic data with different seasonal patterns and no training contamination}\n\\begin{adjustbox}{width=0.69\\textwidth,center}\n\\begin{tabular}{c||c|ccccc|c}\n\\textbf{Seasonality} &\\textbf{Model}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} &\\textbf{FPR} & \\textbf{NAB Score} & \\textbf{Root Cause Recall} \\\\ \\hline\n\n\\multirowcell{4}{Random seasonality} \n& OC-SVM & 0.4579 &\t0.9819 &\t0.6245 &\t0.0097 &\t-8.6320 & - \\\\\n& Isolation Forest & 0.0325 &\t\\textbf{1.0000} &\t0.0630 &\t0.2646 &\t-51.606 &- \\\\\n& MSCRED & 0.8000 &\t0.8451 &\t0.8219 &\t0.0019 &\t0.7495 & \\textbf{0.7533} \\\\\n& RSM-GAN & \\textbf{0.8462} &\t0.9167 &\t\\textbf{0.8800} &\t\\textbf{0.0015} &\t\\textbf{0.8598} & 0.6333 \\\\ \\hline\n\n\\multirowcell{4}{Daily seasonality} \n& OC-SVM & 0.1770 &\t\\textbf{1.0000} &\t0.3008 &\t0.0532 &\t-9.5465 &  - \\\\\n& Isolation Forest & 0.1387 &\t\\textbf{1.0000} &\t0.2436 &\t0.0710 &\t-13.107 & - \\\\\n& MSCRED & 0.7347 &\t0.7912 &\t0.7619 &\t0.0033 &\t0.3775 & \\textbf{0.7467} \\\\\n& RSM-GAN & \\textbf{0.9012} &\t0.7935 &\t\\textbf{0.8439} &\t\\textbf{0.0010} &\t\\textbf{0.5175} & 0.6717 \\\\ \\hline\n\n\\multirowcell{4}{Daily and weekly \\\\ seasonality} \n& OC-SVM & 0.1883 &\t\\textbf{0.9487} &\t0.3142 &\t0.0400 &\t-6.9745 &  - \\\\\n& Isolation Forest & 0.1783 &\t\\textbf{0.9487} &\t0.3002 &\t0.0428 &\t-7.5278 &  - \\\\\n& MSCRED & 0.6548 &\t0.7143 &\t0.6832 &\t0.0036 &\t0.2712 &\t\\textbf{0.6217} \\\\\n& RSM-GAN & \\textbf{0.9000} &\t0.6750 &\t\\textbf{0.7714} &\t\\textbf{0.0008} &\t\\textbf{0.5461} & 0.4650 \\\\ \\hline\n\n\\multirowcell{4}{Weekly and monthly \\\\ seasonality \\\\ with holidays} \n& OC-SVM & 0.2361 &\t\\textbf{0.9444} &\t0.3778 &\t0.0425 &\t-1.7362 & - \\\\\n& Isolation Forest & 0.2783 &\t0.8889 &\t0.4238 &\t0.0321 &\t-1.0773 &  - \\\\\n& MSCRED & 0.0860 &\t0.7059 &\t0.1534 &\t0.0983 &\t-5.1340 & 0.6067 \\\\\n& RSM-GAN & \\textbf{0.6522} &\t0.8108 &\t\\textbf{0.7229} &\t\\textbf{0.0063} &\t\\textbf{0.5617} & \\textbf{0.8667} \\\\ \\hline\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:seasonality}\n\\end{table*}\n\n\n\n\\begin{table*}[htb!]\n\\caption{Model performance on encryption key and synthetic two-period seasonal MTS with medium contamination}\n\\begin{adjustbox}{width=0.64\\textwidth,center}\n\\begin{tabular}{c||c|ccccc|c}\n\\textbf{Dataset} &\\textbf{Model}&\\textbf{Precision}& \\textbf{Recall} & \\textbf{F1} &\\textbf{FPR} & \\textbf{NAB Score} & \\textbf{Root Cause Recall} \\\\ \\hline\n\n\\multirowcell{4}{Encryption \\\\ key} \n& OC-SVM & 0.1532 &\t0.2977 &\t0.2023 &\t0.0063 &\t-17.4715& - \\\\\n& Isolation Forest & 0.3861 &\t\\textbf{0.4649} &\t0.4219 &\t0.0028 &\t-6.9343\t& - \\\\\n& MSCRED & 0.1963 &\t0.2442 &\t0.2176 &\t0.0055 &\t-1.1047\t& 0.4709 \\\\\n& RSM-GAN & \\textbf{0.6852} &\t0.4405 &\t\\textbf{0.5362} &\t\\textbf{0.0011} &\t\\textbf{0.2992}\t& \\textbf{0.5093} \\\\ \\hline\n\n\\multirowcell{4}{Synthetic} \n& OC-SVM & 0.6772 &\t0.9185 &\t0.7772 &\t0.0038 &\t-2.7621 & - \\\\\n& Isolation Forest & 0.7293 &\t\\textbf{0.9610} &\t0.8221 &\t0.0033 &\t-2.2490 & - \\\\\n& MSCRED & 0.6228 &\t0.7403 &\t0.6746 &\t0.0043 &\t0.2753 &\t0.6600 \\\\\n& RSM-GAN & \\textbf{0.8884} &\t0.8438 &\t\\textbf{0.8649} &\t\\textbf{0.0010} &\t\\textbf{0.8986} &\t\\textbf{0.7870} \\\\ \\hline\n\n\\end{tabular}\n\\end{adjustbox}\n\\label{tab:realworld}\n\\end{table*}\n\n\\subsection{Seasonality Adjustment Assessment}\n Next, we assess the performance of our proposed attention mechanism, assuming no training contamination exists. In the first experiment, synthetic MTS includes 2 months of data, sampled per minute, with only random seasonality. Daily and weekly seasonality patterns are added at each further step. In the last experiment, we simulate $3$ years of hourly data, and add special patterns to illustrate holiday effect in both training and test data. The test set of each experiment is contaminated with $10$ random anomalies. Comparing the results in Table \\ref{tab:seasonality}, RSM-GAN shows consistent performance due to the attention adjustment strategy. All the other baseline models, especially MSCRED's performance deteriorate with increased complexity of seasonal patterns.\nIn the last experiment in Table \\ref{tab:seasonality}, all of the abnormalities injected to holidays are misidentified by the baseline models as anomalies, since no holiday adjustment is incorporated in those models and thus, low precision and high FPR has emerged. In RSM-GAN, multiplying the binary vectors of holidays with the attention weights enables accountability for extreme events and leads to the best performance in almost all metrics. \n\n\n\n\n\n\n\n\n\\subsection{Performance on Real-world Dataset}\n\\noindent This section evaluates our model on a real-world encryption key dataset that has both daily and weekly seasonality. To be comprehensive, we also create a synthetic dataset with similar seasonality patterns and 10 anomalies in the training set. \nFrom Table \\ref{tab:realworld}, we make the following observations: 1) RSM-GAN consistently outperforms all the baseline models for anomaly detection and root cause identification recall in both datasets. 2) Not surprisingly, for all the models, performance on the synthetic data is better than that of encryption key data. It is due to the excessive irregularities and noise in the encryption key data. \n\\begin{figure}[bt]\n  \\centering \n  \\includegraphics[width=\\columnwidth]{Figures\/idps_final.png}\n   \\caption{Anomaly score assignment on encryption key data}\n   \\label{fig:final_plot}\n\\end{figure}\n3) Figure \\ref{fig:final_plot} illustrates the anomaly scores assigned to each time point in test dataset by each algorithm, with the bottom plot presenting the ground truth. As we can see, even though isolation forest has the highest recall rate, it also detects many false positives not related to the actual anomaly windows, leading to negative NAB scores. 4) By comparing our model to MSCRED in Figure \\ref{fig:final_plot}, we can see that MSCRED not only has much higher FPR, but it also fails to capture some anomalies. \n\n\n\n\n\n\n\\section{Conclusion}\nIn this work, we proposed a GAN-based AD framework to handle contaminated and seasonality-heavy multivariate time-series. RSM-GAN leverages adversarial learning to accurately capture temporal and spatial dependencies in the data, while simultaneously training an additional encoder to handle training data contamination. The novel attention mechanism in the recurrent layers of RSM-GAN enables the model to adjust complex seasonal patterns often found in the real-world data. We conducted extensive empirical studies and results show that our architecture together with a new score assignment and causal inference lead to an exceptional performance over advanced baseline models on both synthetic and real-world datasets.\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn order to tackle the problem of quantum gravity, instead of studying the full theory of general relativity, it is possible to study simpler models. One such model is pure 3d gravity, which describes a simplified universe with only 2 spatial dimensions and 1 dimension of time and without matter. Since classical 3d gravity is a topological theory (it does not have local degrees of freedom), its quantum theory is much more tractable as was originally noticed by Witten \\cite{Witten:1988hc}. Since then, the model has been studied in various other manners, including using Loop Quantum Gravity techniques \\cite{Freidel:2002hx,Noui:2004iy}. Several directions can be considered from there. One could use the techniques developed to consider a four-dimensional theory and therefore follow the LQG developments. Or it is possible to try and couple 3d gravity to matter, in order to get a more complete model.\n\nThis last direction is however rather difficult since the main property of 3d gravity, namely its topological nature, is generically lost when coupling to matter. In the context of Loop Quantum Gravity, no complete model of 3d gravity coupled to matter, even a simple scalar field, is known \\cite{Date:2011bg} \\footnote{There is however substantial work trying to use matter as a clock \\cite{Giesel:2012rb,Bilski:2017sze}. In that case, the scalar field is used to fix the gauge and the resulting theory is formulated as a diffeomorphism invariant theory. This actually evades the problem of Dirac observable we mention a bit later.}. This is partially due to difficulties in quantizing scalar fields in LQG \\cite{Thiemann:1997rq,Ashtekar:2002vh,Kaminski:2005nc,Kaminski:2006ta}, partially due to difficulties in constructing Dirac observables \\cite{Dittrich:2004cb} but also simply to the difficulties in writing the Hamiltonian constraints involving an inverse metric \\cite{Thiemann:1996ay,Livine:2013wmq}.\n\nIt does not mean that no reasonable conjecture is known. A surprising number of elements, at least from an LQG perspective \\cite{Freidel:2005bb,Freidel:2005me,Ashtekar:1998ak}, converge towards the idea that spacetime in 3d quantum gravity is best described by a non-commutative manifold when coupled to matter. In this regard, non-commutative field theory (see for instance \\cite{Szabo:2001kg}) would be the right effective field theory to describe quantum gravity phenomena, at least in three dimensions. This new non-commutative structure is particularly interesting because it seems to be specific to quantum gravity phenomena and as such, it does provide potential insights for studying the full 4d theory. Our goal in this paper is therefore to work towards the goal of developing a rigorous, non-perturbative theory of 3d quantum gravity coupled to matter (most probably just a scalar field) in the context of LQG. If such a theory can be developed, we will finally be able to test the conjectures regarding the non-commutative structure of spacetime, at least in 3d.\n\nIn this paper and as a first step in this project, we will study the quatum theory of matter coupled to 3d \\textit{linear} gravity. The \\textit{linear} term here refers to the fact that we will consider a simplification on the gravity side, by considering an abelian gauge group (rather than the usual local Lorentz invariance). This model is inspired by Smolin's remark on the $G \\rightarrow 0$ limit of gravity (where $G$ is Newton's constant) \\cite{Smolin:1992wj}. This model, called the $\\mathrm{U}(1)^3$ model, corresponds to the usual linearized gravity theory but expressed in a diffeomorphism invariant manner. This simplification might seem quite drastic, especially in 3d for which linearized gravity is quite trivial. Still, it does serve two purposes. First, pure 3d gravity, which has been studied so far, can be considered a simplification on the matter side. Here, we are trying to keep matter but rather simplify the gravity side in order to get new insights. Second, as we will see, and perhaps unsurprisingly, this linear theory is exactly solvable and exactly quantizable (at least with a few assumptions on the topology). The way it is solved however is interesting. Indeed, by writing every expressions in a diffeomorphism invariant manner, we will get formulas that are starting points for the full theory, either by deforming them accordingly, or as initial point for a perturbative study. On top of these expected benefits, we will also get interesting results and insights on how quantum matter and quantum spacetime interacts. In particular, our work reveals more precisely the role of the BF representation \\cite{Dittrich:2014wpa,Bahr:2015bra} of the holonomy-flux algebra with respect to the solutions of the theory but also the role of unconventional representations (inspired from \\cite{Koslowski:2007kh,Sahlmann:2010hn,Koslowski:2011vn}) in the construction of the field operators.\n\nThe main result of this paper is that, in this simplified setting of a scalar field coupled to 3d linear gravity, two sectors entirely decouple. One of the sector correspond to the matter sector. Its structure is exactly equivalent to the free scalar field though expressed in a diffeomorphism invariant way. The second sector roughly corresponds to gravity and is governed by equations similar to BF theory. This separation is possible because we can write the equivalent of creation and annihilation operators of the free field theory, with the additional property of commuting with all the constraints. The first sector correspond to the states explored by the ladder operators while the second sector correspond to the part on which the constraints act. This separation allows the definition of an explicit exact (though trivial) quantum theory. It is noteworthy however that the scalar field operators (the field operator and its canonically conjugated momentum) cannot be expressed in the natural representations of the algebra we found, even though the ladder operators can. The problem is linked to the definition of the inverse of the determinant of the triad, a problem widely encountered in LQG \\cite{Thiemann:1996ay,Livine:2013wmq}. It is possible to solve this problem in this simplified context by appealing to representations that are peaked on classical solutions of the Gau\u00df constraints. This result might indicate a possible route for solving similar problems in non-linear or 4d theories.\n\nThe paper is organized as follows. The first section gives a bird eye view on the ideas of the paper, staying quite general but still giving more technical details than this introduction. The second section is devoted to the classical study of the theory, in particular the decoupling of the two sectors classically. The third section is concerned with the quantization of the theory. Two approaches are provided: the naive approach that correspond to the previous study and a second approach that allows the development of all the fundamental operators. Finally, the last section discusses various implications of the results with regard to future work.\n\n\\section{Overview}\n\nThe model we intend to study in the end is 3d quantum gravity coupled to matter. More specifically here, we want to couple a scalar field to gravity in a quantum theory. For this, we can start from the standard action:\n\\begin{equation}\nS[e,A,\\phi] = \\int_{\\mathcal{S}} \\left(\\alpha \\epsilon_{IJK} e^I \\wedge F^{JK}[A] + \\frac{\\Lambda}{6} \\epsilon_{IJK} e^I \\wedge e^J \\wedge e^K + \\frac{1}{2} \\star \\mathrm{d}\\phi \\wedge \\mathrm{d}\\phi + \\frac{m^2}{2} \\star \\phi \\wedge \\phi \\right) .\n\\end{equation}\nHere, $\\mathcal{S}$ is the spacetime manifold. $e$ is the triad. It is an $\\mathbb{R}^3$-valued $1$-form that can be interpreted as an $\\mathfrak{su}(1,1)$-valued one using the Levi-Civita symbol. $A$ is the spin connection. It is naturally an $\\mathfrak{su}(1,1)$-valued one form. $F[A]$ is then its curvature. $\\phi$ is the scalar field. $\\alpha$, $\\Lambda$ and $m$ are coupling constants. $\\alpha$ contains the gravity coupling constant $G$ and is, up to numerical factors $\\frac{1}{G}$. $\\Lambda$ is the cosmological constant and $m$ is the mass of the field. Finally, $\\star$ is the Hodge dual associated to the metric constructed out of the triad. We will choose the signature $(-\\ +\\ +\\ +)$, which goes with the sign in front of the mass term. There is a slight subtlety here. Normally, if $g$ is the metric and $\\omega$ is a $p$-form, then:\n\\begin{equation}\n(\\star \\omega)_{\\mu_1 ... \\mu_{n-p}} = \\frac{1}{p! \\sqrt{|\\det g|}} \\omega_{\\nu_1 ... \\nu_p} \\epsilon^{\\nu_1 ... \\nu_p \\rho_1 ... \\rho_{n-p}} g_{\\mu_1 \\rho_1} ... g_{\\mu_{n-p} \\rho_{n-p}}.\n\\end{equation}\n$\\epsilon^{\\mu\\nu\\rho}$ is not a tensor here and is simply the Levi-Civita symbol (it is a tensor multiplied by a density). Namely, $\\epsilon^{012} = 1$ and all the other terms can be deduced by full anti-symmetry. But we have used the first order expression for the action which uses $\\det e$ and not the square-root of the determinant of the metric, which are equal only up to a sign. Here, we will rather use the following expression, which also solves the sign problem:\n\\begin{equation}\n(\\star \\omega)_{\\mu_1 ... \\mu_{n-p}} = \\frac{1}{p! (\\det e)} \\omega_{\\nu_1 ... \\nu_p} \\epsilon^{\\nu_1 ... \\nu_p \\rho_1 ... \\rho_{n-p}} g_{\\mu_1 \\rho_1} ... g_{\\mu_{n-p} \\rho_{n-p}}.\n\\end{equation}\n\nAs we discussed, one can hope that this theory is exactly quantizable (or at least in some special cases like $m = 0$). It is however rather difficult because of a few road-blocks:\n\\begin{itemize}\n\t\\item The gauge group is non-abelian. This leads to various difficulties when constructing well-defined version of operators.\n\t\\item The classical theory is not always solvable. For instance, a simple homogeneous scalar field coupled to 3d quantum gravity does not have an exact solution linking the volume of the universe to the value of the field. Though this is not an argument against the existence of a quantum version of the model exists, it is a noteworthy difficulty.\n\t\\item Even in the classical case of point particles coupled to 3d gravity, the exact solution is rather difficult to implement and involves a lot of book-keeping. \\cite{tHooft:1992izc}\n\\end{itemize}\nThe main idea of this paper is then to study a simpler model. We will study a scalar field coupled to \\textit{linear} gravity. This model is taken from Lee Smolin work \\cite{Smolin:1992wj}. It can be understood as a limit $G \\rightarrow 0$ (that is $\\alpha \\rightarrow \\infty$) of usual gravity with the additional constraint that $\\frac{A}{G}$ (or $\\alpha A$) is constant. This leads to the following (detailed) action:\n\\begin{eqnarray}\nS[e,A,\\phi] &=& \\int_{\\mathcal{S}} \\Big[ \\frac{\\alpha}{2} \\epsilon_{IJK} \\epsilon^{\\mu\\nu\\rho} e_\\mu^I (\\partial_\\nu A_\\rho^{JK} - \\partial_\\rho A_\\nu^{JK}) + \\frac{\\Lambda}{6} \\epsilon_{IJK} \\epsilon^{\\mu\\nu\\rho} e_\\mu^I e_\\nu^J e_\\rho^K \\nonumber \\\\\n&-& \\frac{1}{12} \\epsilon_{IJK} \\epsilon^{\\mu\\nu\\rho} e_\\mu^I e_\\nu^J e_\\rho^K \\left( e^\\sigma_M e^\\tau_N \\eta^{MN} \\right) \\partial_\\sigma \\phi \\partial_\\tau \\phi - \\frac{m^2}{12} \\epsilon_{IJK} \\epsilon^{\\mu\\nu\\rho} e_\\mu^I e_\\nu^J e_\\rho^K \\phi^2 \\Big] \\mathrm{d}^3 x .\n\\end{eqnarray}\nIn this writing, $\\epsilon_{IJK}$ is the standard Levi-Civita symbol. $\\epsilon^{\\mu\\nu\\rho}$ is not a tensor though, and follows the same convention as the one we used for defining the Hodge star. Also, we have used the standard notation of $e^\\mu_I$ to write the inverse of the triad.\n\nIn practice, we see that this amounts to removing the non-abelian term from the curvature of $A$. Everything else is left untouched. This theory is particularly interesting because, while still diffeomorphism invariant, with some natural constraints, it is equivalent to the free scalar field. Indeed, assuming that $\\mathcal{S} \\simeq \\mathbb{R}^3$, that the various fields behave properly at infinity (vanish quickly at infinity with their derivatives or converge at infinity for the triad), and that the triad is invertible everywhere (which we have more or less assumed when writing its inverse), then we can solve the equations of motion. They are:\n\\begin{itemize}\n\t\\item $\\mathrm{d} e^I = 0$ for all $I$. This means that, since $\\mathcal{S}$ is simply connected, there is a collection of fields $\\Psi^I$ such that $e^I = \\mathrm{d} \\Psi^I$.\n\t\\item The usual equation of motion for the scalar field on a curved background: $\\star \\mathrm{d} (\\star \\mathrm{d} \\phi) - m^2 \\phi = 0$.\n\t\\item For $A$, we get:\n\t\\begin{equation}\n\t\\frac{\\alpha}{2} \\epsilon_{IJK} \\epsilon^{\\mu\\nu\\rho} F_{\\nu\\rho}^{JK}[A] + (\\det e) \\Lambda e^\\mu_I = (\\det e)\\left[ \\frac{1}{2} e^\\mu_I \\left( g^{\\sigma\\tau} \\partial_\\sigma \\phi \\partial_\\tau \\phi + m^2 \\phi^2 \\right) - e^\\sigma_I \\partial_\\sigma \\phi \\partial^\\mu \\phi \\right].\n\t\\end{equation}\n\tThis equation always has a solution as long as the right term has a vanishing divergence, which is just the conservation of energy.\n\\end{itemize}\nWe see then, that $A$ is completely fixed by the rest of the fields, that the equation on $\\phi$ are correct as soon as we can show that the space is flat. This is actually not always true. Indeed, all we have is: $e^I = \\mathrm{d} \\Psi^I$ and $e$ is invertible. This translates to $\\epsilon_{IJK} \\mathrm{d}\\Psi^I \\wedge \\mathrm{d}\\Psi^J \\wedge \\mathrm{d}\\Psi^K \\neq 0$ which means that the transformation from $\\mathcal{S}$ to $\\mathbb{R}^3$ encoded by $\\Psi$ is \\textit{locally} invertible. This sadly does not imply global invertibility. It should be noted however that this is part of the space of solutions. And when it is globally invertible, then is true that space is flat and we get the standard free field theory.\n\nSo we still get something interesting: the free scalar field is an entire sector of our theory. At this stage, it is quite unclear if this sector can be quantized independently from the others, but it is surely a fair assumption. We have a theory, therefore, that is diffeomorphism invariant and still contains the free scalar field. We should notice here similarities with parametrized field theory (PFT) \\cite{Kuchar:1989bk,Kuchar:1989wz,Varadarajan:2006am}. And indeed, working with PFT really corresponds to directly working with $\\Psi^I$. Compared to PFT, in addition to using directly the triad, we will also develop new directions for quantizing such a theory.\n\nAs the goal at this point is to write the corresponding quantum theory, we should be able to find quantities more or less equivalent to the creation and annihilation operators in standard quantum field theory. Indeed, if the free scalar field is an entire sector of the theory, this sector should be in correspondence with the usual solutions. We expect in particular corresponding ladder operators acting in this sector, though these quantities should probably be amended to accommodate the new symmetries.\n\nWhat do we expect? A nice way to look at this is to consider an even simpler theory. Let's study a simple harmonic oscillator, that we can describe by the following action:\n\\begin{equation}\nS = \\int \\left(\\frac{1}{2}m\\dot{x}^2 - \\frac{1}{2}kx^2\\right) \\mathrm{d}t .\n\\end{equation}\nLet's write this in a Hamiltonian manner. The momentum is:\n\\begin{equation}\np = m\\dot{x} .\n\\end{equation}\nThis leads to the following Hamiltonian:\n\\begin{equation}\nH = \\frac{p^2}{2m} + \\frac{kx^2}{2}.\n\\end{equation}\nIf we define $\\omega = \\sqrt{\\frac{k}{m}}$, we can now write:\n\\begin{equation}\nH = \\frac{p^2}{2m} + m \\frac{\\omega^2 x^2}{2}.\n\\end{equation}\nNow let's define the complex quantity:\n\\begin{equation}\na = \\sqrt{\\frac{m\\omega}{2}} x  + \\mathrm{i} \\frac{p}{\\sqrt{2m\\omega}}\n\\end{equation}\nAnd we finally have:\n\\begin{equation}\nH = \\omega a \\overline{a} .\n\\end{equation}\nIt is now well-known that $a$ and $\\overline{a}$ becomes creation and annihilation operators in the quantum theory.\n\nLet's now turn to a diffeomorphism invariant version of this problem, starting with:\n\\begin{equation}\nS = \\int \\left(\\frac{1}{2}m\\frac{\\dot{x}^2}{\\dot{t}} - \\frac{1}{2}kx^2 \\dot{t}\\right) \\mathrm{d}s ,\n\\end{equation}\nwhere now $t$ is a variable depending on the parameter $s$ and all derivatives are taken with respect to $s$. A reparametrization will leave the action invariant which is therefore promoted to a diffeomorphism invariant one. We now have two momenta $p_x$ and $p_t$. And a complete Hamiltonian analysis will reveal that they must now satisfy a (first class) constraint which is:\n\\begin{equation}\np_t + \\frac{p_x^2}{2m} + \\frac{kx^2}{2} = 0 ,\n\\end{equation}\nwhich is quite unsurprisingly the Shcr\u00f6dinger equation (in its classical form). The interesting question though is can we adapt the $a$ quantity so that it commutes with this constraint?\n\nYes we can. The commutator of the current $a$ and our constraint is nearly zero already. In fact, the commutator with $p_t$ is zero but there is a constant (which is just the quanta of energy) for the second part. We must therefore add a term that does not commute with $p_t$. There are various ways to do that. The most interesting to us, is to just consider the time dependent expression for $a$. Indeed, $a$ follows the following equation of motion:\n\\begin{equation}\n\\frac{\\mathrm{d}a}{\\mathrm{d}t} = - \\mathrm{i}\\omega a .\n\\end{equation}\nAs a consequence:\n\\begin{equation}\na(t) = \\left( \\sqrt{\\frac{m\\omega}{2}} x  + \\mathrm{i} \\frac{p}{\\sqrt{2m\\omega}} \\right) \\mathrm{e}^{-\\mathrm{i}\\omega t} .\n\\end{equation}\nTaken without modification, and by interpreting the $t$ as the conjugate to $p_t$, this quantity directly commutes with the constraint. This observation is what motivates our construction for the full system.\n\nOur goal will be to reexpress the usual creation and annihilation operators in standard quantum field theory, so that the quantities linked to position and time can be reinterpreted in function of our new variables (the triad and the connection). If such a quantity can be constructed, it is by definition equal to the creation and annihilation operators when the gauge is fixed. But if it also commutes with the constraints, as our small study suggests, then it is a gauge-unfixed version of these operators and are really the natural operators in the diffeomorphism invariant world.\n\nWhat we need to do then, is to get the Hamiltonian version of our problem. Then we will need to extract all the interesting operators as we just illustrated. This is what we do in the next section.\n\n\\section{Classical model}\n\n\\subsection{Hamiltonian analysis}\n\nOk, we now have the action we want to study. Let's start the Hamiltonian analysis proper. There are various mathematical difficulties we will just ignore for now. Namely, there are questions surrounding the behaviour of the fields at infinity or the various possible topologies for $\\mathcal{S}$ the spacetime manifold. We will concentrate on the simplest possibility. All the other possibilities will just create a richer theory for which we will have neglected various sectors.\n\nWe will assume that $\\mathcal{S}$ is homeomorphic to $\\mathbb{R}^3$. We will also assume that all the matter fields vanish at infinity. Granted all this, we choose some decomposition of $\\mathcal{S}$ as $\\mathbb{R}\\times\\Sigma$ with corresponding coordinates $(t,\\sigma)$. $t$ will be our time variable and $\\sigma$ will be the coordinates on the spatial slice $\\Sigma$. We do assume that $\\Sigma$ is homeomorphic (and even diffeomorphic) to $\\mathbb{R}^2$ but not necessarily a flat slice though. We also make the strong assumption that $\\Sigma$ is spacelike with respect to the metric and nowhere degenerate. This last assumption is reasonable though as, in a hamiltonian analysis, we are interested in parametrizing the space of solutions which should correspond to the variables on a Cauchy slice of spacetime.\n\nThis allows the following writing:\n\\begin{equation}\nS[e,A,\\phi] = \\int_\\mathbb{R} L \\mathrm{d}t,\n\\end{equation}\nwith:\n\\begin{eqnarray}\nL &=& \\int_{\\Sigma} \\Big[ \\frac{\\alpha}{2} \\epsilon_{IJK} \\epsilon^{\\mu\\nu\\rho} e_\\mu^I (\\partial_\\nu A_\\rho^{JK} - \\partial_\\rho A_\\nu^{JK})+ \\frac{\\Lambda}{6} \\epsilon_{IJK} \\epsilon^{\\mu\\nu\\rho} e_\\mu^I e_\\nu^J e_\\rho^K \\nonumber \\\\\n&-& \\frac{1}{12} \\epsilon_{IJK} \\epsilon^{\\mu\\nu\\rho} e_\\mu^I e_\\nu^J e_\\rho^K \\left( e^\\sigma_M e^\\tau_N \\eta^{MN} \\right) \\partial_\\sigma \\phi \\partial_\\tau \\phi - \\frac{m^2}{12} \\epsilon_{IJK} \\epsilon^{\\mu\\nu\\rho} e_\\mu^I e_\\nu^J e_\\rho^K \\phi^2 \\Big] \\mathrm{d}^2 \\sigma .\n\\end{eqnarray}\nFrom there, we proceed as usual: define the momenta, reverse the expressions that can be, keep the rest as primary constraints. The details of the computation can be found in appendix \\ref{app:hamil}. Once all this is done, we can write the Legendre transform of the Lagrangian which is the Hamiltonian.\n\nAfter some computations (detailed in the appendix), we finally get:\n\\begin{eqnarray}\nH &=& \\int_{\\Sigma} \\Big[ \\frac{1}{2} \\partial_0 A_0^{IJ} B^0_{IJ} + \\frac{1}{2} \\partial_0 A_a^{IJ} \\left( B^a_{IJ}  - 2 \\alpha \\epsilon_{IJK} \\epsilon^{ab} e_b^K\\right) + X^\\mu_I \\partial_0 e_\\mu^I - \\frac{1}{2} A_0^{JK} \\left(- 2 \\alpha \\epsilon_{IJK} \\epsilon^{ab} \\partial_b e_a ^I \\right)  \\nonumber \\\\\n&-& e_0^I \\Big(\\alpha \\epsilon_{IJK} \\epsilon^{ab} F_{ab}^{JK}[A] + \\Lambda n_I - \\frac{1}{2} n_I h^{cd} \\partial_c \\phi \\partial_d \\phi - \\frac{m^2}{2} n_I \\phi^2 - \\frac{n_I}{2 \\det h} \\Pi^2 \\nonumber \\\\\n&-& \\frac{n_J \\eta^{JK} \\epsilon^{cd} \\epsilon_{IKL} e_d^L}{\\det h} \\Pi \\partial_c \\phi \\Big) \\Big] \\mathrm{d}^2 \\sigma ,\n\\end{eqnarray}\nwith the following primary constraints:\n\\begin{equation}\n\\left\\{\\begin{array}{rcl}\nX^0_I &=& 0, \\\\\nB^0_{IJ} &=& 0, \\\\\nX^a_I &=& 0, \\\\\nB^a_{IJ} &=& 2\\alpha \\epsilon_{IJK} \\epsilon^{ab} e_b^K.\n\\end{array}\\right.\n\\end{equation}\nHere, summations on small latin indices cover only spatial coordinates. Capital latin indices do cover the $3$ dimensions. $X$ is the natural conjugate with respect to $e$, $B$ the conjugate with respect to $A$ and $\\Pi$ the conjugate of $\\phi$. We have also used the following notations in the Hamiltonian:\n\\begin{itemize}\n\t\\item $h_{ab}$ is the induced metric on $\\Sigma$ and can be written as $h_{ab} = e_a^I e_b^J \\eta^{IJ}$. Due to our assumptions, it is spacelike. $h^{ab}$ is the corresponding inverse metric.\n\t\\item $n_I$ is the natural normal to $\\Sigma$. It is a vector valued density and reads: $n_I = \\frac{1}{2} \\epsilon_{IJK} \\epsilon^{ab} e_a^J e_b^K$.\n\\end{itemize}\n\nFrom there, we can pursue the constraint analysis. After some lengthy, but straightforward, computations (see appendix \\ref{app:hamil}), we get the following system of constraints:\n\\begin{equation}\n\\left\\{\\begin{array}{rcl}\n0 &=& X^0_I, \\\\\n0 &=& B^0_{IJ}, \\\\\n0 &=& X^a_I, \\\\\n0 &=& B^a_{IJ} - 2\\alpha \\epsilon_{IJK} \\epsilon^{ab} e_b^K, \\\\\n0 &=& -\\alpha \\epsilon_{IJK} \\epsilon^{ab} F_{ab}^{JK}[A] - \\Lambda n_I + \\frac{1}{2} n_I h^{cd} \\partial_c \\phi \\partial_d \\phi + \\frac{m^2}{2} n_I \\phi^2 + \\frac{n_I}{2 \\det h} \\Pi^2 + \\frac{n_J \\eta^{JK} \\epsilon^{cd} \\epsilon_{IKL} e_d^L}{\\det h} \\Pi \\partial_c \\phi, \\\\\n0 &=& 2 \\alpha \\epsilon_{IJK} \\epsilon^{ab} \\partial_b e_a^I.\n\\end{array}\\right.\n\\end{equation}\nIt can then be separated into first and second class constraints. We get two sets of second class constraints which are the equivalent of the simplicity constraints in 3d \\cite{Charles:2017srg}:\n\\begin{equation}\n\\left\\{\\begin{array}{rcl}\n0 &=& X^a_I, \\\\\n0 &=& B^a_{IJ} - 2\\alpha \\epsilon_{IJK} \\epsilon^{ab} e_b^K.\n\\end{array}\\right.\n\\end{equation}\nAnd we get a system of first class constraints:\n\\begin{equation}\n\\left\\{\\begin{array}{rcl}\n0 &=& X^0_I, \\\\\n0 &=& B^0_{IJ}, \\\\\n0 &=& \\partial_b B^b_{IJ}, \\\\\n0 &=& \\alpha \\epsilon_{IJK} \\epsilon^{ab} F_{ab}^{JK}[A] + \\Lambda \\tilde{n}_I - \\frac{1}{2} \\tilde{n}_I \\tilde{h}^{cd} \\partial_c \\phi \\partial_d \\phi - \\frac{m^2}{2} \\tilde{n}_I \\phi^2 - \\frac{\\tilde{n}_I}{2 \\det \\tilde{h}} \\Pi^2 - \\frac{\\tilde{n}_J \\eta^{JK} \\epsilon^{cd} \\epsilon_{IKL} \\tilde{e}_d^L}{\\det \\tilde{h}} \\Pi \\partial_c \\phi. \n\\end{array}\\right.\n\\end{equation}\nwhere the tilded quantitites are constructed out of $B$ rather than $e$.\n\nThis allows the computation of the Dirac brackets:\n\\begin{equation}\n\\left\\{\\begin{array}{rcl}\n\\{e^I_0(x), X_J^0(y)\\}_D &=& -\\delta^I_J \\delta(x-y),\\\\\n\\{A^{IJ}_0(x), B_{KL}^0(y)\\}_D &=& -(\\delta^I_K \\delta^J_L - \\delta^I_L \\delta^J_K) \\delta(x-y),\\\\\n\\{A^{IJ}_a(x), e^{K}_b(y)\\}_D &=& \\frac{1}{2\\alpha \\det h} \\epsilon_{ab} \\epsilon^{IJK} \\delta(x-y),\\\\\n\\{A^{IJ}_a(x), B_{KL}^b(y)\\}_D &=& -\\delta_a^b (\\delta^I_K \\delta^J_L - \\delta^I_L \\delta^J_K) \\delta(x-y),\\\\\n\\{\\phi(x), \\Pi(y)\\}_D &=& -\\delta(x-y),\n\\end{array}\\right.\n\\end{equation}\nall other (non-fundamental) brackets being zero (including brackets dealing with $X_I^a$). With these brackets, it is rather obvious that the second class constraints commute with all the other constraints. Interestingly, they can be solved, and the system can finally be rewritten as:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\n0 &=& \\alpha \\epsilon_{IJK} \\epsilon^{ab} F_{ab}^{JK}[A] + \\Lambda n_I - \\frac{1}{2} n_I h^{cd} \\partial_c \\phi \\partial_d \\phi - \\frac{m^2}{2} n_I \\phi^2 - \\frac{n_I}{2 \\det h} \\Pi^2 - \\frac{n_J \\eta^{JK} \\epsilon^{cd} \\epsilon_{IKL} e_d^L}{\\det h} \\Pi \\partial_c \\phi, \\\\\n0 &=& \\epsilon^{ab} \\partial_b e_a^I,\n\\end{array}\n\\right.\n\\end{equation}\nwith the following brackets:\n\\begin{equation}\n\\left\\{\\begin{array}{rcl}\n\\{A^{IJ}_a(x), e^{K}_b(y)\\} &=& \\frac{1}{2\\alpha \\det h} \\epsilon_{ab} \\epsilon^{IJK} \\delta(x-y),\\\\\n\\{\\phi(x), \\Pi(y)\\} &=& -\\delta(x-y).\n\\end{array}\\right.\n\\end{equation}\nThe $B$ variables have been removed thanks to the second class constraints and the time component variables have been removed as they decouple from the rest and can be trivially solved. We now have the Hamiltonian formulation of our problem.\n\nHow is this theory supposed to be linked to the free field theory? It is quite obvious that the constraint on the triad really carries the information that space is flat. There are a few subtleties linked to the problem of global invertibility we mentionned earlier but appart from this, it should be interpreted as the fact that the integral of $e$ is a vector that embed of surface $\\Sigma$ into $\\mathbb{R}^3$. The second constraint is familiar in its form (it is really the Einstein equation) but only set the value of the spin connection $A$. Apart from topological obstructions (which we avoided by choosing the simplest case), this equation always has a solution. So, where is the dynamics of the field encoded?\n\nThe point we have to remember is that the dynamics do not impose anything on a given Cauchy surface. As a consequence, $\\phi$ and $\\Pi$ are completely free. The only constraint will come from the evolution in time which should be encoded here as an action of the diffeomorphism constraints (they can be constructed out of the Einstein equation by projecting using $e$ and $n$). Therefore, the dynamics is not encoded in a constraint \\textit{per se} but rather in their action. The constraint must be contained in the brackets with the curvature constraints. Because the equivalence has been established using the equations of motion earlier, we won't dwell into the equivalence here, which would require a careful analysis of possible gauge fixation. Rather, we will admit that this Hamiltonian theory should at least contain the free field theory and try from there to construct interesting quantities. In particular, we will study in the next section if it is possible to construct the equivalent of the creation and annihilation operators.\n\n\\subsection{Creation and annihilation operators}\n\\label{sec:basic_ops}\n\nSo we are looking for operators that should reduce in the correct gauge fixing to the standard creation and annihilation operator for the scalar field. In the diffeomorphism invariant context though, we expect them to commute with the constraints but still preserve a nice algebra among them, as was suggested on our simple harmonic oscillator study.\n\nThe difficulty resides in that the space manifold $\\Sigma$ is not necessarily flat. The expression must therefore be adapted. We can go about two methods of construction. A first method would be to take advantage of the fact that $\\Sigma$, though not flat, is supposed to be a Cauchy surface. This means that the field in the entire spacetime can be reconstructed from $\\Pi$ and $\\phi$ on the surface. The creation and annihilation operators could then be deduced as coefficient of the Fourier transform. This method would actually work (and it will be explored in section \\ref{subsec:Fourier} to prove a couple of interesting properties) but is more complicated than necessary for now. A second idea is just to make a simple ansatz and check that the resulting operators have the correct algebra, among themselves but also with the constraints.\n\nLet's go back to the standard free field theory for a moment. We have the following action:\n\\begin{equation}\nS = -\\int \\frac{1}{2} \\left(\\eta^{\\mu \\nu} \\partial_\\mu \\phi \\partial_\\nu \\phi + m^2 \\phi^2\\right) \\mathrm{d}^2 x \\mathrm{d} t.\n\\end{equation}\nThis action leads to the following Hamiltonian:\n\\begin{equation}\nH = \\frac{1}{2} \\int \\left( \\Pi^2 + (\\vec{\\nabla} \\phi)^2 + m^2\\phi^2 \\right) \\mathrm{d}^2 x ,\n\\end{equation}\nwhere, once again $\\Pi$ is conjugate to $\\phi$. Normally, we define:\n\\begin{equation}\na_{\\vec{k}} = \\frac{1}{\\sqrt{4\\pi\\omega_{\\vec{k}}}}\\int \\left(\\omega_{\\vec{k}} \\phi  + \\mathrm{i} \\Pi \\right) \\exp\\left( -\\mathrm{i} \\vec{k}\\cdot \\vec{x} \\right) \\mathrm{d}^2 x ,\n\\end{equation}\nwhere $\\omega_{\\vec{k}} = \\sqrt{\\vec{k}^2 + m^2}$. This allows the simple expression:\n\\begin{equation}\nH = \\int \\omega_{\\vec{k}} \\overline{a_{\\vec{k}}} a_{\\vec{k}} \\mathrm{d}^2 k .\n\\end{equation}\nAnd of course, we have the well-known algebra:\n\\begin{equation}\n\\left\\{\\begin{array}{rcl}\n\n\\{a_k, \\overline{a_{k'}}\\} &=& \\mathrm{i} \\delta(k-k') , \\\\\n\\{H, a_k\\} &=& -\\mathrm{i} \\omega_k a_k , \\\\\n\\{H, \\overline{a_k}\\} &=& \\mathrm{i} \\omega_k \\overline{a_k} .\n\n\\end{array}\\right.\n\\end{equation}\n\nCan we have a similar algebra with the coupling to linear gravity? The problem comes from the Hamiltonian which no longer exists but is replaced by a collection of constraints. The curvature constraints (which contain the Einstein equation projected on $\\Sigma$) are however local. We can show the problem with this in the non-gravitational case, by looking at the commutator not with the Hamiltonian $H$ but rather with $H(x) = \\frac{1}{2}\\left(\\Pi^2 + (\\vec{\\nabla} \\phi)^2 + m^2\\phi^2\\right)$ which is the integrand. We get:\n\\begin{equation}\n\\{H(x), a_k\\} = \\frac{1}{\\sqrt{4\\pi\\omega_k}} \\left(- \\vec{\\nabla}\\phi \\cdot \\vec{k} - \\mathrm{i} m^2 \\phi + \\omega_k \\Pi\\right) \\exp\\left( -\\mathrm{i} \\vec{k}\\cdot \\vec{x} \\right) .\n\\end{equation}\nThe resulting expression is not integrated over space, depends on the derivatives of $\\phi$ and cannot simply be expressed in terms of the creation and annihilation operators. How can we solve these problems?\n\nWhat must happen is similar to what we have seen in the case of the harmonic oscillator: the curvature of  $A$ in the curvature constraint will not commute with the operators and will exactly compensate. This is possible if some part of the creation-annihilation operators uses the triad. The natural way to do this, is to use the integral of the triad as a position operator.\n\nSo, let's start from this kind of expressions:\n\\begin{equation}\na_k = \\int \\left(f(k,\\sigma,e,A) \\phi + g(k,\\sigma,e,A) \\Pi\\right)\\mathrm{d}^2\\sigma .\n\\end{equation}\nThis is just the most generic linear expression. Can we go further? Well somewhat yes. We want two additionnal properties:\n\\begin{enumerate}\n\t\\item The expression should be covariant with respect to local gauge transforms.\n\t\\item The expression should be covariant (or even invariant) with respect to diffeomorphism transforms.\n\\end{enumerate}\nConcerning the first point, we do expect some covariance. Basically, $k$ should be expressed in some local reference frame and when it is changed, $k$ should change meaning some covariance for $a_k$. In the linear gravity scenario though, the reference frames cannot change by gauge transform (an interpretation of this is that only infinitesimal changes have been kept). We therefore expect full invariance. This leads to the simple condition that $a_k$ should commute with the Gau\u00df constraint ($\\mathrm{d}e = 0$). As $e$ is invariant under Gau\u00df transforms, then this means that $a_k$ can depend on $A$ only through its curvature.\n\nSomething similar can be said for diffeomorphism invariance. In principle, in the full theory, we only expect some kind of covariance. One problem for instance is that the integral of (parallel transported) $e$ depends on the path and so the annihilation operator could be linked to some integration path choice. In that case, diffeomorphism transform might lead to some transformation of the operators. We are in the linear gravity case though. And in that case, it is way easier to solve. The integral of $e$ does not depend on the choice of path (thanks to the Gau\u00df constraint). So we can make similarly the reasonnable assumption that $a_k$ should be invariant under diffeomorphism transforms.\n\nThis leads to the following expression:\n\\begin{equation}\na_k = \\int \\left(\\tilde{f}(k,\\sigma,e,F[A]) \\phi + \\tilde{g}(k,\\sigma,e,F[A])\\Pi\\right)\\mathrm{d}^2\\sigma .\n\\end{equation}\nwith the additional constraint that $a_k$ commutes with the curvature constraints. We can make one additional assumption: that $a_k$ does not depend on $A$ at all. This seems reasonable enough since we don't really see how this would enter the equation anyway and the standard creation operator doesn't have any dependence on curvature (at least for scalars).\n\nSo, we have the following working hypothesis. The annihilation operator has the following form:\n\\begin{equation}\na_k = \\int \\left(h_1(k,\\sigma,e) \\phi + h_2(k,\\sigma,e)\\Pi\\right)\\mathrm{d}^2\\sigma .\n\\end{equation}\nAnd:\n\\begin{equation}\n\\{D_I, a_k\\}_D = 0.\n\\end{equation}\nA nice addition is to use our guess about the depency in the triad for the position operators.. We offer the following ansatz:\n\\begin{equation}\na_k = \\frac{1}{\\sqrt{2}\\pi}\\int \\left(A(k,e,\\sigma) k^I n_I \\phi + \\mathrm{i} B(k,e,\\sigma) \\Pi\\right) \\mathrm{e}^{- \\mathrm{i} \\vec{k} \\cdot \\int^{\\sigma} \\vec{e}} \\mathrm{d}^2\\sigma .\n\\end{equation}\nThis expression is directly inspired from the standard expression for the annihilation operator. Let's explain a few bits:\n\\begin{itemize}\n\t\\item The factor $k^I n_I$ is a density. This way $A$ is a scalar. It might not be the right density to put (for instance $\\sqrt{n^I n_I}$ would work too) but this doesn't matter since it can be corrected with the right expression for $A$ (which would then be the ratio between two densities). It is a natural\\footnote{There are other possibilities that reflect this though: for instance $k^I n_I(Q) \\sqrt{n^I n_I}$ where $Q$ is some fixed reference point on the manifold. But once more, this can be done by adjusting $A$, thought this might be taken as some explicit dependancy on $\\sigma$. So let's not forget this possibility later on.} density to consider though since it very much looks like the energy component of $k$.\n\t\\item The integral term $\\int^{\\sigma} \\vec{e}$ is a bit weird to say the least. First, $\\vec{e}$ is simply the triad taken to be a vector-valued one-form. Now the integral only has an end point of coordinates $\\sigma$. But the fact that there is no start point is actually important: we \\textit{cannot} take a specific point as reference. Indeed, the exponential of the triad creates curvature at one point and destroys it at the other. Here, we need an operator that only create curvatures at a specific point.\n\t\n\tThis operator really corresponds to the $\\Psi$ we encountered earlier such that $\\mathrm{d}\\Psi^I = e^I$. Because of this relation ship with the triad, there is still a sense in which the difference of $2$ $\\Psi$ is an integral of the triad. By extension, we use this notation with only one end-point to the integral.\n\t\n\tThere is a way to make this more rigorous for a non-compact spatial slice. Because, all the information is contained in a Dirac bracket, we can consider the action of the integral as the start points goes to infinity. Though the integral is not well-defined, its Dirac bracket still exists and correspond exactly to what we need.\n\\end{itemize}\nIt turns out that the correct values are:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\nA(k,e,\\sigma) &=& 1, \\\\\nB(k,e,\\sigma) &=& 1.\n\\end{array}\n\\right.\n\\end{equation}\nThis leads to the following, and in fact quite familiar, expression:\n\\begin{equation}\na_k = \\frac{1}{\\sqrt{2}\\pi}\\int \\left(k^I n_I \\phi + \\mathrm{i}\\Pi\\right) \\mathrm{e}^{- \\mathrm{i} \\vec{k} \\cdot \\int^{\\sigma} \\vec{e}} \\mathrm{d}^2\\sigma .\n\\end{equation}\nA lengthy - but not difficult - computation shows that indeed (see appendix \\ref{app:operators}):\n\\begin{equation}\n\\{D_I, a_k\\}_D = 0.\n\\end{equation}\nMore interestingly, the algebra of these operators can be computed explicitly. It requires some technology we will develop in the next section.\n\n\n\\subsection{Fourier transform and full algebra}\n\n\\label{subsec:Fourier}\n\nA point must be underlined here: in usual free field theory, the creation and annihilation operators have a nice interpretation as Fourier coefficients of the 3d field solution of the equation of motion. A similar property holds true here, granted a few assumptions.\n\nOur spacetime is $\\mathbb{R}^3$ (this was one of our simplifying assumptions). We also assumed that $\\Sigma$ (the space manifold) is homeomorphic to $\\mathbb{R}^2$. We will go a bit further here and assume that the embedding of $\\Sigma$ into $\\mathbb{R}^3$ given by the integrals of the triads $\\int \\vec{e}$ is a Cauchy surface for the free field theory. This assumption is reasonable: when we choose a slice $\\Sigma$ of spacetime, our goal is not to break diffeomorphism invariance but to parameterize the space of solutions for the problem. It is natural therefore to choose a Cauchy surface to do so. It is even natural to think that if we don't choose a Cauchy surface, the Hamiltonian analysis will not be well-defined. We will leave this question open however and just assume a correct choice of $\\Sigma$.\n\nWhat we mean by this assumption is the following. Let $\\phi : \\mathbb{R}^3 \\mapsto \\mathbb{R}$ be a field that satisfies the standard free scalar field equation:\n\\begin{equation}\n-\\partial_t^2 \\phi + \\Delta \\phi - m^2 \\phi = 0.\n\\end{equation}\nLet's now interpret $\\Sigma$ as a submanifold of $\\mathbb{R}^3$ with embedding given by $\\vec{\\Psi} = \\int \\vec{e}$. We assume that knowing $\\phi$ and its derivative along the normal on this embedding is sufficient (and also necessary) to know $\\phi$ on the whole $\\mathbb{R}^3$. This means that we can now extend naturally some fields on $\\Sigma$ to the whole $\\mathbb{R}^3$ spacetime.\n\nOn the $\\Sigma$ slice, we have two fields we are interested in $\\phi$ and $\\Pi$. $\\Pi$ can naturally be connected to a derivative of $\\phi$ in the time-direction (see appendix \\ref{app:hamil}):\n\\begin{equation}\n\\Pi = -(\\det e)g^{0\\tau} \\partial_\\tau \\phi = -\\vec{n} \\cdot \\vec{\\nabla} \\phi.\n\\end{equation}\nHere, $\\vec{n}$ is the normal density on $\\Sigma$ induced by the triad and $\\vec{\\nabla} \\phi$ is the gradient of $\\phi$ (as a spacetime field) expressed in the coordinates we used for the embedding. This means that $\\phi$ and $\\Pi$ on $\\Sigma$ can naturally be extended to a field on the whole spacetime $\\mathbb{R}^3$. Now, we can use the Fourier transform as usual on $\\mathbb{R}^3$ and get coefficients that will turn out to be the $a_k$ we defined earlier (up to some Dirac deltas factor). But of course, the formula will be more general and apply to any couple of fields we might define on $\\Sigma$.\n\nNow, let's turn back to our expression for $a_k$:\n\\begin{equation}\na_k = \\frac{1}{\\sqrt{2}\\pi}\\int \\left(k^I n_I \\phi + \\mathrm{i}\\Pi\\right) \\mathrm{e}^{- \\mathrm{i} \\vec{k} \\cdot \\int^{\\sigma} \\vec{e}} \\mathrm{d}^2\\sigma .\n\\end{equation}\nOur claim is that, this is (up to a factor we will make explicit shortly) the Fourier coefficients for the extension of $\\phi$ in $\\mathbb{R}^3$ according to the previous rules. There is a rather simple way to check this thanks to linearity. We just have to consider the case of:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\n\\phi(\\sigma) &=& A\\frac{\\delta(\\sigma-\\sigma_0)}{\\sqrt{\\det h}}, \\\\\n\\Pi(\\sigma) &=& B\\delta(\\sigma - \\sigma_0).\n\\end{array}\n\\right.\n\\label{eq:ini}\n\\end{equation}\nWe have put the determinant for $\\phi$, because $\\phi$ is a scalar and we want $A$ not to depend on the choice of coordinates. $\\Pi$ however is a density, and so to have $B$ coordinate independent, the determinant factor should be avoided. In that case:\n\\begin{equation}\na_k = \\frac{1}{\\sqrt{2}\\pi} \\left(\\frac{k^I n_I(\\sigma_0)}{\\sqrt{\\det h(\\sigma_0)}} A + \\mathrm{i}B\\right) \\mathrm{e}^{- \\mathrm{i} \\vec{k} \\cdot \\int^{\\sigma_0} \\vec{e}} .\n\\end{equation}\nLet's now consider a field $\\Phi(x,t)$ solution of the equation of motion in $\\mathbb{R}^3$. We can write it in a general form as follows:\n\\begin{equation}\n\\Phi(\\vec{x}) = \\int \\delta(k^2 + m^2)b_k \\mathrm{e}^{\\mathrm{i}\\vec{k}\\cdot \\vec{x}} \\mathrm{d}^3 k.\n\\end{equation}\nThe $b_k$ are therefore the Fourier coefficients (up to a Dirac delta factor) of $\\Phi$. Let's now consider the plane $\\mathcal{P}$ going through $\\int^\\sigma_0 \\vec{e}$ and tangent to $\\Sigma$ (or more precisely tangent to its embedding) at this point. This plane is spacelike and as such can be used as a Cauchy surface for the field $\\Phi$.\n\nThere is always a Lorentz transformation sending $(1,0,0)$ to the normalized normal of the plane $\\mathcal{P}$, granted the chosen orientation is the same (there is an infinite amount of such transformation but anyone will do, we can for instance take a boost). Let's note such a Lorentz transformation $L$. We can now write a parametrisation of the points of $\\mathcal{P}$ as follows:\n\\begin{equation}\n\\vec{x}_\\mathcal{P}(\\tilde{X}) = \\overrightarrow{L\\triangleright(0,\\tilde{X})}+\\int^{\\sigma_0} \\vec{e}.\n\\end{equation}\nHere we chose the following notation: to a vector $\\vec{z}$ can be associated a 2d spatial vector $\\tilde{z}$ and a time component $z_t$. By extension, any 2d vector will be written $\\tilde{w}$ as we used for the coordinates on the plane denoted $\\tilde{X}$. Also, $\\triangleright$ is used to indicate the action of the Lorentz group onto 3d vectors. We can now write initial conditions on the plane $\\mathcal{P}$ for $\\Phi$:\n\\begin{equation}\n\\forall \\tilde{X}\\in\\mathbb{R}^2,\\ \\left\\{\\begin{array}{rcl}\n\\Phi(\\vec{x}_\\mathcal{P}(\\tilde{X})) &=& A\\delta(\\tilde{X}), \\\\\n-\\overrightarrow{L\\triangleright(1,0,0)}\\cdot\\vec{\\nabla}\\Phi(\\vec{x}_\\mathcal{P}(\\tilde{X})) &=& B\\delta(\\tilde{X}).\n\\end{array}\n\\right.\n\\end{equation}\nThese initial conditions correspond to the values of equation \\ref{eq:ini}. Indeed, thanks to the Minkowski structure of spacetime, nothing can propagate faster than light. With the conditions of equation \\ref{eq:ini}, this translates to $\\Phi(x) = 0$ for any point outside of the lightcone of the point at $\\sigma_0$. Now, the transformations laws under diffeomorphism are completely local which guarantees that $\\Phi$ is a Dirac delta on any Cauchy surface passing through $\\sigma_0$. The fact that $\\Phi$ is a scalar even gives the coefficient of transformation which is $1$. We must however be careful, as the Dirac delta is a density, which is why the determinant is eaten up. A similar result holds for the derivative: it is zero nearly everywhere and locally can be expressed with respect to the gradient on $\\Sigma$ and $\\Pi$. Because, we chose a surface tangent to $\\Sigma$, the gradient does not appear and we can conclude.\n\nWe can now use the standard derivation of $b_k$ in terms of $A$ and $B$. Let $\\vec{k}$ be a 3d vector with $k^2 + m^2 = 0$ and $k_t > 0$. Then, we get:\n\\begin{equation}\n\\int \\Phi(\\vec{x}_\\mathcal{P}(\\tilde{X})) \\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(\\tilde{X})} \\mathrm{d}^2 \\tilde{X} = A\\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(0)}.\n\\end{equation}\nWe can also compute:\n\\begin{eqnarray}\n& & \\int \\Phi(\\vec{x}_\\mathcal{P}(\\tilde{X})) \\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(\\tilde{X})} \\mathrm{d}^2 \\tilde{X} \\nonumber \\\\\n&=& \\int \\int \\delta((k')^2 + m^2)b_{k'} \\mathrm{e}^{\\mathrm{i}\\vec{k'}\\cdot \\vec{x}_\\mathcal{P}(\\tilde{X})} \\mathrm{d}^3 k' \\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(\\tilde{X})} \\mathrm{d}^2  \\tilde{X} \\nonumber \\\\\n&=& \\int \\int \\delta((k')^2 + m^2)b_{k'} \\mathrm{e}^{\\mathrm{i}\\left(L^{-1} \\triangleright (\\vec{k'} - \\vec{k})\\right)\\cdot \\left(L^{-1} \\triangleright\\vec{x}_\\mathcal{P}(\\tilde{X})\\right)} \\mathrm{d}^3 k' \\mathrm{d}^2  \\tilde{X} \\nonumber \\\\\n&=& \\int \\int \\delta((L\\triangleright k')^2 + m^2)b_{L\\triangleright k'} \\mathrm{e}^{\\mathrm{i}(\\vec{k'} - L^{-1} \\triangleright \\vec{k})\\cdot \\left(L^{-1} \\triangleright\\vec{x}_\\mathcal{P}(\\tilde{X})\\right)} \\mathrm{d}^3 k' \\mathrm{d}^2  \\tilde{X} \\nonumber \\\\\n&=& \\int \\int \\delta((k')^2 + m^2)b_{L\\triangleright k'} \\mathrm{e}^{\\mathrm{i}(\\tilde{k'} - (\\tilde{L^{-1} \\triangleright \\vec{k}}))\\cdot \\tilde{X}} \\mathrm{e}^{\\mathrm{i}(\\vec{k'} - L^{-1} \\triangleright \\vec{k})\\cdot \\left(L^{-1} \\triangleright\\vec{x}_\\mathcal{P}(\\tilde{0})\\right)} \\mathrm{d}^3 k' \\mathrm{d}^2  \\tilde{X} \\nonumber \\\\\n&=& (2\\pi)^2 \\int \\delta((k')^2 + m^2)b_{L\\triangleright k'}\\delta(\\tilde{k'} - (\\tilde{L^{-1} \\triangleright \\vec{k}})) \\mathrm{e}^{\\mathrm{i}(\\vec{k'} - L^{-1} \\triangleright \\vec{k})\\cdot \\left(L^{-1} \\triangleright\\vec{x}_\\mathcal{P}(\\tilde{0})\\right)} \\mathrm{d}^3 k' \\nonumber \\\\\n&=& (2\\pi)^2 \\int \\frac{\\delta\\left(k'_t - \\sqrt{\\vec{k'}^2 + m^2}\\right) + \\delta\\left(k'_t + \\sqrt{\\vec{k'}^2 + m^2}\\right)}{2|k'_t|}b_{L\\triangleright k'}\\delta(\\tilde{k'} - (\\tilde{L^{-1} \\triangleright \\vec{k}})) \\mathrm{e}^{\\mathrm{i}(\\vec{k'} - L^{-1} \\triangleright \\vec{k})\\cdot \\left(L^{-1} \\triangleright\\vec{x}_\\mathcal{P}(\\tilde{0})\\right)} \\mathrm{d}^3 k'.\n\\end{eqnarray}\nThis last line splits into two terms.\nFor the first line, the main observation is that:\n\\begin{equation}\n\\delta\\left(k'_t - \\sqrt{\\vec{k'}^2 + m^2}\\right)\\delta(\\tilde{k'} - (\\tilde{L^{-1} \\triangleright \\vec{k}})) = \\delta(\\vec{k'} - L^{-1} \\triangleright \\vec{k})\n\\end{equation}\nas there is a unique vector of square norm $-m^2$ with given spatial support and with positive time component. The second term is more involved. We get:\n\\begin{equation}\n\\delta\\left(k'_t + \\sqrt{(-\\vec{k'})^2 + m^2}\\right)\\delta(\\tilde{k'} - (\\tilde{L^{-1} \\triangleright \\vec{k}})) = \\delta(\\vec{k'} - \\overline{L^{-1} \\triangleright \\vec{k}}) ,\n\\end{equation}\nwhere $\\overline{\\vec{x}}$ is the vector deduced from $\\vec{x}$ by inverting its time component, namely $(-x_t, \\tilde{x})$. This leads to:\n\\begin{eqnarray}\n& & \\int \\Phi(\\vec{x}_\\mathcal{P}(\\tilde{X})) \\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(\\tilde{X})} \\mathrm{d}^2 \\tilde{X} \\nonumber \\\\\n&=& (2\\pi)^2 \\int \\frac{1}{2|k'_t|}\\delta(\\vec{k'} - L^{-1} \\triangleright \\vec{k})b_{L\\triangleright k'} \\mathrm{e}^{\\mathrm{i}(\\vec{k'} - L^{-1} \\triangleright \\vec{k})\\cdot \\left(L^{-1} \\triangleright\\vec{x}_\\mathcal{P}(\\tilde{0})\\right)} \\mathrm{d}^3 k' \\nonumber \\\\\n&+& (2\\pi)^2 \\int \\frac{1}{2|k'_t|}\\delta(\\vec{k'} - \\overline{L^{-1} \\triangleright \\vec{k}})b_{L\\triangleright k'} \\mathrm{e}^{\\mathrm{i}(\\vec{k'} - L^{-1} \\triangleright \\vec{k})\\cdot \\left(L^{-1} \\triangleright\\vec{x}_\\mathcal{P}(\\tilde{0})\\right)} \\mathrm{d}^3 k' \\nonumber \\\\\n&=& \\frac{2\\pi^2}{2(L\\triangleright k)_t}\\left( b_k + b_{\\overline{k}}\\mathrm{e}^{-2\\mathrm{i} \\left(L^{-1}\\triangleright k\\right)_t \\left(L^{-1} \\triangleright\\vec{x}_\\mathcal{P}(\\tilde{0})\\right)_t} \\right)\n\\end{eqnarray}\nSimilarly, we can compute:\n\\begin{equation}\n\\int -\\overrightarrow{L\\triangleright(1,0,0)}\\cdot\\vec{\\nabla}\\Phi(\\vec{x}_\\mathcal{P}(\\tilde{X})) \\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(\\tilde{X})} \\mathrm{d}^2 \\tilde{X} = B\\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(0)},\n\\end{equation}\nand also:\n\\begin{equation}\n-\\int \\overrightarrow{L\\triangleright(1,0,0)}\\cdot\\vec{\\nabla}\\Phi(\\vec{x}_\\mathcal{P}(\\tilde{X})) \\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(\\tilde{X})} \\mathrm{d}^2 \\tilde{X} = -2\\mathrm{i}\\pi^2 (b_k - b_{\\overline{k}}\\mathrm{e}^{-2\\mathrm{i} \\left(L^{-1}\\triangleright k\\right)_t \\left(L^{-1} \\triangleright\\vec{x}_\\mathcal{P}(\\tilde{0})\\right)_t}).\n\\end{equation}\nWe can conclude:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\nb_k &=& \\frac{1}{2\\pi^2}\\left( (L^{-1}\\triangleright k)_t A + iB \\right)\\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(0)}, \\\\\nb_{\\overline{k}} &=& \\frac{1}{2\\pi^2}\\left( (L^{-1}\\triangleright k)_t A - iB \\right)\\mathrm{e}^{-\\mathrm{i}\\overline{\\vec{k}}\\cdot \\vec{x}_\\mathcal{P}(0)}.\n\\end{array}\n\\right.\n\\end{equation}\nNow:\n\\begin{eqnarray}\n(L^{-1}\\triangleright k)_t &=& (L^{-1}\\triangleright \\vec{k})\\cdot\\overrightarrow{(1,0,0,)} \\nonumber \\\\\n&=& \\vec{k}\\cdot(L^{-1}\\triangleright \\overrightarrow{(1,0,0,)}) \\nonumber \\\\\n&=& \\frac{k^I n_I(\\sigma_0)}{\\sqrt{\\det h(\\sigma_0)}},\n\\end{eqnarray}\nwhere we used $n$ divided by its norm as an expression for the normal to $\\mathcal{P}$. Thus:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\nb_k &=& \\frac{1}{2\\pi^2}\\left( \\frac{k^I n_I(\\sigma_0)}{\\sqrt{\\det h(\\sigma_0)}} A + iB \\right)\\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(0)}, \\\\\nb_{\\overline{k}} &=& \\frac{1}{2\\pi^2}\\left( \\frac{k^I n_I(\\sigma_0)}{\\sqrt{\\det h(\\sigma_0)}} A - iB \\right)\\mathrm{e}^{-\\mathrm{i}\\overline{\\vec{k}}\\cdot \\vec{x}_\\mathcal{P}(0)}.\n\\end{array}\n\\right.\n\\end{equation}\nWe can finally rewrite this in the more traditional manner:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\nb_k &=& \\frac{1}{2\\pi^2}\\left( \\frac{k^I n_I(\\sigma_0)}{\\sqrt{\\det h(\\sigma_0)}} A + iB \\right)\\mathrm{e}^{-\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(0)}, \\\\\nb_{-k} &=& \\frac{1}{2\\pi^2}\\left( -\\frac{k^I n_I(\\sigma_0)}{\\sqrt{\\det h(\\sigma_0)}} A - iB \\right)\\mathrm{e}^{\\mathrm{i}\\vec{k}\\cdot \\vec{x}_\\mathcal{P}(0)}.\n\\end{array}\n\\right.\n\\end{equation}\nAnd then:\n\\begin{equation}\nb_k = \\frac{\\mathrm{sgn}(k_t)}{\\sqrt{2}\\pi}a_k,\n\\end{equation}\nand this is true for any $k$ such that $k^2 + m^2 = 0$.\n\nAll this means that, up to a numerical factor, the sign of $k_t$ and a Dirac delta, the $a_k$ coefficients really are the Fourier coefficients of the field we get by specifying the initial conditions of $\\Phi$ and $\\Pi$ on $\\Sigma$ embedded into $\\mathrm{R}^3$. This is especially useful to compute the brackets between the $a_k$ coefficients. Let's compute the following bracket:\n\\begin{eqnarray}\n& & \\{\\delta(k^2 + m^2)a_k, \\delta(k'^2 + m^2)a_{k'}\\} \\nonumber \\\\\n&=& \\delta(k^2 + m^2) \\delta(k'^2 + m^2) \\{a_k, a_{k'}\\} \\nonumber \\\\\n&=& \\delta(k^2 + m^2) \\delta(k'^2 + m^2) \\frac{1}{2\\pi^2} \\int \\int \\{k^I n_I(x) \\phi(x) + \\mathrm{i}\\Pi(x), k'^J n_J(y) \\phi(y) + \\mathrm{i}\\Pi(y)\\} \\mathrm{e}^{- \\mathrm{i} \\vec{k}\\cdot \\int^{x} \\vec{e} - \\mathrm{i}\\vec{k'}\\cdot \\int^{y} \\vec{e} } \\mathrm{d}^2 x \\mathrm{d}^2 y\\nonumber \\\\\n&=& \\delta(k^2 + m^2) \\delta(k'^2 + m^2) \\frac{\\mathrm{i}}{2\\pi^2} \\int \\int \\left( - k^I n_I(x) \\delta(x-y) + k'^J n_J(y) \\delta(x-y) \\right) \\mathrm{e}^{- \\mathrm{i} \\vec{k}\\cdot \\int^{x} \\vec{e} - \\mathrm{i}\\vec{k'}\\cdot \\int^{y} \\vec{e} } \\mathrm{d}^2 x \\mathrm{d}^2 y\\nonumber \\\\\n&=& \\delta(k^2 + m^2) \\delta(k'^2 + m^2) \\frac{\\mathrm{i}}{2\\pi^2} \\int (k' - k)^I n_I \\mathrm{e}^{- \\mathrm{i} (\\vec{k} + \\vec{k'})\\cdot \\int^{x} \\vec{e}} \\mathrm{d}^2 x.\n\\end{eqnarray}\nThough this last form is pretty compact, it is better to expend it back a bit as follows:\n\\begin{eqnarray}\n & \\{\\delta(k^2 + m^2)a_k, \\delta(k'^2 + m^2)a_{k'}\\} = \\nonumber \\\\\n & \\delta(k'^2 + m^2) \\left[ \\delta(k^2 + m^2) \\frac{1}{\\sqrt{2}\\pi} \\int \\left((-\\frac{\\mathrm{i}}{\\sqrt{2}\\pi} \\mathrm{e}^{- \\mathrm{i} \\vec{k'}\\cdot \\int^{x} \\vec{e}})k^I n_I + \\mathrm{i} (\\frac{1}{\\sqrt{2}\\pi}k'^I n_I \\mathrm{e}^{- \\mathrm{i} \\vec{k'}\\cdot \\int^{x} \\vec{e}})\\right) \\mathrm{e}^{- \\mathrm{i} \\vec{k}\\cdot \\int^{x} \\vec{e}} \\mathrm{d}^2 x \\right]\n\\end{eqnarray}\nFrom what we just saw, the term in large square brackets is (up to a numerical factor and a sign) the Fourier coefficient of a field with initial values on $\\Sigma$ given by:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\n\\phi &=& -\\frac{\\mathrm{i}}{\\sqrt{2}\\pi} \\mathrm{e}^{- \\mathrm{i} \\vec{k'}\\cdot \\int^{x} \\vec{e}}, \\\\\n\\Pi &=& \\frac{1}{\\sqrt{2}\\pi}k'^I n_I \\mathrm{e}^{- \\mathrm{i} \\vec{k'}\\cdot \\int^{x} \\vec{e}}. \n\\end{array}\n\\right.\n\\end{equation}\nBut we know such a field: it is simply the field $\\Phi(x) = -\\frac{\\mathrm{i}}{\\sqrt{2}\\pi} \\mathrm{e}^{- \\mathrm{i} k'\\cdot x}$ on the whole $\\mathbb{R}^3$ spacetime. And its Fourier transform is proportional to a Dirac delta $\\delta(k+k')$. From that, we conclude (with the factors correctly computed):\n\\begin{equation}\n\\{\\delta(k^2 + m^2)a_k, \\delta(k'^2 + m^2)a_{k'}\\} = -\\mathrm{i}\\mathrm{sgn}(k_t)\\delta(k'^2 + m^2)\\delta(k+k').\n\\end{equation}\nThis is exactly the kind of algebra we wanted for creation-annihilation operators. It is correctly adapted to the diffeomorphism invariant case as no frame of reference can be preferred. Let's note here that the sign is the reverse from the usual since we have:\n\\begin{equation}\n\\overline{a_k} = -a_{-k}\n\\end{equation}\nwith the extra sign coming from the fact that we put the $\\mathrm{sgn}(k_t)$ factor out of $a_k$.\n\n\\section{Quantization}\n\n\\subsection{First approach}\n\n\\label{sec:firstapproach}\n\nWe can now turn to the quantization of the system. In principle, we should start with some natural construction of the algebra of observables, starting with canonical variables. This is however notoriously difficult for matter coupled to gravity \\cite{Ashtekar:2002vh,Kaminski:2005nc,Kaminski:2006ta}. As a first approach, let's avoid the usual difficulties by choosing another set of fundamental variables.\n\nThe first point to note is that we have the creation and annihilation operators which are quite natural. They are for instance used in the construction of the Fock space and it does make sense to keep them as fundamental. The second point to note is that the creation and annihiliation operators, by construction, commute with the triad operators and with the curvature constraints. They commute with the triad because they do not depend on the connection, and we devoted a large part of this paper (see appendix \\ref{app:operators}) to prove it commutes with the curvature constraints. Conversely, the triad operators and the curvature constraints are particularly interesting as fundamental variables since they are conjugate to each other. Finally, we have proven previously that the $a_k$ can be interpreted as Fourier coefficients (section \\ref{subsec:Fourier}), which means we can reconstruct (at least classically) the field $phi$ and its momentum $\\Pi$. This also means that, classically, if we now the triad and the curvature constraints, we can reconstruct the curvature of the connection everywhere. This is enough to reconstruct the spin connection up to a gauge. Therefore, the following collection:\n\\begin{itemize}\n\t\\item $a_{k}$ for all $k \\in \\mathbb{R}^3$ such that $k^2 + m^2 = 0$ (which contains both creation and annihilation operators based on the sign of $k^0$),\n\t\\item $D_I(x)$ for all $I$ and $x$,\n\t\\item and $e_a^I(x)$ for all $I$, $a$ and $x$\n\\end{itemize}\ngives a complete description of the gauge invariant phase-space. This collection divides into two sectors that commute with each other and that, remarkably, we know how to quantize separately. The creation-annihilation algebra leads to the well-known Fock quantization (with a few caveats). And the algebra of the curvature and triad operators can lead to a quantization around a state similar to the BF vacuum \\cite{Dittrich:2014wpa,Bahr:2015bra} as we will shortly show.\n\nThere is one important point to underline here: all this works only when restricting to the gauge-invariant subspace of the phase space. It is not always possible to solve for this subspace explicitly, and it is not possible for the non-abelian case. In the abelian case however, not only is it possible, it greatly simplifies a number of expressions. Indeed, the algebra between the $D_I$ is only simple if the Gau\u00df constraints is checked. The same thing holds for the brackets between $D_I$ and $a_k$ which in all generality is linear in the Gau\u00df constraints. In general then, we would have to deal with partial gauge-fixing, the choice of path and other niceties. And such a treatment will be \\textit{necessary} for the non-abelian case. However, as a first approach, and when considering our simple linear theory, it is possible to avoid such consideration. And this is what will do in all the constructions from now on.\n\n\\medskip\n\nLet's start with the Fock quantization. We have shown that the creation-annihilation operators respect an algebra similar to the standard one. There is a caveat though, as this algebra is labeled by vectors in $\\mathbb{R}^3$ (rather than $\\mathbb{R}^2$) but with the additional constraint of being on the mass shell. This corresponds to functions living on the two-sheet hyperboloid, with the condition that reflection with respect to the origin gives rise to a complex conjugation.\n\nIf we want to map this algebra onto the usual one, we have to project these functions over the hyperboloid onto the plane $\\mathbb{R}^2$. This can be done quite easily (though not in a covariant way) by considering only one sheet of the hyperboloid (the other one can be recovered by conjugation) and forgetting about the time component of the momentum $k$. For instance, let's restrict to the $k_t > 0$ sheet. We can define:\n\\begin{equation}\nc_{\\tilde{k}} = a_{(\\sqrt{\\tilde{k}^2 + m^2},\\tilde{k})}.\n\\end{equation}\nThe $c$ operators now check an algebra that is even more familiar:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\n\\{c_{\\tilde{k}}, c_{\\tilde{k}'}\\} &=& 0, \\\\\n\\{\\overline{c_{\\tilde{k}}}, \\overline{c_{\\tilde{k}'}}\\} &=& 0, \\\\\n\\{c_{\\tilde{k}}, \\overline{c_{\\tilde{k}'}}\\} &=& 2\\mathrm{i}\\sqrt{\\tilde{k}^2 + m^2} \\delta(\\tilde{k}-\\tilde{k}').\n\\end{array}\n\\right.\n\\end{equation}\nWe notice here an energy factor. This is due to the unusual convention used for the $a$ as we did not divide by the square root of the energy. Though this was natural to preserve a covariant expression, this means that the square of $a$ operators (that is $N_k = a_k^\\dagger a_k$) does not count particles but rather directly counts energy quantas. From there, the usual Fock quantization is known. It is useful however, for the sake of completeness, to develop it in a language closer to our originally found algebra, that is with:\n\\begin{equation}\n\\{\\delta(k^2 + m^2)a_k, \\delta(k'^2 + m^2)a_{k'}\\} = -\\mathrm{i}\\mathrm{sgn}(k_t)\\delta(k'^2 + m^2)\\delta(k+k').\n\\end{equation}\nThis will lead to a more covariant expression more suited to the quantum gravity problem.\n\nWe must start with the one particle Hilbert space $\\mathcal{H}$. First let $\\mathbb{H}$ be the two-sheet hyperboloid embedded in $\\mathbb{R}^3$ defined by:\n\\begin{equation}\nt^2 - x^2 - y^2 = m^2\n\\end{equation}\nwhere $(t,x,y)$ are the coordinates in $\\mathbb{R}^3$. Now, $\\mathcal{H}$ will be the space of functions from $\\mathbb{H}$ into $\\mathbb{C}$ equipped with the following scalar product:\n\\begin{equation}\n\\langle \\psi | \\phi \\rangle = \\int \\delta(k^2 + m^2)\\overline{\\psi}(k)\\phi(k) \\mathrm{d}^3 k.\n\\end{equation}\nThis is the momentum representation for our one-particle. Because, we are interested in real valued fields, we will add the following constraint:\n\\begin{equation}\n\\forall k \\in \\mathbb{R}^3,\\ \\forall \\phi \\in \\mathcal{H},\\ \\overline{\\phi(k)} = -\\phi(-k).\n\\end{equation}\nNote the minus sign corresponding to the fact that $\\overline{a_k} = -a_{-k}$. With this definition $\\mathcal{H}$ is trivially a pre-Hilbertian space. By choosing a plane in $\\mathbb{R}^3$ to parametrize $\\mathbb{H}$, we get however that:\n\\begin{equation}\n\\langle \\psi | \\phi \\rangle = \\int \\frac{1}{2\\sqrt{\\vec{k}^2 + m^2}}\\overline{\\psi}(k)\\phi(k) \\mathrm{d}^2 k.\n\\end{equation}\nThis shows that $\\mathcal{H}$ is isomorphic to $\\mathrm{L}^2(\\mathbb{R}^2)$ with the caveat that the wave-functions must be divided $\\sqrt{2E}$ in the mapping. This factor is actually quite important as it appeared in our algebra for the $a_k$ and this will allow a simpler representation of the creation-annihilation operators.\n\nNow, we define the following sequence of Hilbert spaces:\n\\begin{enumerate}\n\t\\item $\\mathcal{H}_0 = \\mathbb{C}$, the $0$-particle Hilbert space, also called the vacuum Hilbert space,\n\t\\item $\\mathcal{H}_1 = \\mathcal{H}$, the $1$-particle Hilbert space as previously explained.\n\t\\item $\\mathcal{H}_n = \\mathrm{Sym}(\\mathcal{H}^{\\otimes n})$, for $n \\ge 2$, the symmetric part of the tensor product of $n$ copies of $\\mathcal{H}$ and represents the $n$-particle Hilbert space for bosonic particles..\n\\end{enumerate}\nThe Fock space $\\mathcal{H}_\\phi$ is defined by:\n\\begin{equation}\n\\mathcal{H}_\\phi = \\bigoplus_{n\\in\\mathbb{N}} \\mathcal{H}_n .\n\\end{equation}\n\nNow, we can define the creation and annihilation operators $a_k$. There are two cases. First, let's consider $k$ such that $k^2 + m^2 = 0$ and $k_t < 0$. We define $\\hat{a}_k$ by its restriction $\\hat{a}_{k,n}$ on $\\mathcal{H}_n$. For $n \\ge 1$, we define $\\hat{b}_{k,n}$:\n\\begin{equation}\n\\hat{b}_{k,n} : \\left\\{\n\\begin{array}{rcl}\n\\mathcal{H}^{\\otimes n} &\\rightarrow& \\mathcal{H}^{\\otimes (n-1)} \\\\\n| v_1 \\rangle \\otimes | v_2 \\rangle \\otimes \\cdots \\otimes | v_n \\rangle &\\mapsto& \\frac{1}{\\sqrt{n}}\\sum_{i=1}^n v_i(k) | v_1 \\rangle \\otimes | v_2 \\rangle \\otimes \\cdots \\otimes \\widehat{| v_i \\rangle} \\otimes \\cdots \\otimes | v_n \\rangle\n\\end{array}\n\\right.\n\\end{equation}\nAs standard, $\\widehat{| v_i \\rangle}$ means that $| v_i \\rangle$ is omitted from the list. $\\hat{a}_{k,n}$ is the restriction of $\\hat{b}_{k,n}$ to $\\mathcal{H}_n$. For $n=0$, we have:\n\\begin{equation}\n\\hat{a}_{k,0} : \\left\\{\n\\begin{array}{rcl}\n\\mathcal{H}_0 &\\rightarrow& \\mathcal{H}_{0} \\\\\nv &\\mapsto& 0\n\\end{array}\n\\right.\n\\end{equation}\nwhich corresponds to the fact that the vacuum is annihilated by all annihilation operators.\n\nSimilarly, we can define $a_k$ for $k$ such as $k^2 + m^2 = 0$ and $k_t > 0$. This will act in the (algebraic) dual spaces. Let's define $\\hat{b}_{k,n}$:\n\\begin{equation}\n\\hat{b}_{k,n} : \\left\\{\n\\begin{array}{rcl}\n(\\mathcal{H}^\\star)^{\\otimes n} &\\rightarrow& (\\mathcal{H}^\\star)^{\\otimes (n+1)} \\\\\n\\langle v_1 | \\otimes \\langle v_2 | \\otimes \\cdots \\otimes \\langle v_n | &\\mapsto& \\frac{1}{\\sqrt{n+1}}\\sum_{i=1}^{n+1} \\langle v_1 | \\otimes \\langle v_2 | \\otimes \\cdots \\otimes \\langle u | \\otimes \\langle v_i | \\otimes \\cdots \\otimes \\langle v_n | ,\n\\end{array}\n\\right.\n\\end{equation}\nwith:\n\\begin{equation}\n\\forall | v \\rangle \\in \\mathcal{H},\\ \\langle u | v \\rangle = v(k).\n\\end{equation}\n$\\hat{a}_{k,n}$ is the restriction of $\\hat{b}_{k,n}$ to $\\mathcal{H}_n$. This concludes the matter sector.\n\n\\medskip\n\nFor the gravity sector, we have two sets of observables. We have the curvature constraints which, as long as we don't restrict to the constraint surface, are legitimate observables. We will write $D_I(x)$ from now on and remember that they are densities. And we have the triad $e_a^I(x)$. They are not exactly conjugate. The conjugate arise when we integrate them along a line (possibily starting from infinity as mentioned in section \\ref{sec:basic_ops}). Then $\\int^{P(\\sigma)} e^I$ is conjugate to $D_I(x)$ and commutes with the $a$ operators. When we integrate, we loose some information. But it is remarkable that we don't loose gauge-invariant information: thanks to gauge-invariance, the integral of $e$ only depends on the end-point of the integral. That means we completely characterize the subspace defined by $de^I = 0$. This is this subspace that we will quantize.\n\nThe curvature constraints $D_I(x)$ are densities while, the integral of the triad acts as a scalar function. This setup is similar to Loop Quantum Gravity where conjugate quantities are carried by dual geometrical constructs. It is in fact exactly equivalent to the usual Loop Quantum Gravity setup except that here, because we have used gauge-invariant quantities, the support is on surfaces and points rather than lines. As a first approach however, we will not quantize in the standard fashion - that is using the Ashtekar-Lewandowski representation or its equivalent -  but will rather consider the equivalent of the BF representation \\cite{Dittrich:2014wpa,Bahr:2015bra}. Indeed, we have two choices: either we start from a vacuum state where $e=0$ everywhere or we start with a vacuum state that has $D_I(x) = 0$ everywhere. The second case is akin to the BF vacuum and is very relevant to our problem: this vacuum state is precisely the solution to the constraints. So let's quickly sum up the construction in the abelian case.\n\nLet's define the Hilbert space $\\mathcal{H}_G$. Let $\\mathcal{R}$ be the space of functions over $\\Sigma$ valued in $\\mathbb{R}^3$ that are zero everywhere except for a finite number of points. Now $\\mathcal{H}_G$ is the space of square integrable functions over $\\mathcal{R}$ equipped with the following scalar product:\n\\begin{equation}\n\\langle \\Psi_1 | \\Psi_2 \\rangle = \\sum_{\\vec{f} \\in \\mathcal{R}} \\overline{\\Psi_1(\\vec{f})} \\Psi_2(\\vec{f}).\n\\end{equation}\nThe sum is well-defined (though possibly infinite) thanks to the square integrable condition. Note that this space can be constructed by a projective limit (as it is standard in Loop Quantum Gravity). In that case, we would have functions depending on $\\mathbb{R}^3$ labels for a finite number of points. Two functions with support on a different set of points would be equivalent (regarding cylindrical consistency) if they do not depend on the labels of the points that are no shared and if the dependency is the same for shared points. This is however not needed here thanks to the combination of two properties. First, because we look at the gauge-invariant subspace, the support is points rather than graph, things are greatly simplified. And because the gauge group is abelian, much simpler expressions can be given still. Nonetheless, the construction is similar in spirit: we have a normalized vacuum state which is:\n\\begin{equation}\n\\Psi_0(f) = \\left\\{\n\\begin{array}{rl}\n1 &\\textrm{if }f = \\vec{0}, \\\\\n0 &\\textrm{otherwise.}\n\\end{array}\n\\right.\n\\end{equation}\nHere $\\vec{0}$ is understood to be the function that is constant over $\\Sigma$ and equal to the vector $\\vec{0}$. Then, excitations can be constructed with the action of the exponential of the integrated triad (which we will construct shortly). The Hilbert space is then the completion of the linear span of these excitations. This means that we have an Hilbertian basis given by the indicator functions once more. A member $\\Psi_f$ of the basis is given for each function $f$ of $\\mathcal{R}$ and is defined by:\n\\begin{equation}\n\\Psi_f(g) = \\left\\{\n\\begin{array}{rcl}\n1 &\\textrm{if}& g = f, \\\\\n0 &\\textrm{if}& g \\neq f.\n\\end{array}\n\\right.\n\\end{equation}\n\nThe operator corresponding to $D_I(x)$ must be regularized. As $D_I(x)$ is a density, it is natural to consider the following integrated quantities: $\\int N(x) D_I(x) \\mathrm{d}^2\\sigma$ where $N$ is some test function. We will therefore define the operator $\\hat{D}_I[N]$. It is defined by its action on the basis in the following manner:\n\\begin{equation}\n\\hat{D}_I[N]\\Psi_f = \\left(\\sum_{P \\in \\Sigma} N(P) f(P)_I\\right)\\Psi_f.\n\\end{equation}\nThis action is not always well-defined but it is on a dense subset of the space (namely the span of states $\\Psi_f$ with functions $f$ that have finitely many non-zero points). We see here that the basis we constructed diagonalizes the $\\hat{D}_I[N]$ operator. Similarly, we can defined the exponentiated operator for the triad. We do not need to regularize this time (except through the integral). Let $\\vec{k}$ be in $\\mathbb{R}^3$ and $P$ on $\\Sigma$. We define $\\hat{E}(\\vec{k}, P)$ by its action of the basis:\n\\begin{equation}\n\\hat{E}(\\vec{k}, P)\\Psi_f = \\Psi_{\\tilde{f}},\n\\end{equation}\nwhere:\n\\begin{equation}\n\\tilde{f}(Q) = \\left\\{\n\\begin{array}{rcl}\nf(Q) &\\textrm{if}& Q \\neq P, \\\\\nf(P)+\\vec{k} &\\textrm{if}& Q = P.\n\\end{array}\n\\right.\n\\end{equation}\nAs such $\\hat{E}(\\vec{k}, P)$ is the quantization of $\\exp \\left(-\\mathrm{i}\\vec{k}\\cdot \\int^P \\vec{e}\\right)$.\n\nNote that the non-exponentiated version of the operator does not exist. In practice, this means we have used the Bohr compactification of $\\mathbb{R}^3$ for the values of the integrals. This can be seen by the fact that the dual (present in eigenvalues of the curvature constraints) is $\\mathbb{R}^3$ equipped with a discrete topology. This trick is handy to circumvent the problem of using non-compact groups. Sadly, the Bohr compactification is only injective for maximally almost periodic groups which the gauge group of the non-abelian theory ($\\mathrm{SU}(1,1)$) is not. This is what prevents the standard Ashtekar-Lewandowski construction for non-compact gauge group. It should be noted however that such an obstruction is not present for the BF vacuum \\cite{Bahr:2015bra}. It might very well be then, that the current construction generalizes to the non-abelian case.\n\nFinally, the kinematical Hilbert space is simply $\\mathcal{H}_G \\otimes \\mathcal{H}_\\phi$ with the operators naturally extended. The solution to the constraints is simply: $(\\mathbb{C} \\Psi_0)\\otimes \\mathcal{H}_\\phi \\simeq \\mathcal{H}_\\phi$ where $\\Psi_0$ is the vacuum for $\\mathcal{H}_G$. It is trivial to see that this space is isomorphic to the standard Hilbert space for a free field theory. Though this construction is interesting to get a feel of how the theory works in the quantum realm, it is not satisfying on at least two accounts:\n\\begin{enumerate}\n\t\\item First, it relies too much on a change of variable. Normally, to get a direct link with the classical theory, one would start with canonical variables and represent them, and then try to express constraints and similar operators. Here, not only have we not done that, it is not even possible to express the original operators. For instance, it is incredibly difficult (if not outright impossible) to extract the curvature operator out of the constraints. Indeed, to do that, we require both the fields operators (which we don't have) and the inverse of the metric (which does not even exist as an operator). Similarly, the natural expression for the momentum operator for the field depends on the normal operator, which does not exist because of the Bohr compactification we used.\n\t\\item Second, it relies heavily on the abelian structure of the theory. All this approach was only possible because we can decouple completely two sectors that we might want to call the gravitational and the matter sector (though the curvature cosntraint has a bit of matter in it). This is not something we can hope for in a non-abelian theory. So the method is way too specific to our case.\n\\end{enumerate}\nIt does not mean it is not useful though: this acts as a guideline. We now know what the theory looks like and what to expect from different constructions.\n\nThe ideal construction however would start from the curvature operator, the triad and the field operators and then get the constraints. At least, it should be possible to reconstruct all these operators. This is however not possible in our case. Indeed, the curvature operator (or the holonomy operator) appears only in the curvature operator for now. As a consequence, we will first need the scalar field operator and the momentum operator to be able to retrieve it. However, from the work done in section \\ref{subsec:Fourier}, we can use the Fourier transform in $\\mathbb{R}^3$ to get expressions of $\\phi$ and $\\Pi$ in terms of the creation and annihilation operators. We get:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\n\\phi(\\sigma) &=& \\int \\delta(k^2 + m^2) \\frac{\\mathrm{sgn}(k_t)}{\\sqrt{2}\\pi}a_k \\mathrm{e}^{\\mathrm{i} \\vec{k} \\cdot \\int^\\sigma \\vec{e}} \\mathrm{d}^3 k, \\\\\n\\Pi(\\sigma) &=& \\int \\delta(k^2 + m^2) (\\vec{k} \\cdot \\vec{n}) \\frac{\\mathrm{sgn}(k_t)}{\\sqrt{2}\\pi}a_k \\mathrm{e}^{\\mathrm{i} \\vec{k} \\cdot \\int^\\sigma \\vec{e}} \\mathrm{d}^3 k.\n\\end{array}\n\\right.\n\\end{equation}\nThe expression of $\\Pi$ is particularly problematic as it relies on the existence of an operator for the normal $n$, which does not exists in our representation.\n\nOne might want to try and use the more standard Ashtekar-Lewandowski representation $\\mathcal{H}_{AL}$. In that case, it is possible to construct a normal operator $n$ in a way similar to the area operator in LQG \\cite{Ashtekar:1996eg}. However, in that case, we face another problem: given a state of the form $|0\\rangle \\otimes |\\phi\\rangle \\in \\mathcal{H}_{AL}\\otimes\\mathcal{H}_\\phi$ where $|0\\rangle$ is the AL vacuum and $|\\phi\\rangle$ is some state in $\\mathcal{H}_\\phi$, we have $\\hat{\\Pi}|0\\rangle \\otimes |\\phi\\rangle = 0$ irrespective of the state $|\\phi\\rangle$. This might be possible to cure, by forgetting about classical expressions and rather concentrating on reproducing the algebra at the quantum level. This would be however surprising since the expression for $\\Pi$ is quite regular involving only exponentials and polynomials in the triad that commute among themselves and should not require regularization.\n\nWe want to suggest another direction in this paper, that we will start exploring in the next section. Though, we do not have a complete proof for a successful construction, the arguments we just laid out fail in this context. This solution, though it seems unnatural at first, has - in hindsight - geometrical justification. The idea is to use the work done by Koslowski and Sahlmann \\cite{Koslowski:2007kh,Sahlmann:2010hn,Koslowski:2011vn} and to develop a representation peaked on a classical non-degenerate spatial metric. Though perfect diffeomorphism invariance (for the vacuum) is lost, there is still a notion of diffeomorphism covariance available and the geometrical interpretation we will offer justifies the choice of a particular background, at least for abelian gravity. We develop this approach in the following section.\n\n\\subsection{Ashtekar-Lewandowski representation peaked on a classical vacuum}\n\n\\label{sec:newrep}\n\nThe difficulty we face is linked to the non-existence of non-exponentiated versions of the triad operators on the Hilbert space. This is quite standard in Loop Quantum Gravity: the standard constructions only allow for one operator out of a conjugated pair to be defined, the other one is only defined through its exponentials. In the usual Ashtekar-Lewandowski representation \\cite{Ashtekar:1996eg,Ashtekar:1997fb,Ashtekar:1998ak} for instance, the holonomy operators are well-defined but only the exponentiated versions are defined. In the BF representation defined by Dittrich \\textit{et al.} \\cite{Dittrich:2014wpa,Bahr:2015bra}, the triad is only defined through its exponentials, but some version of the logarithm of the holonomies are defined\\footnote{There are in fact technical difficulties in this case because of the non-abelian nature of the gauge group. However, the limit for loops going to zero is usually well-defined (though group-valued) and play the same role.}. In our case, we have developed the equivalent of the BF representation, since the conjugate to the triad is defined. Moving to the standard Ashtekar-Lewandowski representation will not help however. Indeed, our problem is not only linked with the possibility of writing a simple triad operator but also the possibility of inverting it, at least to some extent as we want to be able to write the inverse determinant of the spatial metric. And the usual Ashtekar-Lewandowski representation does not allow for that (at least not in any known ways\\footnote{Though Thiemann developed some ideas in this regard \\cite{Thiemann:1996ay}, there are severe questions on whether his approach is successful \\cite{Livine:2013wmq}.}) since the vacuum is degenerate everywhere and all the excited states are degenerate almost everywhere. If we want to write the inverse determinant, we will therefore need a new representation of the holonomy-flux algebra (or of its equivalent in our case - since we considered only the gauge-invariant sector).\n\nIt is noteworthy that some other representations have been discussed already in Loop Quantum Gravity, most notably \\cite{Koslowski:2007kh,Sahlmann:2010hn,Koslowski:2011vn}. This representation is very similar to the Ashtekar-Lewandowski representation, except the vacuum is not peaked on degenerate geometry but rather on a given classical metric. Of course, diffeomorphism invariance of the vacuum is lost, which explains how the LOST theorem \\cite{Lewandowski:2005jk} is evaded, and is replaced by a notion of diffeomorphism covariance. This representation is however very interesting to us because the metric is everywhere non-degenerate for the vacuum. Even for most of the excited states, the metric is non-degenerate and when it is not, it is only degenerate on a finite number of points. As long as we can reproduce the classical algebras correctly, this leads to very natural expressions for the inverse determinant of the metric. However, we have now traded another issue which is the choice of the background metric, which seems a bit counter-productive with regard to the standard Loop Quantum Gravity approach.\n\n\\medskip\n\nBefore tackling this problem however, let's sum up Koslowski's and Sahlmann's approach in \\cite{Koslowski:2007kh,Sahlmann:2010hn,Koslowski:2011vn} and adapt it to our case. The construction uses the dual structure to the one we have done in section \\ref{sec:firstapproach}. In the previous construction, the operators acting on surfaces (the constraints) were diagonal, and excitations were created by acting on points. Here, it is the reverse: the point operators are diagonal and the surface operators create the excitations. This means we need some projective techniques to deal with it correctly.\n\nWe can define a Hilbert space $\\mathcal{H}_\\Delta$ for a given triangulation $\\Delta$ of $\\Sigma$. This Hilbert space is the completion of the span of the basis given by $\\mathbb{R}^3$ labels of the triangles that are non-zero for a only finite number of triangles. We can make this precise in the following manner: let $\\mathcal{F}_\\Delta$ be the space of functions for the triangles of $\\Delta$ into $\\mathbb{R}^3$ such that the values are non-zero for a finite number of triangles. This is the space of labels on the triangulation. The elements of $\\mathcal{H}_\\Delta$ are functions from $\\mathcal{F}_\\Delta$ into $\\mathbb{C}$ that are square integrable for:\n\\begin{equation}\n\\langle \\psi | \\phi \\rangle = \\sum_{f \\in \\mathcal{F}_\\Delta} \\overline{\\psi(f)} \\phi(f).\n\\end{equation}\nThe full (continuous) Hilbert space is defined as:\n\\begin{equation}\n\\mathcal{H}_{KS} = \\left(\\bigcup_{\\Delta} \\mathcal{H}_\\Delta \\right)\\Big\\slash \\sim.\n\\end{equation}\nHere the union is a disjoint union over all possible triangulations of $\\Sigma$. We must now define the equivalence relation~$\\sim$.\n\nFor this, we need the notion of a refinement of a triangulation. A triangulation $\\Delta'$ is a refinement of $\\Delta$ if for any triangle in $\\Delta$ is the union of triangles in $\\Delta'$. We can then map any function of $\\mathcal{F}_\\Delta$ into $\\mathcal{F}_{\\Delta'}$. For $f \\in \\mathcal{F}_\\Delta$, we define $f' \\in \\mathcal{F}_{\\Delta'}$ as:\n\\begin{equation}\nf'(t) = f(T)\\textrm{, with }t \\subseteq T.\n\\end{equation}\nSimilarly, we can write extend a state $\\psi \\in \\mathcal{H}_\\Delta$ into $\\psi' \\in \\mathcal{H}_{\\Delta'}$ as follows:\n\\begin{equation}\n\\psi'(f) = \\left\\{\n\\begin{array}{rl}\n\\psi(g)&\\textrm{if }g'=f, \\\\\n0 &\\textrm{otherwise.}\n\\end{array}\\right.\n\\end{equation}\nWe can finally get to our equivalence relation necessary to define $\\mathcal{H}_{KS}$. Two states $\\psi \\in \\mathcal{h}_\\Delta$ and $\\psi' \\in \\mathcal{H}_{\\Delta'}$ are equivalent if and only if there exists a refinement $\\Delta''$ of both $\\Delta$ and $\\Delta'$ such that the extension of $\\psi$ and $\\psi'$ in $\\mathcal{H}_{\\Delta''}$ are identical. Note that if this is true, it is true for any refinement of both triangulations. Note also that there is always a refinement of both triangulations but there is no guarantee that the extension of $\\psi$ and $\\psi'$ will match.\n\nUp to this point, the definition actually follows the techniques of the BF vacuum in order to adapt the construction to quantities carried by surfaces and points (rather than lines). But what will distinguish $\\mathcal{H}_{KS}$ from both the BF representation and the standard AL representation is the construction of the operators.\n\nFirst, let's start with the simplest operator: the integrated curvature constraint. Let $\\Delta$ be a triangulation of $\\Sigma$ and $\\phi$ a function from the triangles into $\\mathbb{R}^3$ non-zero only a finite number of triangles. If $\\Delta'$ is a refinement of $\\Delta$, we define:\n\\begin{equation}\n\\widehat{\\mathrm{e}^{\\mathrm{i} D[\\phi]}} : \\left\\{\n\\begin{array}{rcl}\n\\mathcal{H}_{\\Delta'} &\\rightarrow& \\mathcal{H}_{\\Delta'} \\\\\n\\psi &\\mapsto& \\psi'\n\\end{array}\n\\right.\n\\end{equation}\nwith:\n\\begin{equation}\n\\psi'(f) = \\psi(f + \\phi).\n\\end{equation}\nThe final sum is done by extending $\\phi$ to $\\Delta'$. This is standard action, completely equivalent, so far, to the one in the AL-representation. This action can be extended on coarser representation. It is compatible with the quotient and therefore carries to whole space $\\mathcal{H}_{KS}$.\n\nSecond, we can consider the triad operator. This is done in two steps. As a first step, let $\\Delta$ be a triangulation. Let's denote$| \\psi_f \\rangle$ the state in $\\mathcal{H}_{\\Delta}$ defined by:\n\\begin{equation}\n\\psi_f(g) = \\left\\{\n\\begin{array}{rl}\n1 &\\textrm{if }f=g,\\\\\n0 &\\textrm{otherwise,}\n\\end{array}\n\\right.\n\\end{equation}\nwith $f \\in \\mathcal{F}_{\\Delta}$. These states form a (Hilbertian) basis of $\\mathcal{H}_{\\Delta}$. We can now define:\n\\begin{equation}\n\\widehat{\\mathcal{O}[\\phi]} | \\psi_f \\rangle = \\sum_{\\sigma \\in \\Sigma} \\phi(\\sigma) \\cdot f(\\sigma) | \\psi_f \\rangle,\n\\end{equation}\nwith $\\phi$ is a function from $\\Sigma$ into $\\mathbb{R}^3$ with finitely many non-zero values. Thus $\\sum_{\\sigma \\in \\Sigma} \\phi(\\sigma) \\cdot f(\\sigma)$ is understood as a sum over these finitely many values and $f(\\sigma)$ is the label for the triangle of $\\Delta$ that $\\sigma$ belongs to\\footnote{In practice, this means that this sums is not well-defined if the point $\\sigma$ fulls on an edge or a vertex of the triangulation. This is not important for us as we can just reduce the domain of the operator.}. We recognize here the definition of the triad operator in the standard AL-representation. But now, as a second step, let's define a background field $\\tilde{e} : \\Sigma \\rightarrow \\mathbb{R}^3$. And consider the following operator:\n\\begin{equation}\n\\widehat{e[\\phi]} = \\int \\phi\\cdot\\tilde{e} + \\widehat{\\mathcal{O}[\\phi]}.\n\\end{equation}\nThis operator trivially has the same algebra but is peaked on a classical configuration for the triad. This is the main difference of the KS representation (compared to the usual AL one).\n\n\\medskip\n\nNow, all this construction relies on a choice of background metric and even, to be more precise, a choice of background triad. This choice seems arbitrary at first, but in our case there is a very natural way to select a class of metrics. Indeed, we have to remember that we are considering the gauge-invariant subspace which translates to the condition:\n\\begin{equation}\n\\mathrm{d}\\mathrm{e}^I = 0.\n\\end{equation}\nThis condition entails that, if we restrict once more to a simply connected manifold, the triad derives from a potential $\\Psi^I$. This functions acts as an embedding of $\\Sigma$ into $\\mathbb{R}^3$ (if the metric is invertible). But it also means that the integrated triad is zero on any closed loops. And this is valid also on the vacuum state. This means that the background triad must satisfy all these conditions and in particular correspond to an embedding into $\\mathbb{R}^3$. Up to topological questions, that we have discarded as we are considering the simplest case, this means that the metric is fixed up to diffeomorphism. This entails in turn that the construction will indeed depend on the metric but once the diffeomorphism constraints will be enforced, diffeomorphism invariance will be restored in a way which is independent from the choice of the initial metric (as long as it is invertible). So, from now on, let's just choose a background embedding into $\\mathbb{R}^3$ and use the triad that derives from it.\n\nLet's turn back to the full representation, including the matter sector. Our goal was to able to write expressions like:\n\\begin{equation}\n\\left\\{\n\\begin{array}{rcl}\n\\phi(\\sigma) &=& \\int \\delta(k^2 + m^2) \\frac{\\mathrm{sgn}(k_t)}{\\sqrt{2}\\pi}a_k \\mathrm{e}^{\\mathrm{i} \\vec{k} \\cdot \\int^\\sigma \\vec{e}} \\mathrm{d}^3 k, \\\\\n\\Pi(\\sigma) &=& \\int \\delta(k^2 + m^2) (\\vec{k} \\cdot \\vec{n}) \\frac{\\mathrm{sgn}(k_t)}{\\sqrt{2}\\pi}a_k \\mathrm{e}^{\\mathrm{i} \\vec{k} \\cdot \\int^\\sigma \\vec{e}} \\mathrm{d}^3 k.\n\\end{array}\n\\right.\n\\end{equation}\nThis suggested that the gravity sector needed a new representation. The Fock space used for matter is however completely equipped for such expressions. We will therefore rather keep it. This leads to the full Hilbert space:\n\\begin{equation}\n\\mathcal{H}_{\\textrm{Full}} = \\mathcal{H}_{KS}\\otimes\\mathcal{H}_\\phi.\n\\end{equation}\nBefore moving to the next section, let's make a final remark: though this representation gives natural inverse operators, in a sense, this does not matter. What matters is the algebra of the operators. In the end, we must find two natural pairs of collections of operators, corresponding to the field and momentum operator on the one hand and to the triad and curvature operator on the other. Moreover, these operator should lead to expressions for the constraints that match the previously found algebra. If the naive inversion fails, this will mean that this technique fails. This is what in the end should guide such construction. And these tests are still to be done with the method we just suggested.\n\n\\section{Discussion \\& Future work}\n\nGranted the previous idea can be made to work, the natural question is whether this can be extended outside of the abelian theory. Indeed, the representation we chose depended on a background which, for the abelian case, can be chosen naturally. This however depended on the resolution of the Gau\u00df constraints. In the non-abelian case, such a procedure might not be that well-defined. A few points are encouraging though: this representation gives a natural understanding of how matter propagates on an (abelian) quantum spacetime. Indeed, as we mentioned early on in this paper, the theory we developed is, at least in some sector, equivalent to a free scalar field theory. With such a theory, spacetime is completely classical. Our theory however is completely quantum mechanical, including spacetime. On the constraint surface, the triad in particular is completely ill-defined (in a quantum mechanical sense) and only the curvature has a precise value. We might wonder how a field might propagate freely here. The answer, according to the construction we have just done, is simple: spacetime really is flat. The degeneracy of the triad does not come from a true quantum degeneracy but rather is caused by the superposition of all the states coming from the action of the diffeomorphism constraints. The final state therefore is a superposition of classical flat space but seen from all possible coordinate systems. This is of course possible only because there are no local degrees of freedom in 3d gravity. Though, it might be possible to extend these techniques to non-abelian 3d gravity, the implications are not quite as clear for the 4d case. An interesting idea, that has been explored almost accidentally in the context of cosmology (see for instance \\cite{mukhanov2007introduction}) as a first approach, is that only local degrees of freedom (that is gravitational waves) are quantum in that sense.\n\n\\medskip\n\nLet's get back to the 3d problem. Even in that case, once we want to get to the full non-abelian theory, a few roadblocks appear. One of the major problem is path-dependency. Indeed, we defined the following operator as a creation operator:\n\\begin{equation}\na_k = \\frac{1}{\\sqrt{2}\\pi}\\int \\left(k^I n_I \\phi + \\mathrm{i}\\Pi\\right) \\mathrm{e}^{- \\mathrm{i} \\vec{k} \\cdot \\int^{\\sigma} \\vec{e}} \\mathrm{d}^2\\sigma .\n\\end{equation}\nThere we used the integral $\\int^\\sigma \\vec{e}$ which did not depend on the path chosen as long as the Gau\u00df constraints were satisfied. A natural extension to the non-abelian case would be:\n\\begin{equation}\nb_k = \\frac{1}{\\sqrt{2}\\pi}\\int \\left(k^I n_I \\phi + \\mathrm{i}\\Pi\\right) \\mathrm{e}^{- \\mathrm{i} \\vec{k} \\cdot \\int^{\\sigma} g \\triangleright \\vec{e}} \\mathrm{d}^2\\sigma ,\n\\end{equation}\nwhere $g$ is the holonomy of the connection along the integration path and acts as parallel transport. Though this expression is gauge-invariant, it depends on the path chosen for the integration, even when the Gau\u00df constraints are checked. This makes the correct generalization quite unclear. Two points should be underlined here however. First, similar problem have been dealt with in the construction of the BF representation and have been solved by a systematic choice of paths for gauge-fixing \\cite{Dittrich:2014wda}. This is moreover close to book-keeping techniques needed for the classical solution of the problem \\cite{tHooft:1992izc} which seems to support such an approach. Second, this problem can be partially recovered in the abelian case, if one wants to define the theory more generally without imposing first the Gau\u00df constraints. This might be needed anyway to be able to check the brackets of all the quantum operators we are interested in from the end of section \\ref{sec:newrep}. This will therefore be an interesting intermediate step to consider.\n\n\\medskip\n\nThe abelian case also relied on the commutativity between the operators $a_k$ and the constraints $D_I$. It would be surprising, to say the least, that such a setup could be possible in the non-abelian case, for the operators $b_k$ and the corresponding constraints $\\tilde{D}_I$. Several scenarios can be envisioned, the most probable to our eyes though is that, though the $b_k$ will not commute with the constraints, it should still be possible to make them into the algebra of creation and annihilation operator for some non-commutative field theory. In that case, they would allow us to write a basis of states on which it is reasonable to to a perturbative study. Ideally of course, some exact cases could be found, like a $m \\rightarrow 0$ limit, one-particle states or maybe some cosmological setup. In any case, the non-commutativity is not a problem as long as we can interpret it to be almost commutative in some limit. This, however, will only be possible if we can develop the full set of operators $\\phi$, $\\Pi$, $e$ and $A$ independently from the techniques we have employed in the commutative case. This means that one of the most important point moving forward is concluding the program opened by section \\ref{sec:newrep}.\n\n\\medskip\n\nLet's mention one last point before wrapping up: the idea of studying the abelian theory as a starting point, possibly for perturbative expansion is not new and was originally introduced by Smolin \\cite{Smolin:1992wj}. In our case however, we wanted it in particular to be able to study the geometry of the quantum spacetime. According to Connes'work (for instance \\cite{Chamseddine:1996zu}), this is better encoded in the Dirac operator governing the propagation of fermions rather than just the metric. A similar approach would then start with fermions coupled to abelian gravity. This is however rather ill-defined at the moment. Indeed, the gauge group does share the same topology as $\\mathrm{SU}(1,1)$, making the distinctions between bosons and fermions less clear. Moreover, it is not completely straightforward how the abelian connection should be coupled to the fermions. This is therefore an interesting point to explore further in future work.\n\n\\section{Conclusion}\n\nIn this paper, we considered a simplified model for 3d quantum gravity coupled to a scalar field. The model was taken from Smolin work \\cite{Smolin:1992wj}, corresponds to a specific $G \\rightarrow 0$ limit of standard 3d gravity, and can be formulated as standard BF theory (coupled to a scalar field in our case) but with an abelian gauge group. In four dimensions, this corresponds to a linearization of gravity but still expressed in a diffeomorphism invariant way. In three dimensions, the theory is still topological, but the dynamics is simplified. We showed in this paper in particular that a full sector of the theory is completely equivalent to a free scalar field, the gravity field only being there to allow for a diffeomorphism covariant formulation. This sector is actually fairly similar to what was already developed with parametrized field theories \\cite{Kuchar:1989bk,Kuchar:1989wz,Varadarajan:2006am}, although in higher dimensions and with a different language.\n\n\\medskip\n\nWe showed furthermore that this equivalence with a free scalar field theory leads to the formulation of a creation-annihilation algebra of operators, even in a diffeomorphism invariant setting. This algebra can in principle be extended to other sectors of the theory as long as the metric is everywhere invertible. Though the natural formulation is a bit different due to diffeomorphism invariance, the algebra is completely equivalent to the standard one for the free scalar field. The interesting point is that all these operators commute with the constraints for the abelian theory. This means they allow the construction of a set of solutions of the constraints, and mirror the fact that the classical abelian gravity theory (coupled to a scalar field) is equivalent, at least in some sector, to the classical free scalar field theory. This also means that these expressions are a good starting point for studying the non-abelian theory, for instance to try and quantize the theory perturbatively. This also allows the construction of a full quantization of the linear theory based on these operators as new variables. The quantum theory splits into two sectors. One is the sector that encodes the various excitation of the scalar field, and can be mapped one to one to the free scalar field theory. The second can be understood as the gravity sectors that more or less decouples in this abelian theory. It can be mapped onto the BF theory and be solved exactly.\n\n\\medskip\n\nThe drawback of such an approach is that some natural operators do not exist or are extremely difficult to construct. In particular, the momentum operator for the scalar field, and the holonomy operator for the gravity field, require the definition of (non-exponentiated) triad operators and an inverse-metric operator. This implies in particular, that even the canonical variables of the theory cannot be expressed simply or may be downright impossible to write. This is not really a specific problem of our approach: we used the equivalent of the BF representation in our construction which has similar difficulties for constructing triad operators or inverse-triad operators. However, in our case, these difficulties become a problem when trying to write a correlation operator for the scalar field for instance, which is a quantity we will eventually want to be able to compute. Using the older and somewhat more standard Ashtekar-Lewandowski representation only partially solves the problem. If it is indeed possible to define a non-exponentiated triad operator, the fact that the metric is degenerate almost everywhere for almost all states create huge problems with our approach which precisely requires the opposite. Moreover, natural expressions for the momentum operator of the scalar field are pathological, even though they only require exponential and polynomial terms in the triads, which should not need any regularization for the quantum case.\n\n\\medskip\n\nWe offered a possible way out. Though the construction needs to be studied more thoroughly, the drawbacks of the previous two approaches disappear. The idea is to construct a representation peaked on a given classical state for the spatial metric. This idea was explored by Koslowski and Sahlmann \\cite{Koslowski:2007kh,Sahlmann:2010hn,Koslowski:2011vn} as an equivalent to condensed state around a classical configuration. Though strict diffeomorphism invariance of the vacuum was lost, a sens of diffeomorphism covariance can still be retained. However, if this breaking was natural in their case, it seems more dubious when studying the theory from a more fundamental standpoint. We showed however that a specific vacuum can be selected using the Gau\u00df constraints in the linear case and corresponds to a flat space. Because the vacuum is nowhere degenerate, all the problems with the previously mentioned representations are lifted. Interestingly, the construction also allows a very nice interpretation of how the spacetime on which a free scalar field propagates is recovered in a setup where the triad is supposed to be completely degenerate in a quantum sense. In fact (in the abelian case), the classical spacetime is there all along and the degeneracy only comes from the superposition of all the diffeomorphism equivalent way of describing the system.\n\n\\medskip\n\nFinally, we left several questions open for further inquiry. Most notably, as we just said, the new representation we offered should be studied further. Indeed, even though the straightforward problems have been lifted, the study of the construction of the full operator set is still to be done. We left it ou however because a full and complete study would include a more complete treatment of the Gau\u00df constraints which we just assume to be satisfied. Lifting this condition requires dealing with gauge fixing, choice of path when integrating, etc. These points must be considered at some point as they are needed for the non-abelian theory but were left out of this first investigation. Similarly, we have left out all questions regarding the various possible sectors of the theory, the role of topology, the possible restrictions when considering compact spaces, etc. Though this is certainly worth investigating on its own merit, our goal was to get a first grap on how to develop a non-abelian theory. In this regard, though all this is very important, it will most probably be quite different when changing the Lorentz gauge group.\n\n\\section*{Acknowledgements}\n\nI would like to thank Stefan Hohenegger for the numerous discussions that helped and guided this project, and without whom none of this would have been possible. I would also like to thank John Barrett for the various conversations that launched the initial idea for this paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe envelope instability as a space-charge driven collective \ninstability presents a potentially great danger in high intensity\naccelerators by causing beam size blow up\nand quality degradation.\nIt has been studied theoretically~\\cite{ingo1,jurgen,ingo3,davidson,okamoto,fedotov0,fedotov,lund} and \nexperimentally~\\cite{tiefenback,gilson,groening} since 1980s. \nIn recent years, there was growing interest in further understanding\nthis instability and other structural resonances~\\cite{jeon1,fukushima,li0,li,ingo2,jeon2,oliver,ingoprab17,ito,yuan,ingo4}.\nSome of those studies were summarized in a recently published monograph~\\cite{ingobook}.\nHowever, most of those theoretical studies were based on a two-dimensional\nmodel. Three-dimensional macroparticle simulations were carried out\nfor a bunched beam under the guidance of the two-dimensional\nenvelope instability model~\\cite{ingo2,ingoprab17}.\nIt was found in reference~\\cite{ingo2} that the instability stopband from the \n3D macroparticle simulation\nis broader than that from the 2D envelope model.\nFurthermore, the effect of the longitudinal synchrotron motion has not been systematically\nstudied in those macroparticle simulations and is missed in \nthe 2D envelope instability model. \nIn this paper, we proposed a three-dimensional\nenvelope instability model in periodic focusing channels.\nSuch a model can be used to systematically study the effect of longitudinal\nsynchrotron motion on the instability stopband for a bunched beam.\nIt can also be used to explore the stability in a fully\n3D parameter space and to provide guidance for 3D macroparticle simulations.\n\nThe organization of this paper is as follows: after the introduction, we \nreview the 2D envelope instability model in Section II; we present\nthe 3D envelope instability model in Section III; we present numerical\nstudy of the envelope instability in a periodic transverse solenoid\nand longitudinal RF focusing channel in Section IV; \nwe present numerical study of \nthe envelope instability in a periodic transverse quadrupole and longitudinal\nRF focusing channel in Section V; and draw conclusions in Section VI.\n\n\\section{Two-dimensional envelope instability model}\n\nFor a two-dimensional coasting beam subject to external periodic focusing\nforces and linear space-charge forces, the two-dimensional envelope equations \nare given as\\cite{kv,jurgen}:\n\\begin{eqnarray}\n\t\\frac{d^2 X}{d s^2} + k_x^2(s) X - \\frac{K\/2}{X+Y} - \\frac{\\epsilon_x^2}{X^3} & = & 0 \\\\\n\t\\frac{d^2 Y}{d s^2} + k_y^2(s) Y - \\frac{K\/2}{X+Y} - \\frac{\\epsilon_y^2}{Y^3} & = & 0 \n\t\\label{2denv}\n\\end{eqnarray}\nwhere $X$ and $Y$ are horizontal and vertical rms beam sizes\nrespectively, $k_x^2$ and $k_y^2$ represent the external periodic focusing forces, \n$\\epsilon_x$ and $\\epsilon_y$ denote unnormalized rms emittances, and \n$K$ is the generalized perveance associated with the space-charge strength \ngiven by:\n\\begin{eqnarray}\n\t K & = & \\frac{q I}{2 \\pi \\epsilon_0 p_0 v_0^2 \\gamma_0^2}\n\\end{eqnarray}\nwhere $I$ is the current of the beam, $q$ is the charge of the particle,\n$\\epsilon_0$ is the vacuum permittivity, $p_0$ is the momentum\nof the reference particle, $v_0$ is the speed of the reference particle,\nand $\\gamma_0$ is the relativistic factor of the reference particle.\n\nThe above equations can be linearized with respect to a periodic solution\n(i.e. matched solution) as:\n\\begin{eqnarray}\n\tX(s) & = & X_0(s) + x(s)  \\\\\n\tY(s) & = & Y_0(s) + y(s) \n\\end{eqnarray}\nwhere $X_0$ and $Y_0$ denote the periodic matched envelope solutions \nand $x$ and $y$ denote small perturbations\n\\begin{equation}\n\tx(s) \\ll X_0(s), \\ \\ \\ \\ y(s) \\ll Y_0(s)\n\\end{equation}\nThe equations of motion for the small perturbations are given by:\n\\begin{eqnarray}\n\t\\frac{d^2 x(s)}{d s^2} + a_1(s) x(s) + a_{12}(s) y(s) = 0  \\\\\n\t\\frac{d^2 y(s)}{d s^2} + a_2(s) y(s) + a_{12}(s) x(s) = 0  \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\ta_{12}(s) & = & 2K\/(X_0(s) + Y_0(s))^2  \\\\\n\ta_1(s) & = & k_x^2(s) + 3 \\epsilon_x^2\/X_0^4(s) + a_{12}(s)  \\\\\n\ta_2(s) & = & k_y^2(s) + 3 \\epsilon_y^2\/Y_0^4(s) + a_{12}(s)  \n\\end{eqnarray}\nWith $\\xi = (x,x',y,y')^T$, the prime denotes derivative with respect to $s$, \nand $T$ denotes the transpose of a matrix, \nthe above equations can be rewritten in matrix notation as:\n\\begin{eqnarray}\n\t\\frac{d \\xi}{d s} & = & A_4(s) \\xi(s)\n\t\\label{2deq}\n\\end{eqnarray}\nwith the periodic matrix\n\\begin{eqnarray}\n\tA_4(s) & = & \\begin{pmatrix}\n\t\t0 & 1 & 0 & 0 \\\\\n\t\t-a_1(s) & 0 & -a_{12}(s) & 0 \\\\\n\t\t0 & 0 & 0 & 1 \\\\\n\t\t-a_{12}(s) & 0 & -a_2(s) & 0 \\\\\n\t\t \\end{pmatrix}\n\\end{eqnarray}\nLet $\\xi(s) = M_4(s) \\xi(0)$ be the solution of above equation,\nsubstituting this equation into \nEq.~\\ref{2deq} results in\n\\begin{eqnarray}\n\t\\frac{d M_4(s)}{ds} & = & A_4(s) M_4(s)  \n\\end{eqnarray}\nwhere $M_4(s)$ denotes the $4 \\times 4$ transfer matrix solution of $\\xi(s)$ and\n$M_4(0)$ is a $4\\times 4$ unit matrix. The matrix $A_4(s)$ is a periodic function of $s$\nwith a length of period $L$. Following the Floquet's theorem, the\nsolution of $M_4(s)$ after $n$ lattice periods can be written as\n\\begin{eqnarray}\n\tM_4(s+nL) & = & M_4(s)M_4(L)^n \n\\end{eqnarray}\nThis matrix solution will remain finite as $n->\\infty$, only if all amplitudes \nof the eigenvalues of the matrix $M_4(L)$ be less than or equal to one.\nSince the matrix $M_4(L)$ is a real symplectic matrix, the eigenvalues\nof the matrix occur both as reciprocal and as complex-conjugate pairs.\nTherefore, for stable solutions, \nall eigenvalues of the matrix $M_4(L)$ have to lie on a\nunit circle in the complex plane. \nThe eigenvalues of the matrix $M_4(L)$ can be expressed in polar coordinates as:\n\\begin{eqnarray}\n\t\\lambda & = & |\\lambda| \\exp{(i \\phi)}\n\\end{eqnarray}\nwhere the amplitude $|\\lambda|$ of the eigenvalue gives the growth rate\n(or damping rate) of the envelope eigenmode through one lattice period and the\nphase shift $\\phi$ of the eigenvalue gives the phase of the envelope mode\noscillation through one period.\nFor an unstable envelope mode, there are two possibilities~\\cite{jurgen}:\n\\begin{enumerate}\n\t\\item one or both eigenvalue pairs lie on the real axis: $\\phi_{1,2}=180^\\circ$,\n\t\\item the phase shift angles are equal: $\\phi_1 = \\phi_2$.\n\\end{enumerate}\nThe first case can be seen as a half-integer parametric resonance between\nthe focusing lattice and the envelope oscillation mode. The second\ncase is a confluent resonance between two envelope oscillation modes since they\nhave the same oscillation frequencies. \n\n\\section{Three-dimensional envelope instability model}\nThe 3D envelope equations have been used to study the halo particle formation\nmechanism (e.g. particle-core model) for a bunched beam in high intensity \naccelerators~\\cite{bongardt,allen,qiang0,comunian}.\nThere, the mismatched envelope oscillation resonates with a test particle and\ndrives the particle into large amplitude becoming a halo particle.\nThe mismatched envelope oscillation itself is stable in that case. \nIn this paper, \nwe study the stability\/instability of the mismatched envelope \noscillation itself in periodic focusing channels.\n\nFor a 3D uniform density ellipsoidal beam inside a periodic focusing channel without acceleration, \nthe three-dimensional envelope equations are given as~\\cite{sacherer,ryne}:\n\\begin{eqnarray}\n\t\\frac{d^2 X}{d s^2} + k_x^2(s) X - I_x(X,Y,Z)X - \\frac{\\epsilon_x^2}{X^3} & = & 0 \\\\\n\t\\frac{d^2 Y}{d s^2} + k_y^2(s) Y - I_y(X,Y,Z)Y - \\frac{\\epsilon_y^2}{Y^3} & = & 0   \\\\\n\t\\frac{d^2 Z}{d s^2} + k_z^2(s) Z - I_z(X,Y,Z)Z - \\frac{(\\epsilon_z\/\\gamma^2)^2}{Z^3} & = & 0 \n\t\\label{3denv}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\tI_i(X,Y,Z) = C\\int_0^{\\infty} \\frac{dt}{(e_i^2+t)\\sqrt{(X^2+t)(Y^2+t)(\\gamma^2 Z^2+t)}}\n\\end{eqnarray}\nwhere $X$, $Y$, and $Z$ are horizontal, vertical, and longitudinal rms beam \nsizes respectively, $e_i = X, Y, \\gamma Z$, for $i=x,y,z$, and $C = \\frac{1}{2}\\frac{3}{4\\pi \\epsilon_0}\\frac{q}{mc^2}\\frac{I}{f_{rf} \\beta^2 \\gamma^2}\\frac{1}{5\\sqrt{5}}$. Here, $\\epsilon_0$ is the\nvacuum permittivity, $q$ the charge, $mc^2$ the rest energy of the particle, $c$ the light speed in vacuum, $I$ the average beam current, $f_{rf}$ the RF\nbunch frequency, $\\beta = v\/c$, $v$ the bunch velocity, and the relativistic\nfactor $\\gamma = 1\/\\sqrt{1-\\beta^2}$.\n\nThe above equations can be linearized with respect to periodic solutions\n(i.e. matched solutions) as:\n\\begin{eqnarray}\n\tX(s) & = & X_0(s) + x(s)  \\\\\n\tY(s) & = & Y_0(s) + y(s)  \\\\\n\tZ(s) & = & Z_0(s) + z(s)  \n\\end{eqnarray}\nwhere $X_0$, $Y_0$ and $Z_0$ denote the periodic matched envelope solutions \nand $x$, $y$ and $z$ denote small perturbations\n\\begin{equation}\n\tx(s) \\ll X_0(s), \\ \\ \\ \\ y(s) \\ll Y_0(s), \\ \\ \\ \\ z(s) \\ll Z_0(s)\n\\end{equation}\nThe equations of motion for these small perturbations are given by:\n\\begin{eqnarray}\n\t\\frac{d^2 x}{d s^2} + a_1(s) x(s) + a_{12}(s) y(s) + \\gamma^2 a_{13}(s) z(s) = 0  \\\\\n\t\\frac{d^2 y}{d s^2} + a_{12}(s) x(s) + a_2(s) y(s)  +\\gamma^2 a_{23}(s) z(s) = 0   \\\\\n\t\\frac{d^2 z}{d s^2} + a_{13}(s) x(s) + a_{23}(s) y(s)  + a_3(s) z(s) = 0  \n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\ta_1(s) & = &  k_x^2 + 3 \\epsilon_x^2\/X_0^4 - I_{x}(X_0,Y_0,Z_0) + 3 X_0^2 F_{xx} \\\\\n\ta_{12}(s) & = & X_0 Y_0 F_{xy} \\\\\n\ta_{13}(s) & = & X_0 Z_0 F_{xz} \\\\\n\ta_2(s) & = &  k_y^2 + 3 \\epsilon_y^2\/Y_0^4 - I_{y}(X_0,Y_0,Z_0) + 3 Y_0^2 F_{yy} \\\\\n\ta_{23}(s) & = & Y_0 Z_0 F_{yz} \\\\\n\ta_3(s) & = &  k_z^2 + 3 (\\epsilon_z\/\\gamma^2)^2\/Z_0^4 - I_{z}(X_0,Y_0,Z_0) + 3 \\gamma^2 Z_0^2 F_{zz} \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\tF_{xx} & = & C\\int_0^{\\infty} (X_0^2+t)^{-5\/2}(Y_0^2+t)^{-1\/2}(Z_0^2 \\gamma^2+t)^{-1\/2} dt  \\\\\n\tF_{xy} & = & C\\int_0^{\\infty} (X_0^2+t)^{-3\/2}(Y_0^2+t)^{-3\/2}(Z_0^2 \\gamma^2+t)^{-1\/2} dt  \\\\\n\tF_{xz} & = & C\\int_0^{\\infty} (X_0^2+t)^{-3\/2}(Y_0^2+t)^{-1\/2}(Z_0^2 \\gamma^2+t)^{-3\/2} dt  \\\\\n\tF_{yy} & = & C\\int_0^{\\infty} (X_0^2+t)^{-1\/2}(Y_0^2+t)^{-5\/2}(Z_0^2 \\gamma^2+t)^{-1\/2} dt  \\\\\n\tF_{yz} & = & C\\int_0^{\\infty} (X_0^2+t)^{-1\/2}(Y_0^2+t)^{-3\/2}(Z_0^2 \\gamma^2+t)^{-3\/2} dt  \\\\\n\tF_{zz} & = & C\\int_0^{\\infty} (X_0^2+t)^{-1\/2}(Y_0^2+t)^{-1\/2}(Z_0^2 \\gamma^2+t)^{-5\/2} dt  \n\\end{eqnarray}\n\nWith $\\xi = (x,x',y,y',z,z')^T$, \nthe above equations can be rewritten in matrix notation as:\n\\begin{eqnarray}\n\t\\frac{d \\xi}{d s} & = & A_{6}(s) \\xi(s)\n\t\\label{3deq}\n\\end{eqnarray}\nwith the periodic matrix\n\\begin{eqnarray}\n\tA_{6}(s) & = & \\begin{pmatrix}\n\t\t0 & 1 & 0 & 0 & 0 & 0 \\\\\n\t\t-a_1(s) & 0 & -a_{12}(s) & 0 & -\\gamma^2 a_{13}(s) & 0 \\\\\n\t\t0 & 0 & 0 & 1 & 0 & 0 \\\\\n\t\t-a_{12}(s) & 0 & -a_2(s) & 0 & -\\gamma^2 a_{23}(s) & 0 \\\\\n\t\t0 & 0 & 0 & 0 & 0 & 1 \\\\\n\t\t-a_{13}(s) & 0 & -a_{23}(s) & 0 & -a_{3}(s) & 0 \\\\\n\t\t \\end{pmatrix}\n\\end{eqnarray}\nLet $\\xi(s) = M_{6}(s) \\xi(0)$, substituting this equation into Eq.~\\ref{3deq}\nresults in\n\\begin{eqnarray}\n\t\\frac{d M_6(s)}{ds} & = & A_6(s) M_6(s)  \n\\end{eqnarray}\nwhere $M_6(s)$ denotes the $6\\times 6$ transfer matrix solution of \n$\\xi(s)$ and $M_6(0)$ is a $6\\times 6$ unit matrix. \nThe above ordinary differential equation can be solved using the matched\nenvelope solutions and numerical integration.\nSimilar to the 2D envelope instability model, the stability of these envelope \nperturbations is determined by the eigenvalues of the transfer matrix $M_6(L)$\nthrough one lattice period.\nFor the envelope oscillation to be stable, \nall eigenvalues of the $M_6(L)$ have to stay on the unit circle.\nThe amplitude of the eigenvalue gives the envelope mode growth \n(or damping) rate through one lattice period, while the phase of \nthe eigenvalue yields the \nmode oscillation frequency. When the amplitude of any eigenvalue is\ngreater than one, the envelope oscillation becomes unstable.\n\n\n\\section{Envelope Instability in a periodic solenoid and RF channel}\n\n\\begin{figure}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{sol3d.png}\n\t\\caption{Schematic plot of a periodic solenoid and RF channel. }\n   \\label{sol3d}\n\\end{figure}\n\\begin{figure*}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{sol80rf3d1amp.png}  \n   \\includegraphics*[angle=0,width=210pt]{sol80rf3d2amp.png}  \n   \\includegraphics*[angle=0,width=210pt]{sol100rf3d1amp.png}\n   \\includegraphics*[angle=0,width=210pt]{sol100rf3d2amp.png}\n   \\includegraphics*[angle=0,width=210pt]{sol120rf3d1amp.png}\n   \\includegraphics*[angle=0,width=210pt]{sol120rf3d2amp.png}\n   \\includegraphics*[angle=0,width=210pt]{sol140rf3d1amp.png}\n   \\includegraphics*[angle=0,width=210pt]{sol140rf3d2amp.png}\n   \\caption{The 3D envelope mode amplitudes as a function of \n\tdepressed transverse phase advance with $20$, $40$,\n\t$60$, $80$, $100$, $120$, and $140$ degree zero current \n\tlongitudinal phase advances\n\tfor (a) $80$ degree, (b) $100$ degree, (c) $120$ degree,\n\tand (d) $140$ degree zero current transverse phase advances in\n\ta periodic solenoid-RF channel.\n\t}\n   \\label{sol3damp}\n\\end{figure*}\n\\begin{figure*}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{sol80rf80phs.png} \n   \\includegraphics*[angle=0,width=210pt]{sol100rf100phs.png} \n   \\includegraphics*[angle=0,width=210pt]{sol120rf120phs.png} \n   \\includegraphics*[angle=0,width=210pt]{sol140rf140phs.png} \n   \\caption{The 3D envelope mode phases as a function of \n\tdepressed transverse phase advance with (a) \n\t$80$ degree, (b) $100$ degree, (c) $120$ degree,\n\tand (d) $140$ degree zero current longitudinal\n\tand transverse phase advances in a periodic solenoid-RF channel.\n\t}\n   \\label{sol3dphase}\n\\end{figure*}\n\\begin{figure*}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{cst2d80amp.png}\n   \\includegraphics*[angle=0,width=210pt]{cst2d100amp.png}\n   \\includegraphics*[angle=0,width=210pt]{cst2d120amp.png}\n   \\includegraphics*[angle=0,width=210pt]{cst2d140amp.png}\n   \\caption{The 2D envelope mode amplitudes as a function of \n\tdepressed transverse phase advance\n\tfor (a) $80$ degree, (b) $100$ degree, (c) $120$ degree,\n\tand (d) $140$ degree zero current transverse phase advances \n\tin a periodic solenoid channel.  }\n   \\label{sol2damp}\n\\end{figure*}\n\\begin{figure*}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{cst2d80phs.png}\n   \\includegraphics*[angle=0,width=210pt]{cst2d100phs.png}\n   \\includegraphics*[angle=0,width=210pt]{cst2d120phs.png}\n   \\includegraphics*[angle=0,width=210pt]{cst2d140phs.png}\n   \\caption{The 2D envelope mode phases as a function of \n\tdepressed transverse phase advance\n\tfor (a) $80$ degree, (b) $100$ degree, (c) $120$ degree,\n\tand (d) $140$ degree zero current transverse phase advances \n\tin a periodic solenoid channel.  }\n   \\label{sol2dphase}\n\\end{figure*}\n\\begin{figure}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{sol80rf120phs.png} \n   \\includegraphics*[angle=0,width=210pt]{sol120rf80phs.png} \n\t  \\caption{The 3D envelope mode phases as a function of \nthe depressed transverse phase advance for zero current (a) \n\ttransverse $80$ degree and longitudinal $120$ degree, \n\t(b) transverse $120$ degree and longitudinal $80$ degree\n\tphase advance in a periodic solenoid channel. }\n   \\label{sol80120phase}\n\\end{figure}\n\nWe first studied the envelope instability in a transverse solenoid focusing\nand longitudinal RF focusing periodic channel.\nA schematic plot of this periodic channel is shown in Fig.~\\ref{sol3d}.\nEach period of the channel consists of a $0.2$ meter solenoid and a $0.1$\nmeter RF bunching cavity.\nThe total length of the period is $0.5$ meters.\nThe proton bunch has a kinetic energy of $150$ MeV and normalized rms emittances\nof $0.2$ um, $0.2$ um, and $0.2$ um in horizontal, vertical, and longitudinal\ndirections respectively.\nFigures~\\ref{sol3damp}-\\ref{sol3dphase} show the 3D envelope mode amplitudes and phases\nas a function of transverse depressed phase advance for different\nzero current transverse and longitudinal phase advances.\nAs a comparison, we also show in Figs.~\\ref{sol2damp}-\\ref{sol2dphase}\nthe 2D envelope mode amplitudes and phases as a function of depressed \ntransverse phase advance for the same zero current transverse phase advances. \nHere, the 2D periodic solenoid channel has the same length of period \nas the 3D channel. It is seen that in the 2D periodic solenoid channel, the envelope instability \noccurs when the zero current phase advance is over $90$ degrees. In the\n3D periodic solenoid-RF channel, the envelope instability occurs even with\nthe zero current transverse phase advance $80$ degrees but longitudinal \nphase advance beyond $90$ degrees as shown in Fig.~\\ref{sol3damp} (a2). \nThere is no \ninstability if both the transverse zero current phase advance and the longitudinal\nzero current phase advance are below $90$ degrees as seen \nin Fig.~\\ref{sol3damp} (a1).\nFor the 3D envelope modes, when the longitudinal zero current\nphase advance below $90$ degrees and the transverse zero\ncurrent phase above $90$ degrees as shown in Figs~\\ref{sol3damp} (b1, c1, and d1), \nthe instability stopband\nbecomes broader as the zero current longitudinal phase advance increases.\nThis is probably because the longitudinal synchrotron motion\nhelps bring particles with different depressed transverse tunes into the resonance.\nA faster synchrotron motion might result in more particles falling \ninto the resonance \nand hence a broader instability stopband. \nFor small longitudinal zero current phase advance (e.g. $20$\ndegrees), the 3D envelope mode show the stopband\nsimilar to that of the 2D envelope mode.\nWhen the longitudinal\nzero current phase advance is above $90$ degrees, as shown in \nFig.~\\ref{sol3damp} (b2, c2, and d2), the 3D\nenvelope instability shows more complicated structure and larger instability stopband width\nthan the 2D envelope instability. \n\nIn the 2D periodic transverse solenoid focusing channel, for a coasting beam\nwith equal horizontal and vertical emittances,\nit is seen in Fig.~\\ref{sol2dphase}, the envelope instabilities are due to\nthe $180$ degree half-integer parametric resonance. \nHowever, for a bunched beam, as shown in Fig.~\\ref{sol3dphase}, \nbesides the $180$ degree half-integer\nresonance, there are also confluent resonances where two\nenvelope modes have the same frequencies and resonate with each other.\nThe existence of both instability mechanisms results in more complicated\nstructure as shown in Figs.~\\ref{sol3damp} (b2, c2, and d2).\n\nThe 3D envelope instability shows asymmetry between the\ntransverse direction and the longitudinal direction in \nthe 3D periodic solenoid\nand RF channel. Figure~\\ref{sol80120phase} shows the envelope mode phases as a \nfunction of depressed transverse phase advance for a case with zero current\n$80$ degree transverse phase advance and $120$ degree longitudinal\nphase advance, and a case with zero current $120$ degree transverse phase advance\nand $80$ degree longitudinal phase advance. The envelope mode amplitudes \nfor both cases are shown in Fig.~\\ref{sol3damp} (a2 and c1).\nFor the $80$ degree zero current transverse\nphase advance, there is\nonly one major unstable stopband below $30$ degree depressed transverse \nphase advance\ndue to half-integer parametric resonance as shown in the left plot\nof Fig.~\\ref{sol80120phase}.\nFor the $120$ degree zero current transverse phase advance, there are three \nunstable regions, two due to the half-integer parameter resonance and\none due to the confluent resonance as shown in the right plot of \nFig.~\\ref{sol80120phase}.\nThis asymmetry is probably related to the two degrees of fredom in the\ntransverse plane while only one in the longitudinal direction.\n\n\\section{Envelope Instability in a periodic quadrupole-RF channel}\n\nNext, we studied the 3D envelope instability\nin a periodic transverse quadrupole focusing and longitudinal RF focusing\nchannel for the same bunched proton beam. \n\\begin{figure}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=300pt]{fd3d.png}\n\t\\caption{Schematic plot of a periodic quadrupole and RF channel. }\n   \\label{fd3d}\n\\end{figure}\n\\begin{figure*}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{res0adv80L3d1.png} \n   \\includegraphics*[angle=0,width=210pt]{res0adv80L3d2.png} \n   \\includegraphics*[angle=0,width=210pt]{res0adv100L3d1.png}\n   \\includegraphics*[angle=0,width=210pt]{res0adv100L3d2.png}\n   \\includegraphics*[angle=0,width=210pt]{res0adv120L3d1.png}\n   \\includegraphics*[angle=0,width=210pt]{res0adv120L3d2.png}\n   \\includegraphics*[angle=0,width=210pt]{res0adv140L3d1.png}\n   \\includegraphics*[angle=0,width=210pt]{res0adv140L3d2.png}\n\t   \\caption{The 3D envelope mode amplitudes as a function of \n\tdepressed transverse phase advance with $20$, $40$,\n\t$60$, $80$, $100$, $120$, and $140$ degree zero current \n\tlongitudinal phase advances\n\tfor (a) $80$ degree, (b) $100$ degree, (c) $120$ degree,\n\tand (d) $140$ degree zero current transverse phase advances in\n\ta periodic quadrupole-RF channel.\n\t}\n   \\label{fd3damp}\n\\end{figure*}\n\\begin{figure*}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{res0adv80L80phs.png} \n   \\includegraphics*[angle=0,width=210pt]{res0adv100L100phs.png} \n   \\includegraphics*[angle=0,width=210pt]{res0adv120L120phs.png} \n   \\includegraphics*[angle=0,width=210pt]{res0adv140L140phs.png} \n\t  \\caption{The 3D envelope mode phases as a function of \n\tdepressed transverse phase advance with (a) \n\t$80$ degree, (b) $100$ degree, (c) $120$ degree,\n\tand (d) $140$ degree zero current longitudinal\n\tand transverse phase advances in a periodic quadrupole-RF channel.\n\t}\n   \\label{fd3dphase}\n\\end{figure*}\n\\begin{figure*}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{fd2d80amp.png}\n   \\includegraphics*[angle=0,width=210pt]{fd2d100amp.png}\n   \\includegraphics*[angle=0,width=210pt]{fd2d120amp.png}\n   \\includegraphics*[angle=0,width=210pt]{fd2d140amp.png}\n\t  \\caption{The 2D envelope mode amplitudes as a function of \n\tdepressed transverse phase advance\n\tfor (a) $80$ degree, (b) $100$ degree, (c) $120$ degree,\n\tand (d) $140$ degree zero current transverse phase advances in \n\ta periodic quadrupole channel.  }\n   \\label{fd2damp}\n\\end{figure*}\n\\begin{figure*}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{fd2d80phs.png}\n   \\includegraphics*[angle=0,width=210pt]{fd2d100phs.png}\n   \\includegraphics*[angle=0,width=210pt]{fd2d120phs.png}\n   \\includegraphics*[angle=0,width=210pt]{fd2d140phs.png}\n\t  \\caption{The 2D envelope mode phases as a function of \n\tdepressed transverse phase advance\n\tfor (a) $80$ degree, (b) $100$ degree, (c) $120$ degree,\n\tand (d) $140$ degree zero current transverse phase advances in \n\ta periodic quadrupole channel.  }\n   \\label{fd2dphase}\n\\end{figure*}\n\\begin{figure}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{fd3d80L120phs.png} \n   \\includegraphics*[angle=0,width=210pt]{fd3d120L80phs.png} \n\t\t  \\caption{The 3D envelope mode phases as a function of \n\tdepressed transverse phase advance for zero current (a) \n\ttransverse $80$ degree and longitudinal $120$ degree, \n\t(b) transverse $120$ degree and longitudinal $80$ degree\n\tphase advance in a periodic quadrupole channel. }\n   \\label{quad80120}\n\\end{figure}\nA schematic plot of this periodic channel is shown in Fig.~\\ref{fd3d}.\nEach peroid of the channel consists of a $0.2$ meter focusing quadrupole,\n\ta $0.1$ meter RF focusing cavity, a $0.2$ meter defocusing\n\tquadrupole and another $0.1$\nmeter RF bunching cavity.\nThe total length of the period is $1.0$ meters.\nFigures~\\ref{fd3damp}-\\ref{fd3dphase} show the 3D envelope mode amplitudes and phases\nas a function of transverse depressed phase advance for different\nzero current transverse and longitudinal phase advances.\nAs a comparison, we also show in Figs~\\ref{fd2damp}-\\ref{fd2dphase}\nthe 2D envelope mode amplitudes and phases as a function of the depressed \nphase advance for different zero current phase advances. Here, the \n2D periodic quadrupole channel has the same length of period as the 3D channel.\nIt is seen that in the 2D periodic quadrupole channel, the envelope instability \noccurs when the zero current phase advance is over $90$ degrees. \nThere is no instability when the zero current phase advance is below $90$ \ndegrees. In the\n3D periodic quadrupole-RF channel, the envelope instability occurs even with\nthe zero current transverse phase advance $80$ degrees but the longitudinal \nphase advance beyond $100$ degrees in Fig.~\\ref{fd3damp} (a2). \nThere is no \ninstability if both the transverse zero current phase advance and the longitudinal\nzero current phase advance are below $90$ degrees.\n\tFor the 3D envelope modes, when the longitudinal zero current\nphase advance is below $90$ degrees and the transverse zero\ncurrent phase above $90$ degrees as shown in Fig.~\\ref{fd3damp} (b1, c1, and d1), \n\tthe instability stopband width\nincreases with the increase of the zero current longitudinal phase advance.\nFor small longitudinal zero current phase advance (e.g. $20$\ndegrees), the 3D envelope modes instability stopband\nis similar to that of the 2D envelope modes.\nFor the $100$ degree zero current transverse phase advance case, when\nthe zero current longitudinal phase advance is beyond $90$ degrees,\nthe stopband becomes more complicated and shows multiple stopbands.\nFor the transverse zero current $120$ and $140$ degree\nphase advances, the instability stopbands do not change\nsignificantly with the increase of zero current longitudinal phase\nadvance. \nThis is probably due to the fact that when the transverse\nzero current phase advance is beyond $100$ degrees, most parameter space \n(transverse depressed tune) below $90$ degrees becomes unstable\ncaused by the confluent resonance. Further increasing the\nzero current longitudinal phase advance beyond $90$ degrees \nwill not enlarge that stopband any more.\n\n\nIn the periodic transverse quadrupole focusing channel,\nit is seen in Fig.~\\ref{fd2dphase}, the 2D envelope instabilities are mainly due to\nthe confluent resonance between the two envelope modes when their phases \nbecome equal. This appears still to be valid in the 3D periodic\nquadrupole-RF channel as shown in Fig.~\\ref{fd3dphase}.\n\nThe 3D envelope instability shows asymmetry between the\ntransverse and the longitudinal direction in the 3D periodic quadrupole\nand RF channel too. Figure~\\ref{quad80120} shows the envelope mode phases as a \nfunction of depressed transverse phase advance for a case with zero current\n$80$ degree transverse phase advance and $120$ degree longitudinal\nphase advance, and a case with zero current $120$ degree transverse phase advance\nand $80$ degree longitudinal phase advance. The envelope mode amplitudes \nare shown in Fig.~\\ref{fd3damp} (a2 and c1) for this comparison. \nFor the $80$ degree zero current transverse\nphase advance, there is\nonly one major unstable region around $60$ degree depressed transverse \nphase advance\ndue to the confluent resonance.\nFor the $120$ degree zero current phase advance, there are two \nunstable regions due to two confluent resonances.\n\nIn the above periodic quadrupole and RF channel, we assumed that \nthe two RF cavities have the same longitudinal focusing strength.\nThe longitudinal focusing period is half of the transverse\nfocusing period. This accounts for the absence of the envelope\ninstability for the zero current $80$ degree transverse phase \nadvance and $100$ degree longitudinal phase in the periodic\nquadrupole and RF channel.\nThe envelope instability stopband\nis observed in the periodic solenoid and RF channel with the \nsame zero current phase advances as shown in Fig.~\\ref{sol3damp} (a2). \nThe absence of instability for longitudinal zero current phase advance\n$100$ degrees was also observed in 3D macroparticle simulations in\nreference~\\cite{ingo4}. Now, we break the symmetry of two RF longitudinal \nfocusing cavities, the longitudinal focusing period becomes the\nsame as the transverse focusing period.\nThe envelope instability occurs for these zero\ncurrent phase advances in a periodic quadrupole and RF channel.\nFigures~\\ref{quad80100} show the envelope mode amplitudes and phases as a \nfunction of transverse depressed phase advances with about\n$10\\%$, $20\\%$, and $30\\%$ deviation from the original setting of the two\nRF cavities (one cavity plus that percentage and the other one minus \nthat percentage).\n\\begin{figure}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{res0adv80L100Neqrfamp1.png} \n   \\includegraphics*[angle=0,width=210pt]{res0adv80L100Neqrfphs1.png}\n\t\\caption{The 3D envelope mode (left) amplitudes and (right) phases as a \nfunction of the transverse depressed phase advance \nwith $10\\%$, $20\\%$, and $30\\%$ deviations from the original RF cavity setting\n\tin a periodic quadrupole-RF channel.}\n   \\label{quad80100}\n\\end{figure}\nIt is seen that as the asymmetry between the two RF cavity increases,\nthe instability stopband width also increases. \nBefore breaking of the symmetry of two RF cavities, the longitudinal\nphase advance per longitudinal period is $50$ degrees. After the breaking of the symmetry,\nthe longitudinal period becomes the same as the lattice period\nand the phase advance becomes $100$ degrees. \nSuch a zero current phase advance results in half integer parametric resonance\nas shown in Fig.~\\ref{quad80100}. \n\nIn above 3D periodic solenoid\/quadrupole and RF transport channels, \nwe have assumed that in transverse plane, the zero current phase advances in\nhorizontal direction and the vertical direction are the same.\nFurthermore, the bunch has the same emittances in both horizontal and\nvertical directions. This might imply a two-dimensional \n\ttransverse and longitudinal periodic system (i.e. $r-z$).\nAs a comparison, we also calculated the\nenvelope mode amplitudes and phases for a true\ntwo-dimensional periodic quadrupole\nchannel with different zero\ncurrent phase advances in the horizontal and the vertical direction \n($120$ degrees\nin the horizontal direction and $80$ degrees in the vertical direction). \nFigure~\\ref{fd2d80120} shows the 2D envelope mode amplitudes and\nphases as a function of the depressed horizontal and vertical phase advance.\nComparing the 2D envelope mode amplitudes and phases in above plot with those \nof the 3D envelope mode with the same zero current phase\n\tadvances in Figs.~\\ref{sol3damp} (a2) and \\ref{fd3damp} (a2) \n($80$ degrees in transverse and $120$ in longitudinal)\n\tand Fig.~\\ref{sol3damp} (c1) and \\ref{fd3damp} (c1) \n($120$ degrees in transverse and $80$ in longitudinal),\nwe see that the 2D envelope instability shows somewhat similar structure to\nthe 3D envelope instability in a periodic solenoid-RF channel with\ntransverse zero current phase advance $80$ degrees and longitudinal phase advance $120$ degrees. The major instabilities in both cases are caused\nby the half-integer parametric resonance.\nThe 3D envelope modes in a periodic quadrupole-RF channel shows quite \ndifferent\ninstability stopband from the 2D envelope modes. Also the 3D\nenvelope instability in quadrupole channel is caused by the confluent\nresonance while the 2D asymmetric envelope instability in the quadrupole\nchannel is\nmainly caused by the half-integer parametric resonance.\n\\begin{figure}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{fd2d120-80amp.png} \n   \\includegraphics*[angle=0,width=210pt]{fd2d120-80amp2.png} \n   \\includegraphics*[angle=0,width=210pt]{fd2d120-80phs.png} \n   \\includegraphics*[angle=0,width=210pt]{fd2d120-80phs2.png} \n\t\\caption{The 2D envelope mode (top) amplitudes and (bottom) phases as a \nfunction of depressed phase advance with asymmetric zero current \n\tphase advances ($80$ degrees in one direction and $120$ degrees\n\tin another direction) in a periodic quadrupole channel.}\n   \\label{fd2d80120}\n\\end{figure}\n\nWe also explored 3D envelope instabilities with non-equal\ntransverse zero current phase advances in the horizontal\ndirection and the vertical direction.\nFigure~\\ref{quad3d11012080} shows the 3D envelope mode amplitudes and phases as a \nfunction of the depressed horizontal tune with zero current phase advance \n$120$ degrees\nin the horizontal direction, $110$ degrees in the vertical direction,\nand $80$ degrees in the longitudinal direction in the periodic quadrupole\nand RF channel.\n\\begin{figure}[!htb]\n   \\centering\n   \\includegraphics*[angle=0,width=210pt]{fd3d120-110L80amp.png} \n   \\includegraphics*[angle=0,width=210pt]{fd3d120-110L80phs.png} \n\t\\caption{The 3D envelope mode (left) amplitudes and (right) phases as a \nfunction of the horizontal depressed phase advances \nwith zero current phase advances $120$ degrees in horizontal, $110$ in vertical,\nand $80$ in longitudinal direction\n\tin a periodic quadrupole-RF channel.}\n   \\label{quad3d11012080}\n\\end{figure}\nComparing the above figure with the zero current $120$ degree transverse phase \n\tadvance and $80$ degree longitudinal phase advance case in Fig.~\\ref{fd3damp} (c1),\nwe see that 3D instability stopband from the nonequal transverse focusing \nbecomes broader. \nInstead of one major instability stopband and a minor stopband in\nthe equal transverse phase advance case, now there are four\nstopbands (two major stopbands and two minor stopbands) for the transverse\n$120$ and $110$ degree phase advances.\nBesides the confluent resonance,\nthere also appears a half-integer parametric resonance when the transverse\nsymmetry is broken. Breaking the transverse symmetry results in\nmore resonances of these envelope modes.\n\tThis suggests that keeping the same zero\ncurrent phase advance in both the horizontal and the vertical directions\nmight help reduce the parameter region of the envelope instability.\n\n\\section{Conclusions}\nIn this paper, we proposed a three-dimensional envelope instability\nmodel to study the instability for a bunched beam\nin a periodic solenoid and RF focusing channel and a periodic quadrupole and RF focusing channel.\nThis study showed that when the transverse zero current phase advance \nis below $90$ degrees, the beam envelope can still become unstable if\nthe longitudinal zero current phase advance is beyond $90$ degrees.\nFor the transverse zero current phase advance beyond $90$ degrees, \nthe instability stopband becomes broader with the increase of \nlongitudinal focusing strength and even shows different\nstructure from the 2D case \nwhen the longitudinal zero current phase advance is\nbeyond $90$ degrees. \n\nThe 3D envelope instability shows asymmetry between the longitudinal\nfocusing and the transverse focusing.\nThe instability shows broader stopband when the transverse zero current\nphase advance is beyond $90$ degrees than that when the longitudinal\nzero current phase advance is beyond $90$ degrees.\nIn the 3D periodic quadrupole and RF channel, for the transverse \nzero current phase advance $80$ degree, the envelope modes stay stable \nfor the longitudinal $100$ degree zero current phase advance due to\nthe symmetry of two longitudinal focusing RF cavities. Breaking the symmetry\nof two cavities results in the envelope instability with a finite stopband.\nBreaking the horizontal and vertical focusing symmetry in the transverse \nplane also increases the \nenvelope instability stopband width. This suggests that a\nmore symmetric accelerator lattice design might help reduce the parameter space of the\nenvelope instability.\n\n\\section*{ACKNOWLEDGEMENTS}\nWork supported by the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.\nWe would like to thank Dr. R. D. Ryne for the use of his 2D and 3D envelope codes.  This research used computer resources at the National Energy Research\nScientific Computing Center.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nClassical and quantum systems show new phases with unexpected properties when driven by an external time-dependent periodic field. \nIn many cases, these phases do not have any counterpart in static systems. \nFor example, periodic driving in a one dimensional\nsystems generates chaotic motion of particles \\cite{chaos1,chaos2,chaos3,chaos4}.\nIn molecular systems, temporal changes of microscopic parameters or bias voltage\nmay lead to various exotic non-equilibrium states \\cite{Fainshtein}. The ratio between kinetic and potential energy as well as lattice spacing can be\nvaried in optical lattices to create non-trivial states \\cite{coldatom1, \ncoldatom2}. \nRecently, periodically driven systems have\ngiven rise new phases \\cite{ drive1,drive2,drive3,drive4,drive5,drive6,drive7,drive8,\ndrive9,drive10}, such\nas, Floquet block-states in topological insulator \\cite{ftimat1,ftimat2},\noptical lattices \\cite{ftiopt}, and cold-atom \\cite{fticold} systems.\nTopological insulators have metallic edge states due to the spin-orbit coupling\nand non-trivial topology of the band structure, while sustaining an insulating gap in the bulk. Material such as\nBi\\textsubscript{2}Se\\textsubscript{3} and Bi\\textsubscript{2}Si\\textsubscript{3} shows topologically protected Dirac cone, resulting from the properties of these edge states \\cite{ti1,ti2,ti3}. Moreover, due to spin-momentum locking, back-scattering off potential impurities is prevented. In other words, the definite chirality of the edge states does not allow transitions between the states $|{\\bf k}\\uparrow\\rangle$ and $|-{\\bf k}\\downarrow\\rangle$ assisted by scattering off time-reversal symmetric (non-magnetic) impurities.\n\nThe robustness of these edge mode has been proven theoretically by examining the \nbackscattering due to potential impurity in continuum and lattice model \n\\cite{edge1, edge2, edge3,edge4,edge5,edge6}. The protection of the Dirac \nsurface states against disorder is also seen in various experiments \n\\cite{edge7,edge8,edge9}. Magnetic impurities, on the other hand, do not keep \nthese edge modes intact which results in imperfect quantization of the \nconductance \\cite{mag-im1, mag-im2, mag-im3}. \n \nEven though there are many theoretical and experimental \nstudies of the effects of static  impurities on topological insulator, the effects of dynamical impurities have not been considered previously. \nDynamical local interaction can give rise to various non-linear phenomena such \nas multi-photon dissociation or excitation of\natom or molecule when exposed under strong laser field \\cite{lesser1, \nlesser2,lesser3}.\nIn this article, we consider an atomically sharp time-dependent impurity potential, simulating the conditions of a local, yet extremely focused, monochromatic laser field, affecting a single lattice site. We show the emergence of impurity resonances, in accordance with previously discussed for impurity resonances for stationary conditions \\cite{edge4,edge5,edge6,PhysRevB.94.075401}, which give rise to strong modifications of the local density of electron states near the impurity site, with an increasing number of resonance features with increasing amplitude and frequency of the driving field. While the metallic edge states tend to develop a sharp density peak around the Fermi level for impurities interacting with the edge states, the density gap of the bulk states become increasingly filled up with increasing scattering potential strength, for impurities directly perturbing the bulk sites.\n\n\n\\section{Model and Floquet-Green Function}\nIn this section, we describe the general formalism of the Floquet Green function \nmethod. The Hamiltonian of the full quantum system is a periodic function in \ntime, $H(t) = H(t+\\tau)$, where $\\tau  = 2\\pi\/\\Omega$ is the period of the \nexternal driving field. As a direct consequence of the periodicity, we can use \nthe Floquet theorem \\cite{Floquet1,Floquet2,Floquet3} which can be regarded as \na time domain equivalent to the Bloch theorem.\nDue to the explicit time-dependence of the Hamiltonian, electrons can be excited to different energy states. However, if the time-dependent potential is characterized by a single frequency ($\\Omega$), the energy difference between the final and initial state should be an integer multiple of $\\Omega$. This restriction gives rise to an energy space representation of the Hamiltonian, the corresponding Green function, as well as of operators.\n\nWe define the Floquet Hamiltonian, a Hermitian operator, for the generic time-dependent Hamiltonian\n$H(x,t)$\n\\begin{align}\n H^F(x,t) = H(x,t) - i\\hbar\\frac{\\partial}{\\partial t}\n \t,\n\\end{align}\nwhich gives the Floquet green function written as\n\\begin{align}\n\\Bigl(\n\t\\epsilon - H^F(x,t)\n\t\\Bigr)\n\tG^F(x,x';t,t')=\\delta(x-x')\\delta(t-t')\n\t.\n\\end{align}\nThe time periodicity of the Hamiltonian is inherited in the Floquet Green\nfunction, which becomes periodic in both $t$ and $t'$. Hence, we can Fourier expand both the Green function and Hamiltonian into\n\\begin{subequations}\n\\begin{align}\nH^F(x,t)=&\n\t\\sum_\\gamma\n\t\tH^F_{\\gamma}(x)e^{i\\epsilon_\\gamma t}\n\t,\n\\\\\nG^F(x,x';t,t')=&\n\t\\sum_{\\alpha,\\beta}\n\t\tG_{\\alpha\\beta}^F(x,x')e^{i\\epsilon_\\alpha t-i\\epsilon_\\beta t'}\n\t.\n\\end{align}\n\\end{subequations}\nHere, the quasi-energy $\\epsilon_\\alpha$ is a conserved quantity. In this way, a time-periodic driven system is reduced to an algebraic matrix equation. \nAlthough the Fourier expansion reduces the complexity of the problem, the dimensions of the corresponding Hilbert space is infinite. Therefore, there is no exact analytical closed form of the Green function. In fact, even the exact solution of a two-level system driven with linearly polarized light is not known \\cite{shirley}.\n\nDue to this intrinsic complexity, we approach the Floquet Green function numerically, by considering a harmonic monochromatic time-dependent driving field. The Hamiltonian, then, assumes the form\n\\begin{align}\n \\mathbf{H} = \\mathbf{H}^0 + 2\\mathbf{V}\\cos\\Omega t\n \t,\n\\end{align}\nwhere $\\mathbf{H}^0$ is the time-independent part of the Hamiltonian whereas $\\mathbf{V}$ represents the coupling to the time-dependent driving field. The corresponding Floquet Hamiltonian for the assumed driving field has block tri-diagonal structure, according to\n\\begin{align}\nH^F_{\\alpha\\beta}=\n\t(\\mathbf{H}^0 - \\alpha\\hbar\\Omega)\\delta_{\\alpha\\beta}\n\t+\n\t\\mathbf{V}(\\delta_{\\alpha+1\\beta} + \\delta_{\\alpha-1\\beta})\n\\end{align}\nThe Floquet green function becomes\n\\begin{align}\n(\\mathbf{1}E_\\alpha - \\mathbf{H}^0)G_{\\alpha\\beta}\n\t-\n\t\\mathbf{V}(G_{\\alpha+1\\beta}\n\t+\n\tG_{\\alpha-1\\beta})\n\t=\n\t\\mathbf{1}\\delta_{\\alpha\\beta}\n\t,\n\\end{align}\nwhere $E_\\alpha = \\epsilon +\\alpha\\hbar\\Omega$. This matrix equation can be \nsolved iteratively by using the \\emph{matrix continued fraction} method \n\\cite{martinez1,martinez2}. As a result, we obtain a recursive matrix equation \nfor the Green function\n\\begin{subequations}\n\\begin{align}\n\\Big[\\mathbf{1}E_\\alpha\n\t-\n\t\\mathbf{H}^0\n\t-\n\t\\mathbf{V}_\\text{eff}(E_{\\alpha})\\Big]G_{\\alpha,\\alpha}\n\t=&\n\t\\mathbf{1}\n\t,\n\\\\\n\\mathbf{V}_\\text{eff}(E_{\\alpha})=&\n\t \\mathbf{V}^+_\\text{eff}(E_{\\alpha})\n\t +\n\t \\mathbf{V}^-_\\text{eff}(E_{\\alpha})\n\t .\n\\end{align}\n\\end{subequations}\nHere, the effective potential is given by\n\\begin{align}\n\\mathbf{V}^\\pm_\\text{eff}(E_{\\alpha})=&\n\t\\mathbf{V}\\frac{1}{\\mathbf{C}_{\\alpha\\pm1}\n\t-\n\t\\mathbf{V}\\frac{1}{\\mathbf{C}_{\\alpha\\pm2}\n\t-\n\t\\mathbf{V}\\frac{1}{\\vdots}\\mathbf{V}}\\mathbf{V}}\\mathbf{V}\n\t,\n\\end{align}\nwhere $\\mathbf{C}_{\\alpha} = \\mathbf{1}E_{\\alpha} - \\mathbf{H}^0$.\nAssuming that the impurity sit at the origin $(\\mathbf{r} = 0)$, the effective potential can be written\n\\begin{align}\n\\mathbf{V}^\\pm_\\text{eff}(E_{\\alpha})=&\n\t|0\\rangle\\langle 0|\n\t\t\\frac{1}{\\mathbf{C}_{\\alpha\\pm1}-|0\\rangle\\langle 0|\\frac{1}{\\mathbf{C}_{\\alpha\\pm2}-\\frac{1}{\\vdots}}|0\\rangle\\langle 0|}\n\t|0\\rangle\\langle 0|\n\\nonumber\\\\=&\n\t|0\\rangle\\langle 0|\n\t\t\\frac{1}{\\mathbf{C}_{\\alpha\\pm1}-|0\\rangle\\langle 0|\\mathbf{V}^\\pm_\\text{eff}(E_{\\alpha\\pm1})|0\\rangle\\langle 0|}\n\t|0\\rangle\\langle 0|\n\\nonumber\\\\=&\n\tV^\\pm_\\text{eff}(E_{\\alpha})|0\\rangle\\langle 0|\n\t.\n\\end{align}\nIn general, we compute these effective potentials numerically by setting a maximum frequency $E_M$ above which the effective potentials are zero, that is, $\\mathbf{V}^\\pm_\\text{eff}(E_{m}) = 0 $, for all $E_m>E_M$. Typically we take $m$ to be on the order of 100, which normally is sufficient for convergence.\n\n \n\\section{Topological insulator surface}\nFirst, we will investigate the effect of time-dependent impurity on the edge of three-dimensional topological insulators. The low energy effective model within $k\\cdot p$ approximation  can be described  by\n \\begin{align}\n  H= v\\sum_k\n\\Psi_\\mathbf{k}^\\dag[\\mathbf{k}\\times\\vec{e}_3]\\cdot\\sigma\\Psi_\\mathbf{k}\n \\end{align}\nWe are interested in the local properties of this system such as density of states $N({\\bf r},\\omega)$, which is related to the (retarded) Green function through the identity $N({\\bf r},\\omega)=-\\tr\\im{\\bf G}^r({\\bf r},{\\bf r};\\omega)\/\\pi$. As a function of the effective potential, we write the Green function in momentum space as\n\\begin{align}\nG_{k,k'}=&\n\t\\delta_{kk'}G_{k}^0\n\t+\n\tG^0_k\n\t\\frac{V_\\text{eff} }{1-V_\\text{eff} G^0_0}\n\tG^0_{k'}\n\t,\n\\end{align}\nwhere $G$, $G^0$, and $V$ are implicit functions of the energy.\n \nWe consider the driving potential $ \\mathbf{V}=2A\\delta(\\mathbf{r}-\\mathbf{r}_0)\\sigma^0\\cos\\Omega t$ with frequency $\\Omega $, amplitude $A$, at the position $\\mathbf{r}_0$. In Fig. \\ref{fig:cont1}, we plot the local density of electron states for increasing impurity amplitude $A$, panels (a) through (d). In absence of the impurity, Fig. \\ref{fig:cont1} (a), we retain the typical Dirac-like density of states with vanishing density of state at the Fermi energy, as expected. However, the linear density of states around the Fermi energy is preserved also at finite driving amplitudes, Fig. \\ref{fig:cont1} (b) -- (d). In addition to the linear low energy spectrum, high energy features emerge with increasing $A$. These features, which appear symmetrically around the Dirac point, are direct consequences of the excitations that are generated by the time-dependent potential, much in analogy with the impurity side resonances that are caused by vibrational defects on Dirac materials \\cite{PhysRevLett.110.026802,PhysRevB.87.245404}.\nIn fact, the symmetric appearance of the excitations is caused by the combination of positive and negative scattering potentials $\\bfV^+_\\text{efft}$ and $\\bfV^-_\\text{efft}$, respectively, each of which is responsible for the equally distributed set of excitations on either the valence \\emph{or} the conduction side of the electronic band structure.\nMoreover, the number of excitations increases substantially with increasing $A$, which we understand to be an effect the increased order of $\\mathbf{C}_\\alpha$ that contributes to the effective potential $\\mathbf{V}_\\text{eff}$. Therefore, the Green function picks up an increasing number of higher energy modes. We also notice that the bandwidth of the local density of states increases the stronger the driving force. We refer this to the increasing number of contributing excitations that become available for in the scattering processes and which necessitates a redistribution of the total density onto an increased set of excitations.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{out.pdf}\n \\caption{Local density of electron states at the impurity site with driving frequency $\\Omega=0.50$  for different values of the driving amplitude $A =$ 0.0 (a), 0.5 (b), 1.0 (c), 2.0 (d). }\n\\label{fig:cont1}\n\\end{figure}\n\n\n\n\\section{Kane-Mele model}\n\nIn the second part of this article, we  will consider a lattice, Kane-Mele, model for a topological insulator. The model pertains to a tight-binding Hamiltonian  on a honeycomb lattice which is a straight forward generalization of the Haldane model \\cite{haldane}. Here, we write\n\\begin{align}\nH=&\n\t-t\\sum_{\\langle ij\\rangle\\sigma}c^\\dag_{i\\sigma}c_{j\\sigma}\n\t+\n\ti\\lambda_{so}\\sum_{\\langle\\langle ij\\rangle\\rangle\\sigma}c^\\dag_{i\\sigma}\\sigma^zc_{j\\sigma}\n\t+\n\t\\mu \\sum_{i\\sigma}c^\\dag_{i\\sigma}c_{i\\sigma}\n\t,\n\\end{align}\nwhere $t$ is the nearest-neighbor (NN) hopping integral, $\\lambda $ is the spin-orbit coupling which act as a next-nearest-neighbor hopping element, whereas $\\mu $ is the chemical potential which fixes the number of particle of the system. Here, we consider a half-filled system for which $\\mu=0$. Note that spin-orbit coupling act as a purely imaginary hopping integral and differ by a sign for up and down spin component. We set $t$ as our absolute energy scale, and we have imposed open boundary conditions in the $x$-direction and periodic boundary conditions along the $y$-direction. We calculate $V_\\text{eff}$ iteratively for a lattice size $100\\times50$. We consider the same driving potential as above, $\\mathbf{V}=\\delta(\\mathbf{r}-\\mathbf{r}_0)2A\\sigma^0\\cos\\Omega t$, with the frequency $\\Omega$,  amplitude $A$ at $\\mathbf{r}_0$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{rdos.pdf}\n\\caption{(a) Total density of state of the Kane-Mele model for the  topological insulator with a single impurity at the middle of the lattice. The vertical lines indicate the energy at which local density of state is plotted in the bottom panel. Real space map of the local density of state for different  values of energy  E=0.0 (b), 0.5 (c), 1.0 (d).}\n\\label{fig:kane1}\n\\end{figure}\n\nThe plots in Fig. \\ref{fig:kane1} display the total density of states of the\nKane-Mele model with a single impurity located at the center of the lattice. The\nred vertical line signifies the energy values at which the spacial maps of the local density of state are plotted in panels (b) through (d). The density of states for this tight binding model on a honeycomb lattice vanishes at the Fermi energy and increases linearly away from the Fermi energy. A finite spin-orbit coupling opens up a gap in the spectrum. Due to the non-trivial topological nature of this system, this gap only appears in the bulk while metallic edge states appear at the boundary. In Fig. \\ref{fig:kane1} (b), (c), this can be seen as an enhanced density of states at the edges $x=0$ and $x=100$. Moreover, one should notice the finite density of states around the impurity, a density which oscillates and decays far away from it. At higher energies, however, bulk states appear which displays a uniform density of states throughout the whole system, see Fig. \\ref{fig:kane1} (d). \n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{tmat.pdf}\n\\caption{$T$-matrix of an impurity at the middle of the lattice with driving\n\tamplitude $A = 5.00$. The bulk gap in the spectrum induced by finite spin-orbit coupling reflects in the $T$-matrix. When the driving frequency is small the gap remains intact and no states appear within the gap. With increasing driving frequency states appear in the gap.}\n\\label{fig:tmat1}\n\\end{figure}\n\nThe properties of the spectrum can be better understood by considering the imaginary part of the $T$-matrix, defined by\n\\begin{align}\n{\\cal T}=&\n\t\\frac{V_\\text{eff} }{1-V_\\text{eff} G^0_0}\n\t,\n\\end{align}\nas a function of energy. In Fig. \\ref{fig:tmat1} we plot $-\\im\\, {\\cal T}$ as  a function of the energy for six different frequencies $\\Omega$ of the driving field. The $T$-matrix contains scattering effects of all orders, and the plots in Fig. \\ref{fig:tmat1} show $-\\im\\, {\\cal T}$ for an impurity located at the center of the lattice. Since the bulk of a topological insulator does not have any states available near the Fermi energy, the imaginary part of the $T$-matrix remains gapped for small driving frequencies. Hence, the spectral content of the $T$-matrix only marginally deviates from the expected spectral properties of the pristine lattice. However, huge resonances appear symmetrically outside the bulk gap, providing coherent resonance peaks. These resonances can be thought of as in similar terms as the impurity resonances induced in Dirac materials by particle scattering off  local defects \\cite{AdvPhys.63.1,Nature.403.746,PhysRevLett.104.096804,PhysRevB.81.233405,PhysRevB.85.121103,edge6,PhysRevB.94.075401}.\n\nBy contrasts, it can be noticed from the plots in Fig. \\ref{fig:tmat1} that, with increasing frequency of the driving field, a finite number of states emerge within the bulk gap which tends to become filled up by these states. Simultaneously, the coherent peaks vanish with increased driving frequency as a result of the necessary charge redistribution which follows from the emergence of additional resonances in the spectrum.\n\nNext, we consider the impurity to be located at the edge of the lattice, that\nis, $(x,y)=(0,25)$, resulting in the plots $-\\im\\, {\\cal T}$ shown in Fig.\n\\ref{fig:tmat2} for six different frequencies of the driving field. In stark contrast to the situation discussed, the coherence peaks outside the bulk gap become are turned into dips. The metallic nature of the edge states gives rise to finite scattering states even for small driving frequencies. As the driving frequency is increased, an increased number of Floquet modes contribute to the scattering process, which leads to a non-monotonic oscillation pattern in the $T$-matrix.  \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{tedge.pdf}\n\\caption{$T$-matrix of an impurity at the edge of the lattice with driving amplitude  $A = 0.50$.  Topological Insulators have edge state at the boundary of the sample and a finite no of scattering states appear within the bulk gap. $T$-matrix also, show similar behavior with a finite weight within the gap even for small frequency region.}\n\\label{fig:tmat2}\n\\end{figure}\n\n\n\\section{Local density of states}\n\nIn this section, we consider the effect of impurity on the local density of electron \nstates, something which might be useful for local probing experiments, such as scanning \ntunneling microscopy. First, we focus on the case where the impurity is located\nat the  \ncenter of the lattice. In Fig. \\ref{fig:ldos1}, we plot the local density of electron states at the impurity site for a sequence of \ndifferent values of the amplitude $A$ of the driving field for a fixed frequency $\\Omega$.  \nIn the absence of the impurity, $A=0$, the local density of states at the center (in bulk, far from the edge) of the lattice \nhas a gap (not shown), which is caused by the finite spin-orbit coupling.\nFor a weak driving amplitude, the local density of electron states shows \na hard gap, which is reminiscent of the bulk gap in the unperturbed case.\nSurprisingly, however, even by increasing the driving amplitude by more than an order \nof magnitude, the local density of electron states does not change appreciably within the \ngap. This difference is in stark contrast when compared to the T-matrix results, where a finite weight \nappears when the impurity is located at the center of the lattice. \nThe local density of electron states increases away from the Fermi energy, which can be seen by zooming in inside the gap, while the local density of electron states remains constant at the Fermi energy.  This implies that even though \nthere is a finite number of states appearing within the bulk gap due to impurity scattering, \nthe density of states at the Fermi level does not change. The coherence peaks emerging at the edges of the gap, become less prominent with increasing amplitude of the driving field.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{ldos1.pdf}\n\\caption{Local density of electron states for a single impurity with driving\n\tfrequency $\\Omega=0.50$ at the impurity site which is located in the\n\tmiddle of the lattice for small energy. The inset figure shows the local density of electron states in a large energy range.}\n\\label{fig:ldos1}\n\\end{figure}\n\nIn Fig. \\ref{fig:ldos2}, we plot the local density of electron states for the case where the impurity is located at \nthe edge of the lattice. It is clear that this set-up leads to completely different kinds of features in \nthe local density of electron states, as compared to the previous one.\nThe non-trivial nature of the metallic TI metallic \nedge states, due to the chirality, becomes more significant at the boundary of\nthe lattice. As can be seen in Fig. \\ref{fig:ldos2}, the local density of electron states is not gapped in the absence of impurities, $A=0$. This property is preserved for weak driving fields, $0<A\\lesssim2$, although it is also clear that both the density of states at the Fermi level as well as the sharp resonances ($|\\omega|\\approx1$) become slightly weaker, there is, nonetheless, a finite local density of electron states inside the bulk gap. With even larger amplitudes, the overall low energy local density of electron states becomes simultaneously broadened and suppressed in amplitude.\n\n\\section{Summary and conclusions} \nIn this paper, we have studied the effects of a time-dependent potential \nimpurity in two-dimensional topological insulators. By employing a numerically \nexact  matrix continued fraction method, which takes into account higher order \nFloquet modes, we capture the low energy physics throughout a wide range between \nweak and strong driving fields. This approach is implemented for both continuum \nand lattice models of the topological insulator.\n\nRegarding the continuum model, we find that the low lying energy states remain unaffected by driving fields with weak to moderate strength, whereas the higher lying energy states become strongly modified. The linear property of the local density of electron states disappears with increasing amplitude of the driving field. Many more oscillations emerge in the local density of electron states due to contributions from the higher Floquet modes at larger driving amplitude, as well as a slight increase in the bandwidth.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{ldos2.pdf}\n\\caption{Local density of electrons states for a single impurity with driving frequency $\\Omega = 2.00$ at the impurity site which is located at the edge of the lattice for small energy. Inset figure show local density of electron states in a large energy range.}\n\\label{fig:ldos2}\n\\end{figure}\n\nWe have imposed open boundary condition for the two-dimensional lattice model of\nthe topological insulator along with $x$-direction, which gives metallic edge\nstates. Here, we have analyzed the effect of an impurity at two different\nlocations, namely, at the center and an edge of the lattice. The bare local\ndensity of electron states has insulating and metallic properties at these two lattice sites, respectively. The imaginary part of the $T$-matrix has very different responses for impurities located at these two positions. For a weakly perturbing impurity located at the center of the lattice, the bulk gap of the lattice is preserved in the $T$-matrix, while a finite density grows up within the gap at stronger driving fields caused by increased impurity scattering. For strong enough fields, this density eventually fills up the gap.\nOn the other hand for an impurity located at the edge of the lattice, the $T$-matrix and, hence, the impurity scattering, largely impacts the electronic density around the Fermi energy already for weak driving fields.\n\nWe conclude that despite the apparent simplicity of the model and some\nsimilarities with the physics originating from single impurity interacting with\nthe surfaces states of a three dimensional topological insulator, there is\nincreased complexity in the present set-up which can be directly linked to the\ntime-dependent potential. This should have bearing on issues related to vibrational defects as well as the external driving field we have in mind here. However, since the driving frequencies in the two situations should be expected to differ by orders of magnitude, we can, nonetheless, conjecture that our low frequency results should be applicable to vibrational impurities. We anticipate that studies of the time-dependent using different methods should corroborate our findings.\n\n\\acknowledgments\nWe thank Stiftelsen Olle Engqvist Byggm\\\"astare and Vetenskapsr\\aa det for financial support.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOptimal stopping problems (OSP) with cost depending on the mean of the stopped process arise for instance when the goal is to minimize the variance. In the work by Pedersen and Peskir \\cites{pedersen11, pedersenpeskir16}), the problems of optimal variance stopping and optimal mean-variance stopping  have been investigated in the case where the underlying process is e.g. a Geometric Brownian motion, highlighting connections with portfolio choice. Despite the fact that the OSP of the variance is {\\it time-inconsistent} i.e. for which the value-process is not a supermartingale, they succeeded to solve the problem i.e. derive a variational inequality for the value function with an explicit stopping region.  \nMore recently, the interest in optimal stopping problems with a more general mean-field type interaction such as the dependence on the law of the stopped process increased significantly, due mainly to the connection with the theory of mean-field games and mean-field optimal control (see e.g. \\cites{carmonadelarue1} for a systematic presentation of the topic). The first contribution in this direction is the work by Bertucci \\cite{bertucci} where an optimal stopping problem for a mean-field game is studied using mainly PDE techniques. Then, other results and extensions have been obtained with different techniques in \\cites{carmona2017mean, nutz2018mean, bouveretdumitrescutankov20, dumitresculeutschertankov21} and in the recent papers \\cites{talbitouzizhang21, dumitresculeutschertankov22, agram22osp}. \n\nA powerful tool to study optimal stopping problems is based on the Snell envelope of processes (see \\cites{bismut1977, elkaroui79} and Appendix D in \\cite{karatzasshreve98}). Given the general OSP \n\\begin{equation*}\n\tY_0=\\underset{\\tau\\in \\mathcal{T}_0}{\\sup}\\, \\mathbb{E}\\left[L_\\tau\\right]\n\\end{equation*}\nwhere the reward (a.k.a. the barrier or obstacle) process $(L_t)_{t\\in[0,T]}$ is right continuous with left limits (c{\\`a}dl{\\`a}g) and adapted to a filtration $\\mathbb{F}=\\{\\mathcal{F}_t\\}_{t\\in[0,T]}$ which satisfies the usual conditions, the Snell envelope of the process $L$ is defined by\n\\begin{equation*}\n\tY_t=\\underset{\\tau\\in \\mathcal{T}_t}{\\esssup}\\, \\mathbb{E}\\left[L_\\tau\\,|\\,\\mathcal{F}_t\\right],\n\\end{equation*}\nwhere $\\mathcal{T}_t$ is set of $\\mathbb{F}$-stopping times with values in $[t,T]$. Under mild uniform integrability conditions on $L$, it turns out that $Y$ is the smallest supermartingale that dominates $L$, an important property which implies that, when $L$ has only nonnegative jumps, then $Y$ is continuous and it is optimal to stop when $Y$ hits $L$ i.e. the hitting time\n\\begin{equation}\\label{tau-intro}\n\t\\tau^*_t = \\inf\\{s\\ge t, \\,\\, Y_s=L_s\\}\\wedge T\n\\end{equation}\nis optimal after t. In particular, $\\tau^*:=\\tau^*_0$\nis optimal for $Y_0$.\n\nIn the recent papers \\cites{djehicheeliehamadene, djehichedumitrescuzeng}, this result could be successfully applied to a large class of mean-field OSPs  whose value-process solves a mean-field reflected BSDEs i.e. it is a supermartingale. That class of OSPs includes the following recursive OSP (we ignore the integral term)\n\\begin{equation*}\n\tY_0=\\underset{\\tau\\in \\mathcal{T}_0}{\\sup}\\, \\mathbb{E}\\left[h(Y_{\\tau},\\mathbb{P}_{Y_{s}}\\lvert_{s=\\tau})1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right],\n\\end{equation*}\nwhere the mean-field coupling is in terms of the marginal law of $Y$ {\\it evaluated} at a stopping time i.e. it is not the law of the random variable $Y_{\\tau}$. \nAnother example of a mean-field OSP for which the supermartingale property is preserved is considered in the recent work by Talbi, Touzi and Zhang \\cite{ talbitouzizhang21} where the mean-field OSP for a mean-field diffusion is studied in a weak (or relaxed) formulation i.e. in terms of the joint marginal law of the stopped underlying process $X$ and the survival process $I$ associated with the stopping time. Moreover, the performance function is a deterministic function of the marginal laws of $(X,I)$. They characterized the value function by a dynamic programming equation on the Wasserstein space.\n\nBut, the above result does not immediately apply if e.g. the stopped obstacle is of the form $L_{\\tau}=(X_{\\tau}-\\mathbb{E}[X_{\\tau}])^2$ as in the OSP of the variance of a process $X$ or more generally when $L_{\\tau}=h(X_{\\tau},\\mathbb{P}_{X_{\\tau}})$, for which the value-process is no longer a supermartingale.  Nevertheless, Pedersen and Peskir \\cites{pedersen11, pedersenpeskir16}) could solve the OSP of the variance of an underlying Markov diffusion process  by embedding it into an auxiliary standard OSP whose value function solves a standard variational inequality. \n\n\\medskip\nIn present paper, we consider the following class of time-inconsistent mean-field OSPs beyond the mean-variance case.   \nLet $T>0$ be a finite time horizon, $W$ a Brownian motion defined on a probability space $(\\Omega, \\mathcal{F},\\mathbb{P})$ and $\\mathbb{F}=\\{\\mathcal{F}_t\\}_{t\\in[0,T]}$ the complete Brownian filtration. On the filtered probability space $(\\Omega, \\mathcal{F},\\mathbb{F},\\mathbb{P})$ let $X$ be a diffusion process of mean-field type:\n\\begin{equation*}\nX_t=X_0+\\int_0^tb(s,X_s, \\mathbb{P}_{X_{s}})ds+\\int_0^t \\sigma(s,X_s,\\mathbb{P}_{X_{s}})dW_s.\n\\end{equation*}\nFor a certain $\\mathcal{F}_T$-measurable final condition $\\xi$ and a performance function $h$, we consider the following OSPs:\n\\begin{itemize}\n    \\item[(OSPa)] Optimal stopping of a mean-field diffusion:\n    \\begin{equation}\\label{Y-0-d-intro}\n\tY_0=\\underset{\\tau\\in \\mathcal{T}_0}{\\sup}\\,\\mathbb{E}\\left[h(X_{\\tau},\\mathbb{P}_{X_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right],\n\\end{equation}\n    \n    \\item[(OSPb)] Optimal stopping of a  recursive utility function defined on $(\\Omega, \\mathcal{F},\\mathbb{F},\\mathbb{P})$:\n\\begin{equation}\\label{Y-0-intro}\nY_0=\\underset{\\tau\\in \\mathcal{T}_0}{\\sup}\\,\\mathbb{E}\\left[h(Y_{\\tau},\\mathbb{P}_{Y_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right].\n\\end{equation} \n\\end{itemize}\nThe main purpose of the present work is to show under certain conditions that an optimal stopping time for each of the OSPs \\eqref{Y-0-d-intro} and \\eqref{Y-0-intro}\ncan be charaterized as a hitting time similar to $\\tau^*$ in \\eqref{tau-intro}. A straightforward extension is to consider the combination of \\eqref{Y-0-d-intro} and \\eqref{Y-0-intro} given by\n\\begin{equation}\\label{Y-0-comb-intro}\nY_0=\\underset{\\tau\\in \\mathcal{T}_0}{\\sup}\\,\\mathbb{E}\\left[h(X_{\\tau},\\mathbb{P}_{X_{\\tau}},Y_{\\tau},\\mathbb{P}_{Y_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right].\n\\end{equation}\n\nTo solve the above problems we use a limit approach which consists of introducing a family of interacting Snell envelopes $\\{Y^{i,n}\\}_{i=1}^n$ (see Section \\ref{sec-formulation} for a precise definition for (OSPb) and Section \\ref{ssec-diffusion} for (OSPa)) as approximation of the value-process of the mean-field OSP. For example we approximate the OSP \\eqref{Y-0-intro} with the following family of interacting OSPs:\n\\begin{equation*}\n\tY^{i,n}_0=\\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\, \\mathbb{E}\\left[h(Y^{i,n}_{\\tau},\\frac{1}{n}\\sum_{j=1}^n \\delta_{Y^{j,n}_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\right],\\quad i=1,2,\\ldots,n.\n\\end{equation*}\nThese problems are time-consistent and it can be shown (see Corollary \\ref{opt-i-n}, below) that  it is optimal to stop at the hitting time $\\hat{\\tau}^{i,n}$ at which the value-process (which is now a Snell envelope) $(Y^{i,n}_t)_{t\\geq 0}$ hits the barriere  $(\\mathbb{E}[h(Y^{i,n}_{t},\\frac{1}{n}\\sum_{j=1}^n \\delta_{Y^{j,n}_{t}})\\, \\lvert\\,\\mathcal{F}^i_t])_{t\\in[0,T]}$. In Theorem  \\ref{optimality-tau-i} below we prove that the stopping time\n\\begin{equation}\\label{tau-opt}\n\t\\tau^{*}=\\inf\\{t\\ge 0, \\,\\, Y_t=h(Y_{t},\\mathbb{P}_{Y_{t}})\\}\\wedge T\n\\end{equation}\nis optimal for $Y_0$ given by \\eqref{Y-0-intro}, by showing that it is  the limit in probability of $\\hat{\\tau}^{1,n}$ as $n\\to\\infty$. Thus,  in this time-inconsistent framework an optimal stopping is also given by the usual hitting time. To derive this result, we need to investigate the convergence of $\\{Y^{i,n}_0\\}_{n\\geq1}$ to $Y_0$ (Proposition \\ref{conv-1}) and the convergence of the associated optimal stopping times (Lemma \\ref{conv-os-prob}). To have a more clear exposition, we prove our results in the simplified case of dependence on the mean of the stopped process, and then in Subsection \\ref{ssec-law-dep} we discuss the general case.\n\n\\medskip\nBy embedding this class of OSPs into the ones w.r.t. the set of randomized stopping times which is compact in the Baxter-Chacon topology (cf. \\cite{baxter77}), following many papers including Edgar, Millet and Sucheston \\cite{edgar82}, Arenas \\cite{arenas90} and El Karoui, Lepeltier and Millet \\cite{millet92}, one can show that there exists an optimal randomized stopping time for $Y_0$ without further characterization compared to the explicit optimal stopping time \\eqref{tau-opt}.  \n\n\n\\medskip\nThe paper is organized as follows. In Section \\ref{sec-formulation} we state precisely the simplified version of (OSPb) and the assumptions we need for the remaining part of the section. Then we discuss the well-posedness of the studied system of interacting optimal stopping problems, which is not obvious due to the recursive form of the utility function. In Section \\ref{sec-convergence} we present our main results about the convergence of the family of time-consistent problems. In particular, the convergence of the related optimal stopping times is discussed in Subsection \\ref{ssec-conv-ost}. These results are extended in Subsection \\ref{ssec-law-dep} to the more general case where the mean-field interaction is in terms of the law of the stopped process. Finally, in Section \\ref{ssec-diffusion} we discuss how to apply the suggested techniques to the problem (OSPa) associated to a mean-field  diffusion process. In particular, we obtain the same stopping region of the OSP of the variance as the one derived in \\cite{pedersen11} for Markov diffusion processes, we mentioned above. \n\n\\medskip\n\nThroughout this paper we only consider the one-dimensional Brownian motion and diffusion processes. The generalization to the multidimensional case is straightforward.\n\n\n\\subsection*{Notation.}\nLet $(\\Omega, \\mathcal{F},\\mathbb{P})$ be a complete probability space and $T>0$ a finite time horizon.  $W=(W_t)_{t\\in[0,T]}$  is a standard one-dimensional Brownian motion. We denote by $\\mathbb{F} = \\{\\mathcal{F}_t\\}_{t\\in[0,T]}$ the (completed) natural filtration of the Brownian motion $W$, with $\\mathcal{F}_0=\\{\\emptyset, \\Omega\\}$. In particular, $\\mathbb{F}$ is continuous i.e. for each $t\\ge 0$  $\\mathcal{F}_{t^-}=\\mathcal{F}_t$. Let $\\mathcal{P}$ be the $\\sigma$-algebra on $\\Omega \\times [0,T]$ of $\\mathcal{F}_t$-progressively measurable sets. \nNext, we introduce the following spaces. \\smallskip\n\\begin{itemize} \n    \\item $\\mathcal{T}_t$ is the set of $\\mathbb{F}$-stopping times $\\tau$ such that\n    $\\tau \\in [t,T]$ a.s. \\medskip\n    \\item \\textcolor{black}{$L^2(\\mathcal{F}_T)$ is the set of random variables $\\xi$ which are $\\mathcal{F}_T$-measurable and $\\mathbb{E}[|\\xi|^2]<\\infty$.} \\medskip\n    \\item $\\mathcal{S}^2$ is the set of real-valued $\\mathcal{P}$-measurable processes $y$ for which \\newline $\\|y\\|^2_{\\mathcal{S}^2} :=\\mathbb{E}[\\underset{ u\\in[0,T]}{\\sup} |y_u|^2]<\\infty$.  \\medskip\n    \\item $\\mathcal{S}_{c}^{2}$ is the space of $\\mathcal{S}^{2}$-valued continuous  processes. This space is complete and separable.  \\medskip\n\\begin{comment}\n    \\item $\\mathbb{L}_{\\beta}^2$, $\\beta\\ge 0$, is the set of $\\mathbb{F}$-adapted, real-valued and continuous processes $y$ such that $\\|y\\|^2_{\\mathbb{L}_{\\beta}^2} :=\\underset{\\tau\\in\\mathcal{T}_0}{\\sup}\\mathbb{E}[e^{2\\beta  \\tau}|y_{\\tau}|^2]<\\infty$. $\\mathbb{L}_{\\beta}^2$ is a Banach space (see Theorem 22 in \\cite{dellacheriemeyer82}, \n    pp. 83 for the space $\\mathbb{L}^1$). We set $\\mathbb{L}^2=\\mathbb{L}_{0}^2$. \\\\\n    \n \\item $\\mathcal{H}^{p,d}$ is the set of $\\mathcal{P}$\n   \\medskip\n    \\item $\\mathcal{B}(\\mathbb{R}^d)$ is the Borel $\\sigma$-algebra on $\\mathbb{R}^d$.  \\medskip\n\\end{comment}\n    \\item $C([0,T];\\mathbb{R})$ is the  space of continuous functions over $[0,T]$ endowed with the supremum norm. It is a separable Banach space.  \\medskip\n    \\item $\\mathcal{P}_2(\\mathbb{R})$ is the set of probability measures with finite second moment, i.e. $\\mu\\in\\mathcal{P}_2(\\mathbb{R})$ if and only if $\\int_\\mathbb{R} \\lvert x\\rvert^2\\mu(dx)<\\infty$. We endow this space with the $2$-Wasserstein metric $\\mathcal{W}_2$, defined by\n    \\begin{equation*}\n    \t\\mathcal{W}_2(\\mu,\\nu) = \\inf\\left\\{\\mathbb{E}[\\lvert X - Y\\rvert^2]^{\\frac{1}{2}}, {Law}(X)=\\mu, {Law}(Y)=\\nu\\right\\},\\quad \\mu,\\nu\\in\\mathcal{P}_2(\\mathbb{R}).\n    \\end{equation*}\n    The space $(\\mathcal{P}_2(\\mathbb{R}),\\mathcal{W}_2)$ is a complete separable metric space (see e.g. Theorem 6.18 in \\cite{villani}).\n\\end{itemize}\n\\medskip\n\\section{Optimal stopping of a  recursive utility function}\\label{sec-formulation}\nConsider the following (simplified) finite horizon optimal stopping problem (OSP) of mean-field type: \n\\begin{equation}\\label{Y-0}\nY_0=\\underset{\\tau\\in \\mathcal{T}_0}{\\sup}\\, \\mathbb{E}\\left[h(Y_{\\tau},\\mathbb{E}[Y_{\\tau}])1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right],\n\\end{equation}\nwhere $h$ is a sufficiently smooth cost (see Assumption \\ref{A1} below) and  $\\xi\\in L^2(\\mathcal{F}_T)$. \n\nThe OSP \\eqref{Y-0} is time-inconsistent i.e. the associated value-process is no longer a supermartingale, due to the presence of expected value (law) of the stopped random variable $Y_{\\tau}$. \n\nWe associate to \\eqref{Y-0} the following value-process:\n\\begin{equation}\\label{Y}\nY_t=\\underset{\\tau\\in \\mathcal{T}_t}{\\esssup}\\, \\mathbb{E}\\left[h(Y_{\\tau},\\mathbb{E}[Y_{\\tau}])1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\,|\\,\\mathcal{F}_t\\right].\n\\end{equation}\n\nWe would like to investigate whether the value-process $Y$ is well-defined i.e. whether there exists a unique solution to \\eqref{Y} and whether there exists a optimal stopping time $\\tau^*$ to the OSP \\eqref{Y-0}:\n\\begin{equation}\\label{stop}\n    \\tau^*=\\underset{\\tau\\in \\mathcal{T}_0}{\\arg\\max}\\, \\mathbb{E}\\left[h(Y_{\\tau},\\mathbb{E}[Y_{\\tau}])1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right].\n\\end{equation}\n\n\\medskip\nWe suggest to solve this problem by using a limit approach based on approximating $\\mathbb{E}[Y_{\\cdot}]$ by its empirical mean $\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\cdot}$ for some suitable sample $Y^{i,n},\\,i=1,2,\\ldots,n$ of 'interacting' value-processes which solve a system of standard OSPs.\n\n\\medskip\nTo this end we set $W^1=W$ and let $\\{W^i\\}_{i\\ge 1}$ be independent Brownian motions and denote by $\\mathbb{F}^n:=\\{\\mathcal{F}_t^n\\}_{t \\in [0,T]}$ the completion of the filtration generated by $\\{W^i\\}^n_{i=1}$. Thus, for each $1\\le i\\le n$, the filtration generated by $W^i$ is a sub-filtration of $\\mathbb{F}^n$, and we denote by ${\\mathbb{F}}^i:=\\{{\\mathcal{F}}^i_t\\}_{t\\in[0,T]}$ its completion. Let $\\mathcal{T}^n_t$ (resp. $\\mathcal{T}^i_t$) be the set of $\\mathbb{F}^n$ (resp. $\\mathbb{F}^i$) stopping times with values in $[t,T]$. \n\n\\medskip\nConsider the following family of finite horizon stopping problems.\n\\begin{equation}\\label{Y-i-n}\nY^{i,n}_t=\\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\, \\mathbb{E}\\left[h(Y^{i,n}_{\\tau},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\tau})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\,|\\,\\mathcal{F}^i_t\\right],\\quad i=1,2,\\ldots,n,\n\\end{equation}\nand\n\\begin{equation}\\label{Y-i}\nY^{i}_t=\\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\, \\mathbb{E}\\left[h(Y^{i}_{\\tau},\\mathbb{E}[Y^{i}_{\\tau}])1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\,|\\,\\mathcal{F}^i_t\\right],\\quad i\\ge 1,\n\\end{equation}\nwhere $h$ and $\\{\\xi^i\\}_{i\\geq1}$ satisfies the following conditions.\n\n\\begin{Assumption}\\label{A1} The coefficients $h$ and $\\{\\xi^i\\}_{i\\geq 1}$ satisfy the following conditions:\n\\begin{enumerate}[(i)]\n\t\t\\item For each $i\\ge 1$, $\\xi^i\\in L^2(\\mathcal{F}^i_T)$. Moreover, the $\\xi^i$'s are independent copies of $\\xi$, with $\\xi^1=\\xi$; \\\\\n\t\t\\item the function  $h\\colon\\Omega\\times\\mathbb{R}\\times\\mathbb{R}\\to\\mathbb{R}$ \\\\\n  \\begin{itemize} \n      \\item[(ii-a)]\n  is Lipschitz continuous w.r.t. $(y,z)$: there exist two positive constants $\\gamma_1$ and $\\gamma_2$ such that \n\t\t\\begin{equation*}\n\t\t\t\\lvert h(\\omega,y_1,z_1) - h(\\omega,y_2,z_2)\\rvert\\leq \\gamma_1\\lvert y_1 - y_2\\rvert + \\gamma_2\\lvert z_1 - z_2\\rvert, \n\t\t\\end{equation*}\n\t\tfor any $\\omega\\in\\Omega$ and any $y_1,y_2,z_1,z_2\\in\\mathbb{R}$. \\\\\n  \\item[(ii-b)] $h(\\cdot,0,0)\\in L^2(\\mathbb{P})$. \n  \\end{itemize}\n\t\\end{enumerate}\n\\end{Assumption}\n\n\\subsection{Existence and uniqueness of the value-processes}\\label{sec-existence-uniq}\nIn this section we show the well-posedness of the systems \\eqref{Y-i} and \\eqref{Y-i-n}, since in both cases the performance function depends also on the value-process. \n\\begin{theorem}\\label{Y-i-well-def} Suppose that Assumption \\ref{A1} is in force. Assume further that  $\\gamma_1$ and $\\gamma_2$ satisfy\n\t\\begin{equation}\\label{cond_gagb}\n\t\t\\gamma_1^2 + \\gamma_2^2<\\frac{1}{2}.\n\t\\end{equation}\nThen there exists a unique solution in $\\mathcal{S}_c^2$ to each of the systems \\eqref{Y-i-n} and \\eqref{Y-i}.\n\\end{theorem}\n\nThe proof of Theorem \\eqref{Y-i-well-def} is based on  the fixed point argument used in the proof of Theorem 2.1 in \\cite{djehichedumitrescuzeng}. We omit to reproduce it here. \n\n\\begin{corollary}\\label{opt-i-n} For each $i=1,\\ldots n$, the $\\mathcal{F}^i$-stopping time\n\\begin{equation}\\label{tau-i-n}\n    \\hat{\\tau}^{i,n}=\\inf\\left\\{t\\ge 0, \\,\\, Y^{i,n}_t=\\mathbb{E}[h(Y^{i,n}_{t},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{t})\\, \\lvert\\,\\mathcal{F}^i_t]\\right\\}\\wedge T\n    \\end{equation}\nis optimal for $Y_0^{i,n}$.\n\\end{corollary}\n\\begin{proof} \nSince each of the processes $Y^{i,n}$ is in $\\mathcal{S}_c^2$, by Assumption \\ref{A1} (ii) and Doob's inequality, it follows that the obstacle process $\\mathcal{X}^{i,n}_t:=\\mathbb{E}[h(Y^{i,n}_{t},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{t})\\, \\lvert\\,\\mathcal{F}^i_t]$ is also in $\\mathcal{S}^2$ and thus is of class D. Moreover, since for each $t\\in [0,T]$, both $h(Y^{i,n}_{t},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{t})$ and $\\mathcal{F}^i_t$ are continuous, the optional and predictable projections of $h(Y^{i,n}_{\\cdot},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\cdot})$ w.r.t. $\\mathbb{F}^i$ coincide. This implies that the obstacle process $\\mathcal{X}^{i,n}$ is a.s. continuous.  Therefore, by  e.g. Theorem D.19 in Appendix D in \\cite{karatzasshreve98} or Proposition 1.1.8 in \\cite{pham} (see also \\cite{bismut1977} and \\cite{elkaroui79} for a more general set up), for each $i=1,2,\\ldots,n$, the stopping time $\\tau^{i,n}$ given by \\eqref{tau-i-n} is optimal for $Y^{i,n}_0$.\n\\end{proof}\n\n\n\\section{Convergence results}\\label{sec-convergence}\nThe main aim of this section is to characterize an optimal stopping time for the time inconsistent problem \\eqref{Y-0}. The idea is to exploit the time consistency of the system of interacting optimal stopping problems\n\\begin{equation*}\n\tY^{i,n}_0=\\underset{\\tau\\in \\mathcal{T}^i_t}{\\sup}\\, \\mathbb{E}\\left[h(Y^{i,n}_{\\tau},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\tau})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\right],\\quad i=1,2,\\ldots,n,\n\\end{equation*} to obtain an explicit sequence of optimal stopping times, and then to show that this sequence converges to an optimal stopping time for the OSP \\eqref{Y-0}.\n\\subsection{Convergence of the particle system}\\label{ssec_convergence}\nThis subsection is concerned with the convergence of the process $Y^{i,n}$, solution of the particle system \\eqref{Y-i-n}, to the solution $Y^{i}$ of the mean-field system \\eqref{Y-i}.\n\n\\begin{definition}[Exchangeable r.v.] The random variables $X^1,X^2,\\dots,X^n$ are said to be exchangeable if the law of the random vector $(X^1,X^2,\\dots,X^n)$ is the same as that of the random vector $(X^{\\sigma(1)},X^{\\sigma(2)},\\dots,X^{\\sigma(n)})$ for every permutation $\\sigma$ of the set $\\{1,2,\\dots,n\\}$. We write\n\\begin{equation*}\n\t{Law}(X^1,X^2,\\dots,X^n) = Law(X^{\\sigma(1)},X^{\\sigma(2)},\\dots,X^{\\sigma(n)}).\n\\end{equation*}\n\\end{definition}\nIn the following proposition, we show that the exchangeability property of the final conditions $\\{\\xi^i\\}_{i\\geq1}$, entailed by Assumption \\ref{A1} (i), transfers to the solutions of the systems \\eqref{Y-i-n} and \\eqref{Y-i}.\n\\begin{proposition}[Exchengeability property]\\label{exchangeability} Let Assumption \\ref{A1} hold. Then the processes $\\{Y^{i,n}\\}_{i=1}^n$ solution of the system \\eqref{Y-i-n} are exchangeable. Moreover, the processes $\\{Y^{i}\\}_{i\\geq1}$ solution of the system \\eqref{Y-i} are independent and equally distributed and hence exchangeable.\n\\end{proposition}\n\\begin{proof}\n\tFirst let us focus on  $\\{Y^{i,n}\\}_{i=1}^n$. For any permutation $\\sigma$ of the set $\\{1,2,\\dots,n\\}$, we have for any $t\\in[0,T]$ \n\t\\begin{equation*}\n\t\t\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_t = \\frac{1}{n}\\sum_{j=1}^n Y^{\\sigma(j),n}_t,\n\t\\end{equation*}\n\tand thanks to the uniqueness result in Theorem \\ref{Y-i-well-def} we have\n\t\\begin{equation*}\n\t\t(Y^{1,n},Y^{2,n},\\dots,Y^{n,n}) = (Y^{\\sigma(1),n},Y^{\\sigma(2),n},\\dots,Y^{\\sigma(n),n}),\\quad\\hbox{\\rm a.s.{ }}, \n\t\\end{equation*}\n i.e. the processes $\\{Y^{i,n}\\}_{i=1}^n$ are exchangeable.\n\n\tRegarding the processes $\\{Y^{i}\\}_{i\\geq1}$, from \\eqref{Y-i} we may write $Y_{t}^i=\\varphi(\\xi^i,(W^i_{s})_{0\\le s\\le t})$ for some Borel measurable function $\\varphi$. But, the pairs $(W^i,\\xi^i)$  are independent and  equally distributed. Therefore, the $Y^i$'s are independent and equally distributed and thus exchangeable.\n\\end{proof}\n\\begin{proposition}\\label{conv-1}\n\tAssume that $\\gamma_1$ and $\\gamma_2$ satisfy\n\t\\begin{equation}\\label{cond_gagb_3}\n\t\t\\gamma_1^2 + \\gamma_2^2<\\frac{1}{16}. \n\t\\end{equation}\n\tThen, under Assumption \\ref{A1} we have\n\t\\begin{equation*}\n\t\t\\Lim_{n\\to\\infty}\\mathbb{E}\\left[\\sup_{t\\in[0,T]}\\lvert Y^{i,n}_t - Y^{i}_t\\rvert^2\\right] = 0,\\quad i\\geq1.\n\t\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nFor any $t\\leq T$, we have\n\\begin{equation}\n\\begin{aligned}\n\t\\lvert Y^{i,n}_t - Y^i_t \\rvert& = \\Bigg\\lvert \\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\,\\mathbb{E}\\left[h(Y^{i,n}_{\\tau},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\tau})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\,|\\,\\mathcal{F}^i_t\\right] \\\\\n\t&\\quad- \\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\, \\mathbb{E}\\left[h(Y^i_{\\tau},\\mathbb{E}[Y^i_{\\tau}])1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\,|\\,\\mathcal{F}^i_t\\right] \\Bigg\\rvert\\\\\n\t&\\leq  \\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\, \\Bigg \\lvert \\mathbb{E}\\left[h(Y^{i,n}_{\\tau},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\tau})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\,|\\,\\mathcal{F}^i_t\\right] \\\\\n\t&\\quad-\\mathbb{E}\\left[h(Y^i_{\\tau}, \\mathbb{E}[Y^i_{\\tau}])1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\,|\\,\\mathcal{F}^i_t\\right]\\Bigg\\rvert\\\\\n\t&\\leq \\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\, \\mathbb{E}\\left[\\left\\lvert h(Y^{i,n}_{\\tau},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\tau}) - h(Y^i_{\\tau}, \\mathbb{E}[Y^i_{\\tau}])\\right\\rvert\\,|\\,\\mathcal{F}^i_t\\right]\\\\\n\t&\\leq \\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\, \\mathbb{E}\\left[\\left(\\gamma_1\\lvert Y^{i,n}_{\\tau} - Y^i_{\\tau} \\rvert + \\gamma_2\\lvert \\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\tau} - \\mathbb{E}[Y^i_{\\tau}] \\rvert\\right)\\,|\\,\\mathcal{F}^i_t\\right]\\\\\n\t&\\leq \\mathbb{E}\\left[\\gamma_1\\sup_{s\\in[0,T]}\\lvert Y^{i,n}_{s} - Y^i_{s} \\rvert + \\gamma_2\\sup_{s\\in[0,T]}\\lvert \\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{s} - \\mathbb{E}[Y^i_{s}] \\rvert\\,|\\,\\mathcal{F}^i_t\\right]\n\t\\\\\n\t&\\leq \\mathbb{E}\\left[\tG^{i,n}_T+\\gamma_2\\sup_{s\\in[0,T]}\\lvert \\frac{1}{n}\\sum_{j=1}^n Y^{j}_{s} - \\mathbb{E}[Y^i_{s}] \\rvert\\,|\\,\\mathcal{F}^i_t\\right],\n \\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation*}\n\tG^{i,n}_T:= \\gamma_1\\sup_{s\\in[0,T]}\\lvert Y^{i,n}_{s} - Y^i_{s} \\rvert +  \\frac{\\gamma_2}{n}\\sum_{j=1}^n \\sup_{s\\in[0,T]}\\lvert Y^{j,n}_{s}-Y^j_{s}\\rvert.\n\\end{equation*}\nBy Doob's inequality, we have\n\\begin{equation*}\n\t\\mathbb{E}[\\sup_{t\\in[0,T]} \\lvert  Y^{i,n}_t - Y^i_t  \\rvert^2]\\leq 4\\mathbb{E}\\left[\\left(G^{i,n}_T+\\gamma_2\\sup_{s\\in[0,T]}\\lvert \\frac{1}{n}\\sum_{j=1}^n Y^{j}_{s} - \\mathbb{E}[Y^i_{s}] \\rvert\\right)^2\\right].\n\\end{equation*}\nTherefore,\n\\begin{equation}\\label{sup_ineq}\n\\begin{aligned}\n\t\\mathbb{E}[\\sup_{t\\in[0,T]} \\lvert  Y^{i,n}_t - Y^i_t  \\rvert^2] &\\leq 8\\gamma_2^2\\mathbb{E}\\left[\\Lambda_n\n\\right] + 16\\mathbb{E}\\left[\\gamma_1^2\\sup_{s\\in[0,T]}\\lvert Y^{i,n}_{s} - Y^i_{s} \\rvert^2  \\right.  \\\\   & \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad + \\left. \\frac{\\gamma_2^2}{n}\\sum_{j=1}^n \\sup_{s\\in[0,T]}\\lvert Y^{j,n}_{s} - Y^{j}_{s}\\rvert^2\\right],\n\\end{aligned}\n\\end{equation}\nwhere, since the $Y^i$'s are identically distributed, we have\n\\begin{equation}\\label{Lambda-n}\n\t\\Lambda_n:= \\sup_{s\\in[0,T]} \\lvert \\frac{1}{n}\\sum_{j=1}^n Y^{j}_{s} - \\mathbb{E}[Y^i_s]\\rvert^2= \\sup_{s\\in[0,T]} \\lvert \\frac{1}{n}\\sum_{j=1}^n (Y^{j}_{s} - \\mathbb{E}[Y^j_s])\\rvert^2.\n\\end{equation}\nIn view of the exchangeability of the processes $\\{Y^{i,n}\\}_{i=1}^n$ and $\\{Y^i\\}_{i\\geq1}$ (see Proposition \\ref{exchangeability}), we have \n\\begin{equation*}\n\\mathbb{E}\\left[\\frac{1}{n}\\sum_{j=1}^n \\sup_{s\\in[0,T]}\\lvert Y^{j,n}_{s} - Y^{j}_{s}\\rvert^2\\right]=\\mathbb{E}\\left[\\sup_{t\\in[0,T]} \\lvert  Y^{i,n}_t - Y^i_t  \\rvert^2\\right].\n\\end{equation*}\n\\begin{comment}\\begin{equation*}\n\t\\mathbb{E}\\left[\\left(G^{i,n}_T\\right)^2\\right]\\leq 4(\\gamma_1^2 + \\gamma_2^2)\\mathbb{E}[\\sup_{s\\in[0,T]}\\lvert Y^{i,n}_{s} - Y^i_{s} \\rvert^2] + \\mathbb{E}[\\Lambda_n].\n\\end{equation*}\n\\end{comment}\nThus, from \\eqref{sup_ineq} we obtain\n\\begin{equation*}\n\t\\mathbb{E}[\\sup_{t\\in[0,T]} \\lvert  Y^{i,n}_t - Y^i_t  \\rvert^2]\\leq16(\\gamma_1^2+\\gamma_2^2)\\mathbb{E}[\\sup_{s\\in[0,T]}\\lvert Y^{i,n}_{s} - Y^i_{s} \\rvert^2] +  32\\gamma_2^2\\mathbb{E}[\\Lambda_n],\n\\end{equation*}\nwhere, by \\eqref{cond_gagb_3}, $16(\\gamma_1^2+\\gamma_2^2)<1$. So\n\\begin{equation*}\n\t\\mathbb{E}[\\sup_{t\\in[0,T]} \\lvert  Y^{i,n}_t - Y^i_t  \\rvert^2]\\leq 32(1-16(\\gamma_1^2+\\gamma_2^2))^{-1}\\mathbb{E}[\\Lambda_n].\n\\end{equation*} \nFinally, since the processes $\\{Y^j\\}_{j\\geq1}$ are i.i.d. $C([0,T];\\mathbb{R})$-valued random variables with finite second moments (since they are in $\\mathcal{S}^2$), by the strong law of large numbers for Banach valued r.v. (see Theorem 4.1.1 in \\cite{padgett-taylor06}) and dominated convergence we have  \n\\begin{equation}\\label{L-conv}\n\t\\Lim_{n\\to\\infty} \\mathbb{E}[\\Lambda_n]=0,\n\\end{equation}\nwhich yields the desired result. \n\\end{proof}\n\n\n\\subsection{Convergence of the optimal stopping times}\\label{ssec-conv-ost}\nIn Section \\ref{ssec_convergence} we proved that $Y^{i,n}$ converges to $Y^i$ as $n$ goes to  infinity in the $\\mathcal{S}^2$ norm. This entails the convergence of the values of the OSPs $Y^{i,n}_0$ to $Y^i_0$, as $n\\to\\infty$, for every $i\\geq1$. Furthermore, by Corollary \\ref{opt-i-n}, for each $i=1,\\ldots,n$, the stopping time $\\hat{\\tau}^{i,n}$ given by \\eqref{tau-i-n} is optimal for the OSP $Y^{i,n}_0$, that is\n\\begin{equation}\n    \\hat{\\tau}^{i,n} =\\underset{\\tau\\in \\mathcal{T}^i_0}{\\arg\\max}\\, \\mathbb{E}\\left[h(Y^{i,n}_{\\tau},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\tau})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\right].\n\\end{equation}\n\nFor every $i\\geq 1$, let us introduce the stopping time\n\\begin{equation}\\label{tau-i}\n    \\hat{\\tau}^{i}=\\inf\\{t\\ge 0, \\,\\, Y^{i}_t=h(Y^{i}_{t},\\mathbb{E}[Y^i_{t}])\\}\\wedge T.\n\\end{equation}\nDue to the time-inconsistency of the problem \\eqref{Y-i}, it is not immediate to conclude that $\\hat{\\tau}^{i}$ is optimal, i.e. that it coincides with the optimal stopping $\\tau^{i,*}$ defined by \n\\begin{equation}\\label{stop-i}\n    \\tau^{i,*}=\\underset{\\tau\\in \\mathcal{T}_0}{\\arg\\max}\\, \\mathbb{E}\\left[h(Y^i_{\\tau},\\mathbb{E}[Y^i_{\\tau}])1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\right].\n\\end{equation}\nThe main result of this section is to show that \\eqref{stop-i} actually holds.\nIn particular, since $W^1=W$ and $\\xi^1=\\xi$ , we have $Y^1=Y$ i.e. $Y^1$ is the value-process $Y$ given by \\eqref{Y} and $\\tau^{1,*}$ is the associated optimal stopping time given by \\eqref{stop} i.e. $\\tau^*:=\\tau^{1,*}$.\n\\begin{theorem}\\label{optimality-tau-i}\n\tLet Assumption \\ref{A1} hold and assume that $\\gamma_1$ and $\\gamma_2$ satisfy \\eqref{cond_gagb_3}.\n\n\n\n\tThen for any $i\\geq 1$ the stopping time $\\hat{\\tau}^i$ defined by \\eqref{tau-i} is optimal for the OSP \\eqref{stop-i}.\n\\end{theorem}\n\nTo prove this statement, we rely on the next proposition  about the convergence of optimal stopping times and its corollary.  \n\\begin{proposition}\\label{conv-os-prob}\n    For any $i\\geq 1$, let $\\{\\hat{\\tau}^{i,n}\\}_{n\\geq 1}$ be the sequence of optimal stopping times defined by \\eqref{tau-i-n} and let $\\hat{\\tau}^{i}$ be defined by \\eqref{tau-i}. Let Assumption \\ref{A1} and \\eqref{cond_gagb_3} hold. Then $\\hat{\\tau}^{i,n}$ converges to $\\hat{\\tau}^{i}$ in probability, as $n$ goes to infinity.\n\\end{proposition}\n\\begin{proof}\nTo keep the notation lighter, we hide the dependence on $i$ and set\n    \\begin{equation}\\label{Z}\n        Z^n_t:= Y^{i,n}_t - \\mathbb{E}[h(Y^{i,n}_{t},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{t})\\, \\lvert\\,\\mathcal{F}^i_t],\\quad Z_t = Y^{i}_t - h(Y^{i}_t,\\mathbb{E}[Y^i_t]),\\quad t\\in[0,T].\n    \\end{equation}\n    We notice that, for every $t\\in[0,T]$, $Z^n_t\\geq0$ a.s.  and that $\\hat{\\tau}^n =\\inf\\{t\\ge 0, \\,\\, Z^n_t = 0\\} \\wedge T$. The same holds for $Z$ and $\\hat{\\tau}$.\n    \n    For any $\\varepsilon>0$, we have \n    \\begin{equation}\\label{conv-prob-term}\n        \\mathbb{P}\\left(\\lvert \\hat{\\tau}^n - \\hat{\\tau}\\rvert>\\varepsilon\\right) = \\mathbb{P}\\left(\\hat{\\tau}^n-\\hat{\\tau}>\\varepsilon\\right) + \\mathbb{P}\\left(\\hat{\\tau}-\\hat{\\tau}^n>\\varepsilon\\right).\n    \\end{equation}\n    We first show that $\\mathbb{P}\\left(\\hat{\\tau}^n-\\hat{\\tau}>\\varepsilon\\right)\\to 0$ as $n\\to\\infty$. The event \n    \\begin{equation*}\n        \\left\\{\\hat{\\tau}^n-\\hat{\\tau}>\\varepsilon\\right\\}\n    \\end{equation*}\n    means that $Z^n$ attains $0$ at a time which is  larger than the time $\\hat{\\tau}$  at which $Z$ attains the same level 0 with at least $\\varepsilon>0$. In other words,\n    \\begin{equation}\n        \\mathbb{P}\\left(\\hat{\\tau}^n-\\hat{\\tau}>\\varepsilon\\right) = \\mathbb{P}\\left(\\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z^n_t > 0 ,Z_{\\hat{\\tau}} = 0\\right).\n    \\end{equation}\n    Notice that, for the limit process $Z$, it trivially holds that\n    \\begin{equation}\n        \\mathbb{P}\\left(\\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z_t > 0 ,Z_{\\hat{\\tau}} = 0\\right) = 0.\n    \\end{equation}\n    So, it remains to prove that \n    \\begin{equation}\\label{conv-prob}\n        \\Lim_{n\\to\\infty}\\mathbb{P}\\left(\\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z^n_t > 0 ,Z_{\\hat{\\tau}} = 0\\right) = \\mathbb{P}\\left(\\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z_t > 0 ,Z_{\\hat{\\tau}} = 0\\right).\n    \\end{equation}\n    Indeed, by Assumption \\ref{A1} (ii), we have\n    \\begin{equation*}\n    \\begin{aligned}\n        \\left\\lvert \\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z^n_t - \\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z_t\\right\\rvert\n    & \\leq \\sup_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon}\\lvert Z^n_t - Z_t\\rvert\\\\\n    & \\leq \\sup_{t\\in[0,T]}\\lvert Y^{i,n}_t - Y^i_t\\rvert + \\sup_{t\\in[0,T]}\\lvert \\mathbb{E}[h(Y^{i,n}_{t},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{t})\\, \\lvert\\,\\mathcal{F}^i_t] - h(Y^i_t,\\mathbb{E}[Y^i_t])\\rvert \\\\\n    &\\leq (1 + \\gamma_1)\\sup_{t\\in[0,T]}\\lvert Y^{i,n}_t - Y^i_t\\rvert + \\gamma_2 \\sup_{t\\in[0,T]}\\mathbb{E}[\\sup_{s\\in[0,T]}\\lvert\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{s} - \\mathbb{E}[Y^i_s] \\rvert\\,\\lvert\\, \\mathcal{F}_t^ i].\n    \\end{aligned}\n    \\end{equation*}\nUsing Doob's inequality, we obtain\n    \\begin{equation*}\n    \\begin{aligned}\n\\mathbb{E}\\left[\\left( \\sup_{t\\in[0,T]}\\mathbb{E}[\\sup_{s\\in[0,T]}\\lvert\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{s} - \\mathbb{E}[Y^i_s] \\rvert\\,\\lvert\\, \\mathcal{F}_t^ i] \\right)^2\\right] \\le 4 \\mathbb{E}\\left[\\left(\\sup_{t\\in[0,T]}\\lvert\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{t} - \\mathbb{E}[Y^i_t] \\rvert \\right)^2\\right] \\\\ \\le 8 \\mathbb{E}\\left[\\left(\\frac{1}{n}\\sum_{j=1}^n \\sup_{t\\in[0,T]}\\lvert Y^{j,n}_{t} - Y^j_t\\rvert\\right)^2\\right]+8\\mathbb{E}\\left[\\left(\\sup_{t\\in[0,T]}\\lvert\\frac{1}{n}\\sum_{j=1}^n \\lvert Y^{j}_{t} - \\mathbb{E}[Y^j_t]\\lvert\\right)^2\\right] \\\\ \\le  8\\left[  \\sup_{t\\in[0,T]}\\lvert Y^{i,n}_t - Y^i_t\\rvert^2\\right] + 8\\mathbb{E}[\\Lambda_n],\n    \\end{aligned}\n    \\end{equation*}\n    where the first term of the last inequality  follows from the Cauchy-Schwarz inequality and the exchangeability of the processes $\\{Y^{i,n}\\}_{i=1}^n$ and $\\{Y^i\\}_{i\\geq1}$ (by Proposition \\ref{exchangeability}) and $\\Lambda_n$ is given by \\eqref{Lambda-n}.\n    \nTherefore, we have\n   \\begin{equation*}\n   \\begin{aligned}\n   \t\\mathbb{E}\\left[ \\left\\lvert \\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z^n_t - \\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z_t\\right\\rvert^2\\right]\\leq 2\\left((1 + \\gamma_1)^2 + 8\\gamma_2^2\\right)\\mathbb{E}\\left[  \\sup_{t\\in[0,T]}\\lvert Y^{i,n}_t - Y^i_t\\rvert^2\\right] + 16\\gamma_2^2\\mathbb{E}[\\Lambda_n].\n   \\end{aligned}\n   \\end{equation*}\n    Thus, thanks to Proposition \\ref{conv-1} and  \\eqref{L-conv}, it holds that\n    \\begin{equation}\\label{conv-mean-sup}\n        \\Lim_{n\\to\\infty}\\mathbb{E}\\left[\\left\\lvert \\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z^n_t - \\inf_{0\\leq t\\leq\\hat{\\tau}+\\varepsilon} Z_t\\right\\rvert^2\\right] = 0,\n    \\end{equation}\n    which entails \\eqref{conv-prob}. \n    \n    Let us now focus on the second term on the right hand side of \\eqref{conv-prob-term}. We have\n    \\begin{equation*}\n        \\begin{aligned}\n            \\mathbb{P}\\left(\\hat{\\tau}-\\hat{\\tau}^n>\\varepsilon\\right) &= \\mathbb{P}\\left(\\inf_{0\\leq t\\leq\\hat{\\tau}^n + \\varepsilon} Z_t > 0, Z^n_{\\hat{\\tau}^n} = 0\\right)\\\\\n            & = \\mathbb{P}\\left(\\inf_{0\\leq t\\leq\\hat{\\tau}^n + \\varepsilon} Z_t  - Z_{\\hat{\\tau}^n} > 0, Z^n_{\\hat{\\tau}^n} = 0\\right)\\\\\n            &\\leq \\mathbb{P}\\left(\\inf_{0\\leq t\\leq\\hat{\\tau}^n + \\varepsilon} Z_t  - Z^n_{\\hat{\\tau}^n} > 0\\right)\\\\\n            &\\leq \\mathbb{P}\\left(\\inf_{0\\leq t\\leq\\hat{\\tau}^n + \\varepsilon} Z_t  - \\inf_{0\\leq t\\leq\\hat{\\tau}^n + \\varepsilon} Z^n_{t} > 0\\right)\\\\\n            &\\leq \\mathbb{P}\\left ( \\sup_{t\\in[0,T]}\\lvert Z_t - Z^n_t \\rvert>0\\right).\n        \\end{aligned}\n    \\end{equation*}\n    Now, similarly to the first part of the proof, by combining Assumption \\ref{A1} (ii) with Proposition \\ref{conv-1} we obtain that the sequence of random variables $\\{\\sup_{t\\in[0,T]}\\lvert Z_t - Z^n_t\\rvert\\}_{n\\geq 1}$ converges in probability to zero as $n\\to\\infty$, and thus $\\mathbb{P}\\left(\\hat{\\tau}-\\hat{\\tau}^n>\\varepsilon\\right)\\to 0$ as well.\n\\end{proof}\n\n\\begin{corollary}\\label{conv-os-prob-1}\nWe have, for every $i\\ge 1$,\n\\begin{equation}\\label{BC-1}\n\\lim_{n\\to\\infty}\\mathbb{E}[Y^i_{\\hat{\\tau}^{i,n}}]=\\mathbb{E}[Y^i_{\\hat{\\tau}^{i}}].\n\\end{equation}\nMoreover, up to a subsequence, it holds that \n\\begin{equation}\\label{BC-2}\n\\lim_{n\\to\\infty}\\mathbb{E}\\left[h(Y^{i,n}_{\\hat{\\tau}^{i,n}},\\frac{1}{n}\\sum_{j=1}^n Y^{j,n}_{\\hat{\\tau}^{j,n}})1\\!\\!1_{\\{\\hat{\\tau}^{i,n}<T\\}}+\\xi^i1\\!\\!1_{\\{\n\\hat{\\tau}^{i,n}=T\\}}\\right]=\\mathbb{E}\\left[h(Y^i_{\\hat{\\tau}^i},\\mathbb{E}[Y^i_{\\hat{\\tau}^i}])1\\!\\!1_{\\{\\hat{\\tau}^i<T\\}}+\\xi^i1\\!\\!1_{\\{\\hat{\\tau}^i=T\\}}\\right].\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\n   We derive \\eqref{BC-1} by contradiction. Assume that $\\hat{\\tau}^{i,n}$ converges in probability to $\\hat{\\tau}^{i}$ with $|\\mathbb{E}[Y^i_{\\hat{\\tau}^{i,n}}]-\\mathbb{E}[Y^i_{\\hat{\\tau}^{i}}]|\\ge \\varepsilon >0$ for all $n$. But, then we can extract a subsequence $\\hat{\\tau}^{i,n_k}$ which converges to $\\hat{\\tau}^{i}$ a.s. Since the continuous process  $Y^i$ is in $\\mathcal{S}^2$, by dominated convergence, we arrive at a contradiction.\n   \n   To derive \\eqref{BC-2}, we note that since the process $Y^i$ is continuous and  $\\hat{\\tau}^{i,n}, \\hat{\\tau}^i$  are $\\mathbb{F}^i$-stopping times, it holds that the sequence $(Y^{i,n},\\hat{\\tau}^{i,n})$ converges in probability to $(Y^{i},\\hat{\\tau}^{i})$. Therefore, in view of \\cite{aldous81}, Corollary 16.23, $(\\hat{\\tau}^{i,n}, Y^{i,n}_{\\hat{\\tau}^{i,n}})$ converges in distribution to $(\\hat{\\tau}^{i}, Y^{i}_{\\hat{\\tau}^{i}})$. For each $i\\ge 1$, let $\\{\\hat{\\tau}^{i,n_k}\\}_{k\\geq1}$ be a subsequence of the sequence of stopping times $\\{\\hat{\\tau}^{i,n}\\}_{n\\geq 1}$, which converges a.s. to $\\hat{\\tau}^{i}$. \n   We claim that for every $i\\ge 1$, $\\frac{1}{n_k}\\sum_{j=1}^{n_k} Y^{j,n_k}_{\\hat{\\tau}^{j,n_k}}\\overset{L^1}\\to Y^{i}_{\\hat{\\tau}^{i}}$ as $k\\to \\infty$.\n   Indeed, noting that  $(Y^i,\\tau^i),\\,i=1,2,\\ldots$ are i.i.d., we have\n   $$\\begin{array}{lll}\n   \\mathbb{E}\\left[\\left\\lvert \\frac{1}{n_k}\\sum_{j=1}^{n_k} Y^{j,n_k}_{\\hat{\\tau}^{j,n_k}}- \\mathbb{E}[Y^{i}_{\\hat{\\tau}^{i}}]\\right\\rvert\\right] & \\leq \\mathbb{E}\\left[\\left\\lvert \\frac{1}{n_k}\\sum_{j=1}^{n_k} (Y^{j,n_k}_{\\hat{\\tau}^{j,n_k}}-Y^j_{\\hat{\\tau}^{j,n_k}})\\right\\lvert\\right]+\\frac{1}{n_k}\\sum_{j=1}^{n_k}\\mathbb{E}\\left[\\left\\lvert  Y^j_{\\hat{\\tau}^{j,n_k}}-Y^j_{\\hat{\\tau}^{j}}\\right\\lvert\\right] \\\\ &\\quad + \\mathbb{E}\\left[\\left\\lvert \\frac{1}{n_k}\\sum_{j=1}^{n_k} (Y^j_{\\hat{\\tau}^{j}}-\\mathbb{E}[Y^j_{\\hat{\\tau}^{j}}])\\right\\lvert\\right].\n   \\end{array}\n  $$\n  We have\n  $$\n  \\mathbb{E}\\left[\\left\\lvert \\frac{1}{n_k}\\sum_{j=1}^{n_k} (Y^{j,n_k}_{\\hat{\\tau}^{j,n_k}}-Y^j_{\\hat{\\tau}^{j,n_k}})\\right\\lvert\\right]\\le \\frac{1}{n_k}\\sum_{j=1}^{n_k}\\mathbb{E}\\left[\\underset{t\\in [0,T]}{\\sup}|Y^{j,n_k}_t-Y^j_t|\\right]\\to 0, \\quad k\\to\\infty,\n  $$\n  by the Cesaro Mean Lemma. Moreover, in view of the dominated convergence theorem, for every $j\\ge 1$, $\\mathbb{E}\\left[\\left\\lvert  Y^j_{\\hat{\\tau}^{j,n_k}}-Y^j_{\\hat{\\tau}^{j}}\\right\\lvert\\right] \\to 0$ as $k\\to\\infty$. Thus, again by the Cesaro Mean Lemma, we have \n  $\\frac{1}{n_k}\\sum_{j=1}^{n_k}\\mathbb{E}\\left[\\left\\lvert  Y^j_{\\hat{\\tau}^{j,n_k}}-Y^j_{\\hat{\\tau}^{j}}\\right\\lvert\\right] \\to 0$ as $k\\to\\infty.$\n  \nSince $(Y^i,\\tau^i),\\,i=1,2,\\ldots,$ are i.i.d., the r.v. $Y^i_{\\hat{\\tau}^i},\\,i=1,2,\\ldots$ are i.i.d. By the strong law of large numbers and dominated convergence we have $\\mathbb{E}\\left[\\left\\lvert \\frac{1}{n_k}\\sum_{j=1}^{n_k} (Y^j_{\\hat{\\tau}^{j}}-\\mathbb{E}[Y^j_{\\hat{\\tau}^{j}}])\\right\\lvert\\right]\\to 0$ as $k\\to\\infty$. Therefore, as $k\\to \\infty$, $h(Y^{i,n_k}_{\\hat{\\tau}^{i.n_k}},\\frac{1}{n_k}\\sum_{j=1}^n Y^{j,n_k}_{\\hat{\\tau}^{j,n_k}})1\\!\\!1_{\\{\\hat{\\tau}^{i,n_k}<T\\}}+\\xi^i1\\!\\!1_{\\{\n\\hat{\\tau}^{i,n_k}=T\\}}$ converges almost surely to $h(Y^i_{\\hat{\\tau}^i},\\mathbb{E}[Y^i_{\\hat{\\tau}^i}])1\\!\\!1_{\\{\\hat{\\tau}^i<T\\}}+\\xi^i1\\!\\!1_{\\{\\hat{\\tau}^i=T\\}}$. The claim \\eqref{BC-2} follows by dominated convergence.\n\\end{proof}\n\n\\medskip\n\\begin{proof}[Proof of Theorem \\ref{optimality-tau-i}]\nLet $\\{\\hat{\\tau}^{i,n_k}\\}_{k\\geq 1}$ be a subsequence of the sequence of stopping times $\\{\\hat{\\tau}^{i,n}\\}_{n\\geq1}$, which converges a.s. to $\\hat{\\tau}^{i}$. In view of \\eqref{BC-2} and the optimality of $\\{\\hat{\\tau}^{i,n_k}\\}_{k\\geq 1}$, we have \n\\begin{multline*}\nY_0^i=\\underset{k\\to\\infty}{\\lim}Y_0^{i,n_k}=\\underset{k\\to\\infty}{\\lim}\\mathbb{E}\\left[h(Y^{i,n_k}_{\\hat{\\tau}^{i.n_k}},\\frac{1}{n_k}\\sum_{j=1}^{n_k} Y^{j,n_k}_{\\hat{\\tau}^{j,n_k}})1\\!\\!1_{\\{\\hat{\\tau}^{i,n_k}<T\\}}+\\xi^i1\\!\\!1_{\\{\n\\hat{\\tau}^{i,n_k}=T\\}}\\right] \\\\ =\\mathbb{E}\\left[h(Y^i_{\\hat{\\tau}^i},\\mathbb{E}[Y^i_{\\hat{\\tau}^i}])1\\!\\!1_{\\{\\hat{\\tau}^i<T\\}}+\\xi^i1\\!\\!1_{\\{\\hat{\\tau}^i=T\\}}\\right].\n\\end{multline*}\n\\end{proof}\n\n\\subsection{Extension to the general mean-field case}\\label{ssec-law-dep}\nThe aim of this section is to extend the results we proved in Sections \\ref{sec-existence-uniq} and \\ref{sec-convergence} to the more general case where the performance function depends on the law of the stopped process and not only on its mean. More precisely, we want to solve the OSP\n\\begin{equation}\n\tY_0=\\underset{\\tau\\in \\mathcal{T}_0}{\\sup}\\, \\mathbb{E}\\left[h(Y_{\\tau},\\mathbb{P}_{Y_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right],\n\\end{equation} \nwhere $\\mathbb{P}_{\\zeta}$ denotes the law of the random variable $\\zeta$ under $\\mathbb{P}$. To this end, as in Section \\ref{sec-formulation}, we introduce two families of value-processes: \n\\begin{equation}\\label{Y-i-n-law}\nY^{i,n}_t=\\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\, \\mathbb{E}\\left[h(Y^{i,n}_{\\tau},\\frac{1}{n}\\sum_{j=1}^n \\delta_{Y^{j,n}_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\,|\\,\\mathcal{F}^i_t\\right],\\quad i=1,2,\\ldots,n,\n\\end{equation}\nwhere $\\delta_x$ is the Dirac measure centered at $x\\in\\mathbb{R}$, and\n\\begin{equation}\\label{Y-i-law}\nY^{i}_t=\\underset{\\tau\\in \\mathcal{T}^i_t}{\\esssup}\\, \\mathbb{E}\\left[h(Y^{i}_{\\tau},\\mathbb{P}_{Y_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\,|\\,\\mathcal{F}^i_t\\right],\\quad i=1,2,\\ldots.\n\\end{equation}\nSince now $h$ is also a function of  probability measures, we have to modify Assumption \\ref{A1}(ii) but keep \\ref{A1}(i). \n\n\\begin{Assumption}\\label{A2} The coefficients $h$ and $\\{\\xi^i\\}_{i\\geq 1}$ satisfy\n\t\\begin{enumerate}[(i)]\n\t\t\\item for each $i\\ge 1$, $\\xi^i\\in L^2(\\mathcal{F}^i_T)$. Moreover, the $\\xi^i$'s are independent copies of $\\xi$, with $\\xi^1=\\xi$; \n\t\t\\item $h\\colon\\Omega\\times\\mathbb{R}\\times\\mathcal{P}_2(\\mathbb{R})\\to\\mathbb{R}$ \n\t\t\\begin{itemize}\n\t\t\\item[(ii-a)] is Lipschitz w.r.t. $(y,\\mu)$, i.e. there exist two positive constants $\\gamma_1$ and $\\gamma_2$ such that \n\t\t\\begin{equation*}\n\t\t\t\\lvert h(\\omega,y_1,\\mu_1) - h(\\omega,y_2,\\mu_2)\\rvert\\leq \\gamma_1\\lvert y_1 - y_2\\rvert + \\gamma_2\\mathcal{W}_2(\\mu_1,\\mu_2), \n\t\t\\end{equation*}\n\t\tfor any $\\omega\\in\\Omega$, any $y_1,y_2\\in\\mathbb{R}$ and any $\\mu_1,\\mu_2\\in\\mathcal{P}_2(\\mathbb{R})$.\n\t\t\\item[(ii-b)] $h(\\cdot,0,\\delta_0)\\in L^2(\\mathbb{P})$.\n\t\t\\end{itemize}\n\t\\end{enumerate}\n\\end{Assumption}\n\nWith Assumption \\ref{A2} instead of \\ref{A1}, we can recover Theorem \\ref{Y-i-well-def}, Corollary \\ref{opt-i-n} and Theorem \\ref{optimality-tau-i}, as well as all the intermediate results, for this more general framework by using the following inequalities  satisfied by the $2$-Wasserstein distance:\n\\begin{enumerate}\n\t\\item Given two random variables $X,Y \\in L^2(\\mathbb{P})$,\n\t\\begin{equation*}\n\t\t\\mathcal{W}_2^2(\\mathbb{P}_X,\\mathbb{P}_Y)\\leq\\mathbb{E}[\\lvert X - Y\\rvert^2],\n\t\\end{equation*}\n\tand in particular $\\mathcal{W}_2(\\mathbb{P}_X,\\delta_0)\\leq\\mathbb{E}[\\lvert X\\rvert^2]^{\\frac{1}{2}}$;\n\t\\item for any $x,y\\in\\mathbb{R}^n$, \n\t\\begin{equation*}\n\t\t\\mathcal{W}^2_2\\left(\\frac{1}{n}\\sum_{i=1}^n\\delta_{x^i},\\frac{1}{n}\\sum_{i=1}^n\\delta_{y^i}\\right)\\leq\\frac{1}{n}\\sum_{i=1}^n\\lvert x^i - y^i\\rvert^2.\n\t\\end{equation*}\n\tIn particular, $\\mathcal{W}^2_2\\left(\\frac{1}{n}\\sum_{i=1}^n\\delta_{x^i},\\delta_{0}\\right)\\leq\\frac{1}{n}\\sum_{i=1}^n\\lvert x^i\\rvert^2$.\n\\end{enumerate}\nThis inequality allows to show that the process $t\\mapsto h(Y_t^{i,n},\\frac{1}{n}\\sum_{j=1}^n \\delta_{Y^{j,n}_{t}})$ is continuous a.s., a condition needed to derive a similar result as Corollary \\ref{opt-i-n}.\n\nFor the convergence results we also need the following law of large numbers (see e.g. Lemma 4.1 in \\cite{djehichedumitrescuzeng} for a proof):\n\\begin{lemma} \n\tLet $Y^1,Y^2,\\dots,Y^n$ be solutions to the system \\eqref{Y-i-law} (notice that they are independent and equally distributed with common probability law $Q$). Then, \n\t\\begin{equation*}\n\t\t\\Lim_{n\\to\\infty}\\mathbb{E}\\left[\\widetilde{\\mathcal{W}}^2_2\\left(\\frac{1}{n}\\sum_{i=1}^n\\delta_{Y^i},Q\\right)\\right] = 0,\n\t\\end{equation*} \n where $\\widetilde{\\mathcal{W}}_2$ is the Wasserstein metric on the space of probability measure on the space $C([0,T]; \\mathbb{R})$ endowed with the supremum norm.\n\\end{lemma} \n\n\\section{Optimal stopping of mean-field SDEs}\\label{ssec-diffusion}\nLet us consider the following mean-field extension of the standard optimal stopping problem of one-dimensional a diffusion process $X$.\nFind a stopping time $\\tau^*$ such that \n\\begin{equation}\\label{Y-0-d}\n\\tau^*=\\underset{\\tau\\in \\mathcal{T}_0}{\\arg\\max}\\, \\mathbb{E}\\left[h(X_{\\tau},\\mathbb{P}_{X_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right],\n\\end{equation}\nwhere $X$ is a diffusion process of mean-field type\n\\begin{equation}\\label{mf-sde}\nX_t=X_0+\\int_0^tb(s,X_s, \\mathbb{P}_{X_{s}})ds+\\int_0^t\\sigma(s,X_s,\\mathbb{P}_{X_{s}})dW_s,\\quad t\\in [0,T],\n\\end{equation}\nwhere $X_0$ is square-integrable and independent of $W$. Here, $\\mathcal{F}_t$ is the $\\mathbb{P}$-completion of $\\sigma(X_0,W_s,s\\le t)$.\n\nThe OSP associated with the MF-SDE \\eqref{mf-sde} is\n\\begin{equation}\\label{Y-0-sde}\nY_0=\\underset{\\tau\\in \\mathcal{T}_0}{\\esssup}\\, \\mathbb{E}\\left[h(X_{\\tau}, \\mathbb{P}_{X_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right].\n\\end{equation}\n\nThe particle system to use to solve this OSP is simply the i.i.d. processes $\\{X^i\\}_{i\\geq1}$ which solve \n\\begin{equation}\\label{mf-sde-i}\nX^i_t=X^i_0+\\int_0^tb(s,X^i_s, \\mathbb{P}_{X^i_{s}})ds+\\int_0^t\\sigma(s,X^i_s,\\mathbb{P}_{X^i_{s}})dW^i_s,\\quad t\\in [0,T],\n\\end{equation}\n and vector $(X^{1,n},\\ldots,X^{n,n})$ of $n$ weakly interacting diffusions defined by\n\\begin{equation}\\label{sde-i-n}\nX^{i,n}_t=X_0^{i}+\\int_0^tb(s,X^{i,n}_s, \\frac{1}{n}\\sum_{j=1}^n \\delta_{X^{j,n}_s})ds+\\int_0^t\\sigma(s,X^{i,n}_s, \\frac{1}{n}\\sum_{j=1}^n \\delta_{X^{j,n}_s})dW^i_s,\\quad t\\in[0,T],\n\\end{equation}\nwhere $(X^1_0,W^1)=(X_0,W)$, which implies that $X^1=X$, and $(X_0^i,W^i)$ are independent and equally distributed.\n\nTo the system \\eqref{mf-sde-i} we associate the OSP\n\\begin{equation}\\label{Y-i-sde}\nY^{i}_0=\\underset{\\tau\\in \\mathcal{T}^i_0}{\\esssup}\\, \\mathbb{E}\\left[h(X^{i}_{\\tau},\\mathbb{P}_{X^i_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\right],\\quad i\\ge 1,\n\\end{equation}\nand the associated family of optimal stopping times \n\\begin{equation}\\label{tau-i-sde}\n    \\hat{\\tau}^{i}=\\inf\\left\\{t\\ge 0, \\,\\, Y^{i,n}_t=h(X^{i}_{t},\\mathbb{P}_{X^i_t})\\right\\}\\wedge T.\n    \\end{equation}\nMoreover, to the system \\eqref{sde-i-n} we associate the family of  OSPs\n\\begin{equation}\\label{Y-i-n-sde}\nY^{i,n}_0=\\underset{\\tau\\in \\mathcal{T}^i_0}{\\esssup}\\, \\mathbb{E}\\left[h(X^{i,n}_{\\tau},\\frac{1}{n}\\sum_{j=1}^n \\delta_{X^{j,n}_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi^i1\\!\\!1_{\\{\\tau=T\\}}\\right],\\quad i=1,2,\\ldots,n,\n\\end{equation}\n and the associated family of optimal stopping times \n\\begin{equation}\\label{tau-i-n-sde}\n    \\hat{\\tau}^{i,n}=\\inf\\left\\{t\\ge 0, \\,\\, Y^{i,n}_t=\\mathbb{E}[h(X^{i,n}_{t},\\frac{1}{n}\\sum_{j=1}^n \\delta_{X^{j,n}_{t}})\\, \\lvert\\,\\mathcal{F}^i_t]\\right\\}\\wedge T.\n    \\end{equation}\n\nProvided that $b$ and $\\sigma$ satisfy conditions which yield (see e.g. Theorem 3 in \\cite{meleard08}):\n\\begin{Assumption}\\label{A3}\\quad\n\\begin{itemize}\n\\item[(1)] Each of the $X^i$'s and $X^{i,n}$'s is in $\\mathcal{S}_c^2$,\n\\medskip\n    \\item[(2)] $\n\\underset{n\\to\\infty}{\\lim}\\mathbb{E}\\left[\\underset{t\\in[0,T]}{\\sup}|X^{i,n}_t-X^i_t|^2\\right]=0,\\quad i\\ge 1,$\n\\end{itemize}\n\\end{Assumption}\nbased on the above results, we obtain the following \n\\begin{theorem}\nUnder Assumptions \\ref{A2} and \\ref{A3}, the hitting time\n\\begin{equation*}\n\\tau^*=\\inf \\{s\\ge 0;\\,\\,  Y_s=h(X_s,\\mathbb{P}_{X_s})\\}\\wedge T\n\\end{equation*}\nsatisfies\n\\begin{equation*}\n\\tau^*=\\underset{\\tau\\in \\mathcal{T}_0}{\\arg\\max}\\, \\mathbb{E}\\left[h(X_{\\tau},\\mathbb{P}_{X_{\\tau}})1\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right].\n\\end{equation*}\n\\end{theorem}\n\n\\subsection{Optimal stopping of the variance of a mean-field diffusion}\nFor $h(\\omega,x,\\mu):=(x-\\int_{\\mathbb{R}^d} y \\mu(d y))^2$ and $\\xi\\ge 0$, we obtain an OSP of the variance:\n\\begin{equation}\\label{Y-0-v}\nY_0=\\underset{\\tau\\in \\mathcal{T}_0}{\\sup}\\,\\mathbb{E}\\left[\\left(X_{\\tau}-\\mathbb{E}[X_{\\tau}]\\right)^21\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right].\n\\end{equation}\nSince $h$ is not Lipschitz continuous we cannot directly apply the above results to claim that the hitting time\n\\begin{equation}\\label{var-tau}\n\\tau^*=\\inf \\{s\\ge 0;\\,\\,  Y_s=(X_s-\\mathbb{E}[X_s])^2\\}\\wedge T\n\\end{equation}\nsatisfies\n\\begin{equation}\\label{var-tau-1}\n\\tau^*=\\underset{\\tau\\in \\mathcal{T}_0}{\\arg\\max}\\, \\,\\mathbb{E}\\left[\\left(X_{\\tau}-\\mathbb{E}[X_{\\tau}]\\right)^21\\!\\!1_{\\{\\tau<T\\}}+\\xi1\\!\\!1_{\\{\\tau=T\\}}\\right].\n\\end{equation}\n\n\\medskip We note that when $X$ is one of the Markov processes considered in \\cite{pedersen11}, the stopping region defined by $\\tau^*$ is the same as the one given in Theorems 3.2, 4.1 and 5.1 in \\cite{pedersen11}.\n \n \\medskip \n Below, we provide a proof that the hitting time $\\tau^*$ defined by \\eqref{var-tau} satisfies \\eqref{var-tau-1}. To this end, using the notation above, we need to show similar results as Proposition \\ref{conv-os-prob} and Corollary \\ref{conv-os-prob-1} which follow  provided that \n\\begin{equation}\\label{var-ex-1}\n\\underset{n\\to\\infty}{\\lim}\\mathbb{E}\\left[\\underset{t\\in[0,T]}{\\sup}\\lvert Y^{1,n}_{t}- Y_{t}\\lvert\\right]=0,\\qquad \\underset{n\\to\\infty}{\\lim}\\mathbb{E}\\left[\\underset{t\\in[0,T]}{\\sup}\\lvert Z^{n}_{t}- Z_{t}\\lvert\\right]=0,\n\\end{equation}\nwhere $Z^n$ and $Z$ are defined as in \\eqref{Z} and\n\\begin{equation}\\label{Y-i-n-example}\nY^{1,n}_t=\\underset{\\tau\\in \\mathcal{T}^1_t}{\\esssup}\\, \\mathbb{E}\\left[(X^{1,n}_{\\tau}-\\frac{1}{n}\\sum_{j=1}^nX^{j,n}_{\\tau})^2\\, \\lvert\\,\\mathcal{F}_t\\right],\\quad Y_t=\\underset{\\tau\\in \\mathcal{T}_t}{\\esssup}\\, \\mathbb{E}\\left[(X_{\\tau}-\\mathbb{E}[X_{\\tau}])^2\\, \\lvert\\,\\mathcal{F}_t\\right].\n\\end{equation}\nIn view of the proofs of Propositions \\ref{conv-1} and \\ref{conv-os-prob}, the limits \\eqref{var-ex-1} hold provided that\n\\begin{equation}\\label{var-ex-2}\n\\underset{n\\to\\infty}{\\lim}\\mathbb{E}\\left[\\underset{t\\in[0,T]}{\\sup}\\lvert \\mathbb{E}[(X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t})^2\\, \\lvert\\,\\mathcal{F}_t]-(X_t-\\mathbb{E}[X_t])^2\\lvert \\right]=0.\n\\end{equation}\nLet us show \\eqref{var-ex-2}. We have\n\\begin{equation*}\n\\begin{aligned}\n    &\\underset{t\\in[0,T]}{\\sup} \\lvert \\mathbb{E}[(X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t})^2\\, \\lvert\\,\\mathcal{F}_t]-(X_t-\\mathbb{E}[X_t])^2\\lvert \\\\\n    &= \\underset{t\\in[0,T]}{\\sup} \\lvert \\mathbb{E}[((X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t}) - (X_t-\\mathbb{E}[X_t]))  ((X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t}) + (X_t-\\mathbb{E}[X_t])) \\lvert\\,\\mathcal{F}_t]\\lvert\\\\\n    &\\leq \\underset{t\\in[0,T]}{\\sup}  \\mathbb{E}[\\underset{s\\in[0,T]}{\\sup}\\lvert(X^{1,n}_{s}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{s}) - (X_s-\\mathbb{E}[X_s])\\rvert\\lvert(X^{1,n}_{s}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{s}) + (X_s-\\mathbb{E}[X_s])\\rvert \\lvert\\,\\mathcal{F}_t]\\\\\n    &\\leq \\underset{t\\in[0,T]}{\\sup}  \\left(\\mathbb{E}[\\underset{s\\in[0,T]}{\\sup}\\lvert(X^{1,n}_{s}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{s}) - (X_s-\\mathbb{E}[X_s])\\rvert^2\\lvert\\,\\mathcal{F}_t]\\right)^{\\frac{1}{2}}\\\\\n    &\\quad\\quad \\underset{t\\in[0,T]}{\\sup}\\left(\\mathbb{E}[\\underset{s\\in[0,T]}{\\sup}\\lvert(X^{1,n}_{s}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{s}) + (X_s-\\mathbb{E}[X_s])\\rvert^2\\lvert\\,\\mathcal{F}_t]\\right)^{\\frac{1}{2}},\n\\end{aligned}\n\\end{equation*}\nwhere in the last inequality we used the Cauchy-Schwarz inequality for the conditional expectation. Then, again by Doob's inequalities, we obtain\n\\begin{equation*}\n\\begin{aligned}\n     &\\mathbb{E}\\left[\\underset{t\\in[0,T]}{\\sup}\\lvert \\mathbb{E}[(X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t})^2\\, \\lvert\\,\\mathcal{F}_t]-(X_t-\\mathbb{E}[X_t])^2\\lvert\\right]\\\\\n     &\\leq 4 \\left(\\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert(X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t}) - (X_t-\\mathbb{E}[X_t])\\rvert^2]\\right)^\\frac{1}{2}\\\\\n     &\\quad\\quad \\left(\\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert(X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t}) + (X_t-\\mathbb{E}[X_t])\\rvert^2]\\right)^\\frac{1}{2}\\\\ & \\qquad\\qquad\\qquad\\qquad\\leq C \\left(\\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert(X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t}) - (X_t-\\mathbb{E}[X_t])\\rvert^2]\\right)^{1\/2},\n\\end{aligned}\n\\end{equation*}\nwhere in the last inequality we have used the fact that the term $\\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert(X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t}) + (X_t-\\mathbb{E}[X_t])\\rvert^2]$ is bounded by a constant $C$ which only depends on the  $\\mathcal{S}^2$-norm of $X$, due to Assumption \\ref{A3} (ii) and the exchangeability of the sequence $\\{X^{j,n}\\}^n_{j=1}$. Furthermore, we have\n\\begin{equation*}\n\\begin{aligned}\n     &\\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert(X^{1,n}_{t}-\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t}) - (X_t-\\mathbb{E}[X_t])\\rvert^2] \\\\ &\\le 2 \\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert X^{1,n}_{t}-X_t\\lvert^2]  +2\\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert\\frac{1}{n}\\sum_{j=1}^n X^{j,n}_{t}-\\mathbb{E}[X_t])\\rvert^2]\\\\ & \\le 2 \\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert X^{1,n}_{t}-X_t\\lvert^2] + 4\\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert\\frac{1}{n}\\sum_{j=1}^n (X^{j,n}_{t}-X^{j}_{t})\\rvert^2] +4\\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert\\frac{1}{n}\\sum_{j=1}^n (X^{j}_{t}-\\mathbb{E}[X_t])\\rvert^2].\n\\end{aligned}\n\\end{equation*}\nAgain, by Assumption \\ref{A3} (ii) and the exchangeability of the process $\\{X^{j,n}-X^j\\}^n_{j=1}$,  the first two terms in the last inequality  go 0 as $n$ goes to infinity. Now,  since the processes $\\{X^j\\}_{j\\geq1}$ are i.i.d. $C([0,T];\\mathbb{R})$-valued random variables with finite second moments (since they are in $\\mathcal{S}^2$), by the strong law of large numbers for Banach-valued r.v. (see Theorem 4.1.1 in \\cite{padgett-taylor06}) and dominated convergence, it holds that\n\\begin{equation*}\n    \\underset{n\\to\\infty}{\\lim}\\mathbb{E}[\\underset{t\\in[0,T]}{\\sup}\\lvert\\frac{1}{n}\\sum_{j=1}^n (X^{j}_{t}-\\mathbb{E}[X_t])\\rvert^2]=0.\n\\end{equation*}\nThis finishes the proof of \\eqref{var-ex-2}.\n\n\n\n\\medskip\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}