# Quantum state of Yang-Mills fields in $SU(\infty)$ Quantum Gravity ($SU(\infty)$-QGR)

Houri Ziaeepour

► To cite this version:

Houri Ziaeepour. Quantum state of Yang-Mills fields in $SU(\infty)$ Quantum Gravity ($SU(\infty)$-QGR). Academia Quantum, 2025, 2 (1), pp.1. 10.20935/AcadQuant7579. hal-04523223

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Submitted on 16 Jul 2025

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# Quantum state of fields in SU($\infty$) Quantum Gravity

¹*,<fnref n="2" />* Houri Ziaeepour

houriziaeepour@gmail.com

## abstract

Our Universe is ruled by quantum mechanics and should be treated as a quantum system. SU($\infty$)-QGR is a recently proposed quantum model for the Universe, in which gravity is associated to SU($\infty$) symmetry of its Hilbert space. Fragmentation of its infinite dimensional state due to random quantum fluctuations divides the Universe to approximately isolated subsystems. In addition to parameters of their internal finite rank symmetries, states and dynamics of subsystems are characterized by 4 continuous parameters and the perceived classical spacetime is their effective representation, reflecting quantum states of subsystems and their relative evolution. At lowest order the effective Lagrangian of SU($\infty$)-QGR has the form of Yang-Mills gauge theories for both SU($\infty$) - gravity - and internal symmetries defined on the aforementioned 4D parameter space. In the present work we study more thoroughly some of the fundamental aspects of SU($\infty$)-QGR. Specifically, we clarify the impact of the degeneracies of SU($\infty$) algebra on the construction of the model; describe mixed states of subsystems and their purification; calculate measures of their entanglement to the rest of the Universe; and discuss their role in the emergence of local gauge symmetries. We also describe the relationship between what is called internal space of SU($\infty$) Yang-Mills with the 4D parameter space, and analytically demonstrate irrelevance of the geometry of parameter space for physical observables. Along with these topics, we demonstrate the equivalence of two sets of criteria for compositeness of a quantum system, and show uniqueness of the limit of various algebras leading to SU($\infty$).

**Keywords** quantum gravity; SU($\infty$) symmetry; Yang-Mills theory

## Contents

1 Introduction to SU($\infty$)-QGR 2

1.1 Summary of main results 4

2 Multiple limits of SU($N \to \infty$) and their consequence for SU($\infty$)-QGR 5

3 Order from randomness: Emergence of subsystems 6

3.1 Decomposition to finite rank symmetries 7

3.1.1 Secondary parameters of SU($\infty$) representations of subsystems 9

3.2 Quantum states of subsystems 9

<footnote n="1">¹Institut UTINAM, CNRS UMR 6213, Observatoire de Besançon, Université de Franche-Comté, 41 bis ave. de l'Observatoire, BP 1615, 25010 Besançon, France</footnote>

<footnote n="2">²Mullard Space Science Laboratory, University College London, Holmbury St. Mary, GU5 6NT, Dorking, Surrey, UK</footnote>
<table>
  <tbody>
    <tr>
      <td>3.2.1</td>
      <td>Significance of entanglement</td>
      <td>10</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>Emergence of gauge symmetries</td>
      <td>11</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Purified states of subsystems</td>
      <td>11</td>
    </tr>
    <tr>
      <td>3.4.1</td>
      <td>Physical interpretation of purified states</td>
      <td>12</td>
    </tr>
    <tr>
      <td>3.5</td>
      <td>Emergence of area/size parameter</td>
      <td>13</td>
    </tr>
    <tr>
      <td>3.5.1</td>
      <td>Consistency of the two perceptions of the quantum Universe</td>
      <td>14</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Dynamics over a parameter space</td>
      <td>15</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>SU(∞)-QGR as a Yang-Mills QFT on the parameter space of subsystems</td>
      <td>15</td>
    </tr>
    <tr>
      <td>4.1.1</td>
      <td>Classical spacetime from quantum gravity action</td>
      <td>16</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Equivalence of diffeomorphism of parameter space and SU(∞) gauge transformation</td>
      <td>17</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Perspectives for test and applications of SU(∞)-QGR</td>
      <td>19</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Difference in coherence of astronomical masers as a signature of QGR</td>
      <td>19</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Perspective for further investigations</td>
      <td>20</td>
    </tr>
    <tr>
      <td>A</td>
      <td>A brief review of SU(∞) representations</td>
      <td>21</td>
    </tr>
    <tr>
      <td>B</td>
      <td>No exact description of sine algebra on a 2D surface</td>
      <td>23</td>
    </tr>
    <tr>
      <td>C</td>
      <td>Equivalence of amplitude factorization and algebraic criteria for compositeness</td>
      <td>24</td>
    </tr>
    <tr>
      <td>D</td>
      <td>Homomorphism of SU(∞) and G × SU(∞)</td>
      <td>25</td>
    </tr>
    <tr>
      <td>E</td>
      <td>Entanglement measures</td>
      <td>26</td>
    </tr>
    <tr>
      <td>E.1</td>
      <td>Entanglement entropy</td>
      <td>26</td>
    </tr>
    <tr>
      <td>E.2</td>
      <td>Entanglement negativity</td>
      <td>26</td>
    </tr>
    <tr>
      <td>F</td>
      <td>Conditions for perfect purification</td>
      <td>27</td>
    </tr>
    <tr>
      <td>G</td>
      <td>SU(∞) Yang-Mills QFT on a background spacetime</td>
      <td>29</td>
    </tr>
    <tr>
      <td>H</td>
      <td>Quantum fields and density matrices</td>
      <td>29</td>
    </tr>
  </tbody>
</table>

# 1 Introduction to SU(∞)-QGR

SU(∞)-QGRis a fundamentally quantum approach to cosmology and gravity. Its rationale, foundation and essential structure and properties are investigated in [1, 2]. It is reviewed in [3, 4] and compared with several popular Quantum Gravity (QGR) models in [5].

We begin by a nutshell description of SU(∞)-QGR. The essential axiom of this model is the existence of infinite number of mutually commuting observables in a quantum Universe. Consequently, the Hilbert space $\mathcal{H}_U$ of the Universe represents SU(∞) symmetry group [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. There is no background spacetime in this model and the usual quantization procedure is replaced by non-commutative SU(∞) algebra of linear operators $\hat{O} \in \mathcal{B}[\mathcal{H}_U]$ acting on $\mathcal{H}_U$ [17, 18].

It is demonstrated [1] that as a whole this Universe is static and trivial, in the sense that any state can be transformed to another one by a SU(∞) transformation under which observables are conserved, Nonetheless, quantum fluctuations, that is random application of $\hat{O} \in \mathcal{B}[\mathcal{H}_U]$, lead to approximate fragmentation of its state, and thereby the Hilbert space $\mathcal{H}_U$, and an approximate division to subsystems according to the criteria defined in [19].
We remind that the process of Hilbert space fragmentation of closed quantum systems to subsystems with their own symmetries and behaviour is observed in many condensed matter contexts, see e.g. [22] and references therein. Indeed, this topic is currently under intensive theoretical [20, 21] and experimental [22] investigation, and is reviewed in [23, 24]. Due to many-body nature of these systems and existence of a large number of degrees of freedom, dominance of long range electromagnetic interactions, and isolation of the whole system they can be considered as *mini-universes* which their Hilbert spaces represent $SU(N \to \infty)$ symmetry. Therefore, it is not unreasonable to assume that the Hilbert space $H_U$ of the whole Universe - the only truly isolated many-body quantum system - represents $SU(\infty)$ symmetry and becomes fragmented to subsystems by random quantum fluctuations.

The clustered and fragmented blocks in $\mathcal{B}[H_U]$ represent finite rank internal symmetries, which we collectively call $G$. Moreover, the global $SU(\infty)$ becomes the symmetry of their infinite dimensional environment consisting of other subsystems entangled to them. Hence, the Hilbert space of a subsystem represents $G \times SU(\infty)$ symmetry. As a consequence, representations of $SU(\infty)$ by subsystems become comparable, and a dimensionful area or length parameter is added to the list of parameters characterizing quantum state of subsystems. In addition, a subsystem can play the role of a quantum clock and define a relative time and dynamics à la Page and Wootters [25, 26]. Mutual entanglement of all subsystems, including the clock, with their environment ensures a synchronous relative variation and an arrow of time.

Using Mandelstam-Tamm inequality [27] - the quantum speed limit - it is shown that a classical spacetime emerges as an effective manifestation of the (1+3)-dimensional parameter space of the Hilbert space of $SU(\infty)$ symmetry of the subsystems and their relative dynamics. Moreover, it is proved that signature of the metric of this emergent spacetime is negative. Thus, $SU(\infty)$-QGR relates the Lorentzian geometry of classical spacetime to quantum states of the content of the Universe and their dynamics. However, $SU(\infty)$-QGR does not explore the geometry of the Hilbert space<fnref n="1" /> per se. Rather, it relates the perceived spacetime and its geometry to parameter space of quantum states. It is also shown that the geometry of the parameter space is irrelevant for physical observables and can be gauged out by a $SU(\infty)$ transformation.

A functional constructed from invariants of $SU(\infty)$ and $G$ plays the role of action analogous to those of Quantum Field Theories (QFT's). At lowest order of symmetry invariant traces this effective action is similar to non-Abelian Yang-Mills models for both $SU(\infty)$ and $G$ symmetries. There is however an essential difference: the action is defined on the (3+1)D space of continuous parameters of the model, rather than a classical spacetime. Due to the Yang-Mills nature of interactions, the mediator boson of quantum gravity in $SU(\infty)$-QGR is a spin-1 field. Nonetheless, it is shown that at classical limit - that is when quantum effects of gravity are not detectable - the gravity sector dynamics is ruled by the Einstein-Hilbert action<fnref n="2" />.

The aim of present work is to discuss in more details and clarify some of the technical issues that were not addressed in the previous works, in particular in [2] where properties of $SU(\infty)$-QGR summarized in the above paragraphs are studied. Here we only remind those details which are necessary for understanding of the content of this work. For complementary explanations and demonstrations the reader should refer to [1, 2]. The topics discussed here are necessary both for

<footnote n="1">¹Geometry has been first introduced in the Hilbert space of many-body quantum systems in order to study collective properties, such as deformation of nuclides, see [28] and references therein. Considering an Euclidean $\mathbb{R}^n$ parameter space for the quantum states, they used variation of state and its path in the Hilbert space to define a symmetric 2-tensor as metric and a 2-form as symplectic structure interpreted as *induced geometry* of parameter space. This geometric approach helps to study complicated variations of the quantum state of many-body systems, see e.g. [29] for an application example of this method.</footnote>

<footnote n="2">²The non-commutative algebra of $SU(\infty)$ can be normalized such that it depends on $\hbar/M_P$, where $M_P$ and $\hbar$ are Planck mass and constant, respectively. This normalization is necessary for making pure gravity - $SU(\infty)$ gauge - term of the action to have a dimensionality similar to the scalar curvature in the Einstein-Hilbert action. In the standard classical limit i.e. $\hbar \to 0$ or in absence of gravity, i.e. $M_P \to \infty$ the algebra becomes Abelian and the model trivial.</footnote>
reinforcing the construction and consistency of the model, and for preparing the ground for its applications to topics such as black hole information puzzle and prediction of outcomes of future laboratory tests of quantum gravity. The main results reported here are summarized in Sec. 1.1

In Sec. 2 we clarify the issue of the degeneracy of $SU(N \to \infty)$ algebra and its impact on $SU(\infty)$-QGR's parameter space. Emergence of subsystems, their Hilbert spaces, and entanglement of their states with the rest of the Universe is investigated in Sec. 3. In order to provide explicit and quantitative proof of the Proposition 1 in [2], we quantify the entanglement by calculating several of its measures in Appendix E, and compare the results with QGR models that try to explain gravity as a consequence of entanglement [30, 31, 32]. In Sec. 4 we review dynamics of the Universe and its subsystems according to $SU(\infty)$-QGR and provide an analytical description of the relationship between $SU(\infty)$ symmetry of the Hilbert space and diffeomorphism of its parameter space. Irrelevance of the geometry of parameter space is the subject of Proposition 2 in [2]. Here we investigate and demonstrate this crucial property in more details. Finally, in Sec. 5 we briefly outline an experiment/observation which may be able to detect signature of $SU(\infty)$-QGR with present facilities. We also discuss topics left for future investigations. Appendices include details of arguments, calculations, and reviews of subjects necessary for this work.

## 1.1 Summary of main results

Regarding $SU(\infty)$-QGR model we reinforce demonstration of Propositions 1 and 2 of [2], calculate quantum states of subsystems of the Universe, and investigate their properties:

**Emergence and properties of structures in quantum state of the Universe:** In [2] we employed an operational approach to show how structures may emerge in the quantum state of the Universe. We demonstrated that they can be approximately treated as quantum subsystems. Using those results, here we construct their states in details. In particular, we calculate mixed states of subsystems with finite rank symmetries and their infinite dimensional environment. We quantify their entanglement by determining their mutual information (entanglement entropy) and quantum negativity. We demonstrate the emergence of secondary parameters related to $SU(\infty)$ symmetry and relative dynamics of subsystem. They include two continuous parameters, namely a quantity presenting relative area or size of diffeo-surfaces of $SU(\infty)$ representations by subsystems, and a time variable. In addition, states may depend on two discrete parameters characterizing the orthogonal function basis used for construction of the generators of $SU(\infty)$ algebra. Mixed states of subsystems depend on these parameters and two continuous parameters characterizing $SU(\infty)$ representations. We show that these states are similar to fields in QFT's with gauge symmetry, for example those of the Standard Model (SM). Thus, $SU(\infty)$-QGR relates the origin of the observed gauge symmetries of fundamental interactions to QGR. On the other hand, quantum mixed nature of these states and their dependence on discrete quantum numbers, unrelated to their local symmetries, deviate from pure state of the SM particles. They reflect quantum origin of the 4 continuous parameters, which their effective values is shown in [1, 2] to be the perceived classical spacetime. These features might be testable in future experiments seeking signatures of quantum gravity.

**Purification:** Quantum origin of the 4D continuous parameter space of subsystems is made transparent by purification of their mixed states. We demonstrate that states of subsystems satisfy conditions for perfect purification according to [33]. Moreover, using effective classical metric mentioned above, we show its relationship with the quantum state of environment seen by subsystems. In turn, we show that in accordance with results of [34] the mixed state of environment is perspective dependent and related to the internal symmetries of the subsystem. This is a remarkable result, because in general relativity, the relation between spacetime geometry and matter is established through dynamics, that is
the Einstein-Hilbert action and Einstein equation. By contrast, in $SU(\infty)$-QGR it arises through introduction of a quantum clock and before introducing an action.

**Clarification of relationship between 4 continuous parameters of subsystems and their $SU(\infty)$ symmetry:** Association of 4D continuous parameters to $SU(\infty)$ symmetry of subsystems seems confusing, because representations of $SU(\infty)$ depend on two continuous parameters. We show that purified states are composed of two entangled components, one representing a finite rank symmetry $G$ and the other $SU(\infty)$. Using the fact that sine algebra represents $SU(\infty) + U(1)$, we demonstrate that in presence of multiple representations of $SU(\infty)$, their individual $U(1)$ symmetry breaks (see diagrams (21) and (22)), because one can define a morphism between their algebras - in other words between their associated 2D diffeo-surfaces. A consequence of this morphism is that relative area of these compact surfaces becomes an observable. Moreover, with introduction of a quantum clock and time parameter, quantum states of subsystems will depend on 4 continuous parameters, which are indistinguishable and quantum states are in their coherent superposition. Additionally, arbitrariness of references and clocks dictates that observables must be independent of reparameterization and diffeomorphism of the parameter space. Therefore, we conclude that 2D diffeo-surfaces of $SU(\infty)$ representations are subspaces of the 4D parameter space (see Fig. 1).

**Equivalence of $SU(\infty)$ gauge transformation and diffeomorphism of the parameter space of subsystems:** We provide an analytical description of the relationship between $SU(\infty)$ symmetry of the Hilbert space and diffeomorphism of the space of its continuous parameters. We discuss both the case of the Universe as a whole, where there is only one representation of $SU(\infty)$, and the case of the ensemble of subsystems representing $SU(\infty)$ and their relative dynamics, where diffeo-surfaces of $SU(\infty)$ representations are 2D subspaces of a 4D space. We show that diffeomorphism of this parameter space can be neutralized by a $SU(\infty)$ gauge transformation. Therefore, its geometry is irrelevant for the Lagrangian functional of the model. This demonstration reinforce the proof of Proposition 2 of [2].

Additionally, we demonstrate the following general properties of $SU(\infty)$ symmetry group and criteria for considering a quantum system as being composite. They were necessary for obtaining the results summarized in the previous paragraph.

**Multiple limits of $SU(\infty)$:** We show that the degeneracy of $SU(N \to \infty)$ algebra demonstrated in [14] does not have impact on the construction and properties of $SU(\infty)$-QGR parameter space. Although the sine algebras (42) for $k \neq 2\pi/N$ where $N \in \mathbb{Z}$ are not homomorphic to each others or to $k = 2\pi/N$, they all have the same limiting algebra when $k \to 0$, namely (39) which is homomorphic to the algebra of area preserving diffeomorphism $ADiff(D_2)$ of 2D compact Riemannian surfaces. In the framework of $SU(\infty)$-QGR it is this limit that we call $SU(\infty)$. Thus, there is no ambiguity in the definition of parameters characterizing representations of $SU(\infty)$ in the model.

**Criteria for compositeness:** We show that two sets of conditions used in the literature as indicator of compositeness of a quantum system, namely factorization of density matrix or state vector [35] and factorization of the algebra of observables [19] are equivalent.

## 2 Multiple limits of $SU(N \to \infty)$ and their consequence for $SU(\infty)$-QGR

The group $SU(\infty)$, its representations and algebra, and their properties are briefly reviewed in Appendix A. Variables and symbols related to these entities are defined there.

A particularity of $SU(N \to \infty)$ is its non-uniqueness [14]. There are various ways to see this. For instance, if $N \to \infty$ and $N' \to \infty$ are prime with respect to each other, or more generally for
irrational $k$'s and integer vectors $(\mathbf{m}, \mathbf{n})$ in (42) - the case considered by [12, 13, 14] - structure coefficients of the algebras would be always different, and thereby they would not be homomorphic to each others. In [14] it is proved that for any $k \in \mathbb{C}$, the algebra (43) is homomorphic to $SU(\infty) + U(1)$. Moreover, it is shown that for $k \in \mathbb{R}$, algebras with different values of $k$ are not pairwise homomorphic. Consequently, it seems that $SU(N \to \infty)$ and its algebra have uncountable many degeneracies. Additionally, for irrational $k$ such that $1/k \to \infty$ these algebras do not have a shift symmetry similar to (48) and cannot be homomorphic to $ADiff$ of any orientable Riemann surface. We remind that the proof of $SU(\infty) \cong ADiff(D_2)^3$ in [9, 16] is limited to the case of rational $k$ (or equivalently countable $N$ in (43)). Furthermore, in Appendix B we demonstrate that the description of generators $\hat{K}_m$ of (42) as functionals of $\mathbf{x}$ and $\partial_\mathbf{x}$ proposed in [12] generates this algebra only approximately, and for $k \neq 0$ no other functional of these variables can generate the algebra exactly. This finding confirms the conclusion of [14], asserting that some of (42) algebras are not homomorphic to $ADiff(D_2)$.

In what concerns the application of $SU(\infty)$ in string and matrix (M-)theories as candidate quantum gravity, the absence of its homomorphism with $ADiff(D_2)$ can be considered as a serious problem. Indeed, the motivation for investigating the relationship between $SU(\infty)$ and $ADiff(D_2)$ in the 1980's and 1990's was using $N \times N$ matrices as symplectic approximation of strings worldsheets or more generally membranes [9, 15]. In this context homomorphism of $SU(\infty)$ and $ADiff(D_2)$ is crucial, because quantum gravitational interactions are associated to deformation of $D_2$ surfaces and vis-versa. Thus, homomorphism between geometrical and algebraic operations is crucial.

By contrast, $SU(\infty)$ symmetry of the Hilbert space in $SU(\infty)$-QGR arises because of the assumption of infinite number of mutually commuting observables. Although the homomorphism of $SU(\infty)$ representations with $ADiff(D_2)$ is also important, specially for providing two continuous parameters, it is a subsidiary property. As discussed in details in [2] and in the following sections here, these parameters along with two other continuous variables, namely area or size/distance and time, characterize $SU(\infty)$ and dynamics related aspects of the quantum states of subsystems/particles. On the other hand, it is easily seen that $SU(\infty)$ algebra always depends on two variables, irrespective of the degeneracy of algebra (43) and whether it is homomorphic to $ADiff(D_2)$ or not. Indeed, $SU(N), \forall N$ is a $N$-dim representation of $SU(2)$. As the group manifold of $SU(2)$ is the sphere $S^{(2)}$, its generators in any representation, including $SU(\infty)$, can be described as functions of angular coordinates of sphere, see [6] for details$^4$. Therefore, diffeomorphism of $SU(\infty)$ and $ADiff(D_2)$ is less crucial for $SU(\infty)$-QGR.

In any case, it is important to remind that both (42) and (43) algebras are homomorphic to $SU(\infty)$ only for $1/k \to \infty$. The expression (49) of the generators of (42) also becomes exact in this limit, which corresponds to the algebra (39) and homomorphic to $ADiff(D_2)$. Therefore, although for $k \neq 0$ (42), (43), and (49) present different algebras, they all converge to a unique limit, namely the algebra (39), which is homomorphic to Poisson algebra, and as demonstrated in Appendix A, to $ADiff(D_2)$. Thus, from now on and in the context of $SU(\infty)$-QGR model by $SU(\infty)$ we always mean this unique limit.

### 3 Order from randomness: Emergence of subsystems

In [1, 2] we showed that $SU(\infty)$-QGR Universe as a whole is static and trivial, in the sense that different states can be transformed to each others by application of $SU(\infty)$ members, under

<footnote>$^3$In this work we use the symbol $\cong$ for homomorphism, because the symbol $\simeq$ used for this purpose in mathematics literature is used for numerical approximation in physics literature.</footnote>

<footnote>$^4$It is easy to see that even without using association with $ADiff(D_2)$, algebras (43) and (42) also depend on two approximately continuous variables. Indeed, generators $\hat{K}_m$ with $\sqrt{2\pi/N}\mathbf{m} \to (\infty, \infty)$ are dominant by their number. As fractional numbers are dense in $\mathbb{R}$, vectors $\sqrt{2\pi/N}\mathbf{m}$ are approximately continuous 2-vectors.</footnote>
which according to the axioms of the model the physics is invariant. Only when the Universe is divided to approximately isolated subsystems, structures and relative dynamics arise from their interactions and entanglement. In this section we study in more details quantum state of a subsystem and its relationship with the rest of the Universe, formed by the ensemble of other subsystems.

Consider a quantum system with a $N$-dimensional Hilbert space representing $SU(N)$ group. Assuming $N \gg 1$, the Cartan decomposition of $SU(N)$ [36] can be used to describe the symmetry, and thereby the Hilbert space, as tensor product of smaller rank Lie groups:

$$ \{|\psi_1\rangle\} \times \{|\psi_2\rangle\} \cdots \times \{|\psi_n\rangle\} \qquad (1) $$

where the $n_i$-dimensional set $\{|\psi_i\rangle\}$ is a basis for the $i^{th}$ component of the Cartan decomposition. Notice that (1) has the same structure as the Hilbert space of an n-body system. In general, $N$, the number of components $n$, and dimensions $n_i$ can be infinite or even innumerable. In the case of $SU(\infty)$-QGR $N = n = \infty$.

The Cartan decomposition of $SU(\infty)$ group is reviewed in Appendix A of [2]. It is shown that it can be decomposed to tensor product of infinite number of $SU(K)$, i.e $SU(\infty) \cong \bigotimes^{\infty} SU(K)$ for any $K$, including infinity. More generally, $SU(\infty)$ can be written as tensor product of any finite rank compact Lie group $G$ [16]. Thus, there is a basis similar to (1) for the Hilbert space of the Universe $\mathcal{H}_U$ with $n_1.n_2.\cdots.n_i \to \infty$ components, corresponding to the decomposition $SU(\infty) = G^1 \times G^2 \times \cdots$, where $n_i$ is dimension of representation $\{|\psi_i(k_i)\rangle\}$ for the group $G^i$. The $d_i$ dimensional set $\{k_i\}$ indicates space of parameters characterizing representation of $G^i$. With this decomposition the state of the Universe $|\Psi_U\rangle$ can be expanded as:

$$ |\Psi_U\rangle = \sum_{\{k_i=1,\cdots,d_i\}, i=1,\cdots,n \to \infty} \mathcal{A}(k_1k_2\cdots k_n, \cdots) |\psi_{k_1}\rangle \times |\psi_{k_2}\rangle \times \cdots \times |\psi_{k_n}\rangle \cdots \qquad (2) $$

Notice that we have simplified the notation by using $k_i$ as an index. Thus, $|\psi_{k_i}\rangle \equiv |\psi_i(k_i)\rangle$.

Description of a basis as a tensor product does not necessarily mean that the corresponding quantum system consists of subsystems [19]. In [1, 2] we used algebraic criteria introduced in [19] and random quantum fluctuations to explain how the concepts of locality and approximate division to subsystems arise in the Universe. An equivalent but more explicit criterion for definition of subsystems using a tensor product basis, as in (2), is the factorization of amplitudes $A_{k_1k_2\cdots k_n}$. States for which these amplitudes can be decomposed to factors, depending only on a limited number of parameters that characterize $\bigotimes_i^{n_i} |\psi_{k_i}\rangle$, are called *separable* or *product states* [35]. They present partial localization in the Hilbert space.

The factors $A_{k_1k_2\cdots k_n}$ depending on a single $k_i$ expand subspaces with no entanglement. By contrast, factors depending on $n' \to \infty$ parameters expand inseparable and fully nonlocal$^5$ subspaces of $\mathcal{H}_U$. In Appendix C we demonstrate that factorization of state vector or density matrix is equivalent to the algebraic criteria defined in [19] for the compositeness of a quantum system.

## 3.1 Decomposition to finite rank symmetries

In $SU(\infty)$-QGR when dealing with the whole Universe it is not useful to consider a tensor decomposition like (2) for its state $|\Psi_U\rangle$, because it does not survive a $SU(\infty)$ transformation, which preserves physical observables. Indeed, in [2, 1] we demonstrated that due to this global symmetry $|\Psi_U\rangle$ is trivial - the vacuum - except for the dependence of $SU(\infty)$ representation on the topology of its diffeo-surface. Nonetheless, states like $|\Psi_U\rangle$ can have very different components. Specifically, in [2, 1] we showed that quantum fluctuations, that is random application

<footnote>$^5$Unless locality in spacetime is explicitly mentioned, here *locality* means in the Hilbert space, according to the definition given in Appendix C.</footnote>
of operators $\hat{O} \in \mathcal{B}[\mathcal{H}_U]$ on $|\Psi_U\rangle$, induce clustering of components. In particular, a completely coherent state in an arbitrary basis changes to a less coherent state with more clustered and unbalanced components. In such state, subsets of elements with very different amplitudes from their neighbours can be approximately considered as subsystems. Hence, the state $|\Psi_U\rangle$ can be approximately factorized to a tensor product of subspaces of $\mathcal{H}_U$ where each factor represents a smaller rank symmetry.

Consider one of the factors in (2) and assume that it represents a finite rank symmetry $G$. The ensemble of remaining factors is still infinite dimensional and represents $SU(\infty)$ [16]. To prove this property we use decomposition of $SU(\infty)$ to a finite rank $SU(K)$, i.e. $SU(\infty) \cong \bigotimes^{\infty} SU(K)$, $K \in \mathbb{Z}^+ + 2$. For any finite rank compact Lie group $G$ one can find $K \in \mathbb{Z}^+ + 2$, such that $SU(K') \subset G \subset SU(K)$, $K' < K$, and the following relations are satisfied:

$$SU(\infty) \cong \bigotimes^{\infty} SU(K') \subset G \times \bigotimes^{\infty} SU(K') \subset G \times \bigotimes^{\infty} SU(K) \subset SU(K) \times \bigotimes^{\infty} SU(K) \cong SU(\infty) \quad (3)$$

Moreover, this chain shows that for any finite rank $G$:

$$G \times SU(\infty) \cong SU(\infty) \quad (4)$$

The difference between $SU(\infty)$'s in the two extremes of (3) is similar to a line without any specific point on it - analogous to the left extreme of (3) - and a line with a finite number of selected points - analogous to the right extreme of (3). In both cases the set of points constituting the line are the same. But, in the second case some points are interpreted, employed or perceived differently. A detailed demonstration of (4) is provided in Appendix D.

We call the basis that expands states representing $G$ symmetry $\{|\psi_G(k_G)\rangle\}$, where $k_G$ collectively represents parameters of the representation of $G$ by this basis. For finite rank Lie groups the number of independent $k_G$'s, that is the dimension $d_G$ of parameter space $\{k_G\}$ is finite. For instance, for $G = SU(2)$, $k_G = (l', m')$, $l' \in \mathbb{Z}^+$, $-l' \le m \le l'$. For a fixed $l'$, corresponding to a super-selected representation of $SU(2)$, the dimension $d_G = 2l' + 1$.

Following the tensor decomposition (3), the state of the Universe can be expanded as:

$$|\Psi_U\rangle = \sum_{\{k_G\}, (\eta, \zeta, \dots)} A(k_G; \eta, \zeta, \dots) |\psi_G(k_G)\rangle \times |\psi_\infty(\eta, \zeta, \dots)\rangle \quad (5)$$

and the corresponding density matrix:

$$\hat{\rho}_U = \sum_{\{k_G, k'_G\} \quad (\eta, \zeta, \eta', \zeta', \dots)} A(k_G; \eta, \zeta, \dots) A^*(k'_G, \eta', \zeta', \dots) \hat{\rho}_G(k_G, k'_G) \times \hat{\rho}_\infty(\eta, \zeta, \eta', \zeta', \dots) \quad (6)$$

$$\hat{\rho}_G(k_G, k'_G) = |\psi_G(k_G)\rangle\langle\psi_G(k'_G)|, \quad \hat{\rho}_\infty(\eta, \zeta, \eta', \zeta', \dots) = |\psi_\infty(\eta, \zeta, \dots)\rangle\langle\psi_\infty(\eta', \zeta', \dots)| \quad (7)$$

$$\sum_{\{k_G\} \quad (\eta, \zeta, \dots)} |A(k_G; \eta, \zeta, \dots)|^2 = 1 \quad (8)$$

where states $|\psi_\infty(\eta, \zeta, \dots)\rangle$ constitute a basis for $SU(\infty)$ factor of $\mathcal{H}_U$ and continuous parameters $(\eta, \zeta)$ characterize generators of $SU(\infty)$, see Appendix A for details. Extension dots indicate secondary parameters defined in the following subsection. When they are irrelevant or do not contribute to the discussion we replace them with extension dots $'' \dots''$. Notice that we have used an arbitrary basis for $G$ and $SU(\infty)$ factors. Consequently, in general off-diagonal elements of the density matrix are not zero. Moreover, despite the tensor product structure of the basis in (5) and (6), amplitudes $A(k_G; \eta, \zeta, \dots)$ are not exactly factorizable, because a global $SU(\infty)$ transformation changes the tensor product decomposition of $|\Psi_U\rangle$ and $\hat{\rho}_U$. Therefore, isolation of a subsystem is always an approximation.
### 3.1.1 Secondary parameters of SU($\infty$) representations of subsystems

Secondary parameters in the basis $|\psi_{\infty}(\eta, \zeta; \cdots)\rangle$ are those that do not affect a single representation of SU($\infty$), but can be involved in characterization of SU($\infty$) represented by Hilbert spaces of multiple quantum systems/subsystems interacting with each others. One of these parameter $r$ indicates the relative area or a size scale for diffeo-surfaces of SU($\infty$) representations in the model with respect to that of a reference. As discussed in Appendix A, the algebra ADiff($D_2$) is homomorphic to SU($\infty$) + U(1), where U(1) presents scaling of the diffeo-surface. This symmetry is irrelevant - unobservable - for the state of a single quantum system representing SU($\infty$) irreducibly, such as $\hat{\rho}_U$. On the other hand, area or size scale of diffeo-surfaces of multiple (sub)systems representing SU($\infty$) becomes a relative observable. This is the case when the Universe is divided to subsystems each representing SU($\infty$) symmetry.

Another secondary parameters is time $t$ characterizing the quantum state of a clock subsystem. In [2, 1] we showed that in this way subsystems acquire a relative dynamics à la Page & Wootters [25, 26] or equivalent approaches [37]. As time measurement may be POVM rather than projective, in SU($\infty$)-QGR quantum states of subsystems are in general in coherent superposition of 4 continuous parameters $x \equiv (\eta, \zeta, r, t)$, where $(\eta, \zeta)$ are parameters of SU($\infty$) representation. Assuming compact ranges for the last two parameters, i.e. using the fact that $\mathbb{R} \cong U(1)$, or non-compact values for $\eta, \zeta$, the 4 parameters become indistinguishable components of a 4D vector $x \equiv (x^i, i = 0, \cdots, 3) \in S_4(\mathbb{R}^4)$, of a compact or non-compact parameter space that we call $\Xi$. Logically, observables of subsystems should be invariant under reparameterization of $\Xi$. Moreover, Proposition 2 of [2] demonstrates that the geometry of $\Xi$ is irrelevant for observables and can be modified by a SU($\infty$) transformation under which the physics is invariant. This property will be further discussed in Sec. 4.2

As generators of ADiff($D_2$) $\cong$ SU($\infty$) act on continuous functions of $(\eta, \zeta)$, they are usually expanded with respect to a set of orthogonal functions as basis. These functions may in turn depend on a set of usually discrete parameters that we collectively call $\ell$. For example, for a sphere basis $\ell$ corresponds to the spherical harmonic modes $(l, m)$, see appendices of [1] for a brief review. For a torus basis, as discussed in Appendix A, $\ell = \mathbf{m}$, where $\mathbf{m}$ is an integer 2D vector. We emphasize that $\ell$ is not an independent parameter of the model, because as it is shown in [1], the algebra SU($\infty$) and its generators can be formally expressed with respect to two continuous parameters only.

## 3.2 Quantum states of subsystems

To extract from the factorized $\hat{\rho}_U$ in (6) a state that represents only the group $G$, we trace out its SU($\infty$) representing factor:

$$ \hat{\rho}_G \equiv \text{tr}_{\infty} \hat{\rho}_U = \sum_{\substack{k_G, k'_G \\ \{y\}}} A_G(k_G; y) A_G^*(k'_G, y) \hat{\rho}_G(k_G, k'_G), \quad y \equiv (\eta, \zeta, \cdots) \qquad (9) $$

The amplitude $A_G$ differs from $A$ in (6) by a normalization factor, which may in general depends on $(\eta, \zeta, \cdots)$. We call the Hilbert space generated by $\hat{\rho}_G$ density matrices $\mathcal{H}_G$. It represents the symmetry group $G$, because the basis $\hat{\rho}_G(k_G, k'_G)$ represents this symmetry. Nonetheless, amplitudes $A_G(k_G; \eta, \zeta, \cdots)$ and their complex conjugate $A_G^*$ depend on parameters $(\eta, \zeta, \cdots)$, which are not related to the $G$ symmetry. An observer ignoring their quantum origin interprets them as being related to a classical environment. On the other hand, application of any operator $\hat{O} \in \mathcal{B}(\mathcal{H}_U) \cong SU(\infty)$ changes $\hat{\rho}_U$, and consequently the dependence of $A_G(k_G; \eta, \zeta, \cdots)$ on the environment parameters $(\eta, \zeta, \cdots)$. Therefore, $\hat{\rho}_G$ presents the mixed quantum state of an open subsystem. To show explicitly the mixedness of $\hat{\rho}_G$ it is enough to demonstrate that $\hat{\rho}_G^2 \neq \hat{\rho}_G$. It
straightforward to show that:

$$ \hat{\rho}_G^2 = \sum_{\{\{y\}, \{\{y'\}\}\}} B(y, y') \sum_{\{k_G\}, \{k'_G\}} A_G(k_G; y) A_G^*(k'_G; y') \hat{\rho}_G(k_G, k'_G) \quad (10) $$

$$ B(y, y') = \sum_{\{k''_G\}} A_G^*(k''_G; y) A_G(k''_G; y') \quad (11) $$

Comparison of (10) and (9) clearly shows that $\hat{\rho}_G^2 \neq \hat{\rho}_G$, except in the limit where amplitudes $A_G$ do not depend on the value of $y$. Thus, $\hat{\rho}_G$ is in general a mixed state and the subsystem representing $G$ is entangled to the rest of the Universe.

In a similar manner $\hat{\rho}_G(k_G)$ factor in $\hat{\rho}_U$ can be traced out to find the contribution of $SU(\infty)$ symmetry in $\hat{\rho}_U$ after separating a subsystem with $G$ symmetry:

$$ \hat{\rho}_{\infty} \equiv \text{tr}_G \hat{\rho}_U = \sum_{\{k_G\}, \{(\eta, \zeta, \dots)\}, \{(\eta', \zeta', \dots)\}} A_{\infty}(k_G; \eta, \zeta, \dots) A_{\infty}^*(k_G; \eta', \zeta', \dots) \hat{\rho}_{\infty}(\eta, \zeta; \eta', \zeta', \dots) \quad (12) $$

We call the Hilbert space generated by $\hat{\rho}_{\infty}$ states $\mathcal{H}_{\infty}$. Using analogous expressions to (10 - 11) for $\hat{\rho}_{\infty}$, it is straightforward to show that $\hat{\rho}_{\infty}$ is also a mixed state. Moreover, it depends on external parameters $k_G$, which is not related to $SU(\infty)$ symmetry. In Appendix E we calculate several entanglement measures for $\hat{\rho}_G$ and $\hat{\rho}_{\infty}$.

### 3.2.1 Significance of entanglement

For various reasons the above results are interesting. They show that although $\mathcal{H}_U$, and thereby $\hat{\rho}_U$, are infinite dimensional and after tracing out a finite number of its components the resulting state $\hat{\rho}_{\infty}$ is still infinite dimensional, it is no longer pure. Indeed, application of any $\hat{O} \in G \subset SU(\infty)$ to $\hat{\rho}_U$ changes the dependence of $A_{\infty}(k_G; \eta, \zeta, \dots)$ on $k_G$, which is the environment parameter for $\hat{\rho}_{\infty}$. Therefore, the environment itself is an open subsystem of the Universe. The only closed - and thereby static - quantum system is the Universe itself. This observation is an explicit demonstration of the Proposition 1 in [2], and confirms the entanglement of every subsystem with the rest of the Universe.

Another interesting aspect of $\hat{\rho}_{\infty}$ is its dependence on the parameters of the traced symmetry $G$, which as we have discussed in [2] (and further in the following sections) can be considered as an approximately isolated subsystem - a matter field. In [1, 2] we related the variation of $\hat{\rho}_{\infty}$ to an affine parameter and an effective metric, which we identified as the metric of the perceived classical spacetime:

$$ ds^2 \equiv \Lambda ds_{WY}^2 = \Lambda \text{tr}(\sqrt{\delta\hat{\rho}_{\infty}}\sqrt{\delta\hat{\rho}_{\infty}^{\dagger}}) \equiv g_{\mu\nu}(y)dy^{\mu}dy^{\nu} \quad (13) $$

The constant $\Lambda$ is a dimensionful scaling constant and $ds_{WY}^2$ is the separation between $\hat{\rho}_{\infty}$ and $\hat{\rho}_{\infty} + \delta\hat{\rho}_{\infty}$ according to the Wigner-Yanase skew information [38, 39]. Using Mandelstam-Tamm inequality [27] - also called quantum speed limit - we proved that the effective metric $g_{\mu\nu}$ is Lorentzian and has negative signature. Thus, it is reasonable to interpret it as the perceived classical spacetime, that we call $\Xi_{sp}$ to distinguish it from the parameter space $\Xi$.

The dependence of $\hat{\rho}_{\infty}$ and its variation $\delta\hat{\rho}_{\infty}$ on the matter parameter $k_G$ means that in $SU(\infty)$-QGR relationship between the effective metric $g_{\mu\nu}$ and state of matter emerges at constructional level. By contrast, in Einstein gravity metric tensor and affine parameter are related to matter and its state dynamically, that is through the Einstein-Hilbert action and Einstein equation. Of course, the single equation (13) is not enough for a full determination of $g_{\mu\nu}$ tensor and dynamics is needed for this purpose.
### 3.3 Emergence of gauge symmetries

In quantum mechanics and quantum information inaccessible or unobservable degrees of freedom are usually traced out. Hence, we can interpret separation/tagging of a finite dimensional subspace representing group $G$ from $H_U$ and tracing out its infinite dimensional complement subspace as isolation of a subsystem due to the inaccessibility of its environment.

According to (3), it is possible to factorize infinite number of $G$ factors from $SU(\infty)$. Each factor can be considered as one realization of a subsystem representing $G$. Moreover, $H_G$ may be a reducible representation of $G$. In this case $\hat{\rho}_G$ can be interpreted as many-body quantum state of an ensemble of subsystems each representing $G$ irreducibly. In both cases subsystems are not completely isolated and interact with each others through application of operators $\hat{O}_G \in B[H_G]$. They also interact with each others and with other types of subsystems with different internal symmetries through their shared $SU(\infty)$ symmetry/gravity.

In (9) operators $\hat{O}_G$ only affect $\hat{\rho}_G$. Therefore, they can be performed independently for every value of the environment parameters $(x; \ell)$. By rearranging summations in (9) the density matrix $\hat{\rho}_G$ can be written as:

$$ \hat{\rho}_G = \sum_{\substack{\{k_G\},\{k'_G\} \\ \{(x;\ell)\}}} \hat{\rho}_G(k_G, k'_G; x; \ell) \quad (14) $$

$$ \hat{\rho}_G(k_G, k'_G; x; \ell) \equiv A_G(k_G; x; \ell) A_G^*(k'_G; x; \ell) \hat{\rho}_G(k_G, k'_G) \quad (15) $$

Operators $\hat{\rho}_G$ have the structure of local quantum fields in a 4D Yang-Mills QFT with group $G$ as their gauge symmetry, in the sense that application of the members of $G$ symmetry projects them to the space generated by $\hat{\rho}_G^6$. We emphasize on the locality of such operations, because they can be performed independently for each set of parameters $(x; \ell)$ a subsystem. In [5, 2] we showed that in order to have a dynamics invariant under reparameterization of $\Xi$, the operator $\hat{\rho}_G$ must transform similar to a spin-1 field (see also Sec. 4.). This completes the demonstration of the analogy between $\hat{\rho}_G(k_G, k'_G; x; \ell)$ and Yang-Mills QFT's. However, there are several differences between $\hat{\rho}_G(k_G, k'_G; x; \ell)$ and fields in models without quantum gravity. In $SU(\infty)$-QGR the locality of finite rank $G$ symmetry arises at construction level. This is in contrast to QFT's in which the dynamics - the Lagrangian - is designed such that it be invariant under local application of symmetry operators. Additionally, in QFT's there is no discrete parameters analogous to $\ell$, we call mode quantum numbers of $SU(\infty)$ symmetry - quantum gravity.

### 3.4 Purified states of subsystems

In [2, 1] we used the global $SU(\infty)$ symmetry and how it affects subsystems to demonstrate that full symmetry of subsystems is $G \times SU(\infty)$. In this section we use purification of the mixed state $\hat{\rho}_G$ to obtain this symmetry from a quantum point of view rather than algebraic properties of the model.

Although tracing of $SU(\infty)$ related factors in (6) demonstrates how a local Yang-Mills structure associated to a finite rank symmetry arises for subsystems, it erases the quantum origin of $SU(\infty)$ and dynamics related parameters $x; \ell$. Nonetheless, with the prior knowledge that the origin of these parameters in $\hat{\rho}_G$ is the global $SU(\infty)$ symmetry of the Universe, we can extend $H_G$ by a Hilbert space representing $SU(\infty)$. The new Hilbert space $H_{G_\infty}$ represents $G \times SU(\infty)$ and its states can be expanded as:

$$ |\Psi_{G_\infty}\rangle \equiv \sum_{\{k_G\}; (\eta, \zeta, \cdots)} A_{G_\infty}(k_G; \eta, \zeta, \cdots) |\psi_G(k_G)\rangle \times |\psi_\infty(\eta, \zeta, \cdots)\rangle \quad (16) $$

<footnote n="6">⁶We remind that the notion of density matrix in quantum mechanics and notion of quantum field in QFT are related. Appendix H review of their relationship.</footnote>
where $|\psi_{\infty}(\eta, \zeta, \cdots)\rangle$ is the same as in (5). Notice that $|\Psi_{G_{\infty}}\rangle$ can be also considered as purification of $\hat{\rho}_{\infty}$. Although the state $|\Psi_{G_{\infty}}\rangle$ looks like the state of the Universe $|\Psi_{U}\rangle$ in (5), they do not present exactly the same quantum system for following reasons.

In (16) the group $SU(\infty)$ acts only on the $|\psi_{\infty}(\eta, \zeta, \cdots)\rangle$ factor of the basis. Thus, in comparison with the state $|\Psi_{U}\rangle$ in (5), or equivalently its density matrix $\hat{\rho}_{U}$ in (6), the global $SU(\infty)$ symmetry is broken. The difference between $|\Psi_{G_{\infty}}\rangle$ and $|\Psi_{U}\rangle$ becomes more transparent if we express $SU(\infty)$ as $SU(N \to \infty)$. Indeed, in $|\Psi_{G_{\infty}}\rangle$ the $SU(\infty)$ symmetry of the appended Hilbert space should be read as $SU((N - d_G) \to \infty)$. In addition, the basis of the appended subspace $|\psi_{\infty}(\eta, \zeta, \cdots)\rangle$ is not necessarily the same as that used for obtaining $\hat{\rho}_{G}$. Therefore, amplitudes $A_{G_{\infty}}(k_G; \eta, \zeta, \cdots)$ are arbitrary up to the unitarity constraint:

$$ \sum_{\substack{k_G \\ \{(\eta, \zeta, \cdots)\}}} |A_{G_{\infty}}(k_G; \eta, \zeta, \cdots)|^2 = 1 \qquad (17) $$

and faithfulness constraint [33], that is tracing over extended subspace must recover $\hat{\rho}_{G}$:

$$ \mathrm{tr}_{\infty} \hat{\rho}_{G_{\infty}} = \hat{\rho}_{G}, \qquad \hat{\rho}_{G_{\infty}} \equiv |\Psi_{G_{\infty}}\rangle\langle\Psi_{G_{\infty}}| \qquad (18) $$

These conditions are not sufficient to uniquely relate amplitudes $A_{G_{\infty}}$'s to $A_G$'s. This is not a surprise as the degeneracy of purification is well known through Schödinger-HJW theorem [40, 41, 42] asserting that any mixed state can be purified to many unitarily equivalent pure states. This is a consequence of the arbitrariness of both the auxiliary space used for the extension of the Hilbert space, and the basis chosen for this subspace. Although here we have chosen a fixed auxiliary Hilbert space based on our prior knowledge about how $\hat{\rho}_{G}$ was obtained, as mentioned earlier, the basis remains arbitrary. It is well known that the lack of information about the basis - the reference frame - leads to decoherence [43]. Specifically, because the basis $|\psi_{\infty}(\eta, \zeta, \cdots)\rangle$ is orthogonal to the G-related factor and can be chosen or modified arbitrarily, the global $SU(N \to \infty)$ coherence symmetry is broken to $G \times SU((N - d_G) \to \infty)$. Nonetheless, if we treat the two subspaces representing G and $SU((N - d_G) \to \infty)$ as subsystems, they remain entangled because the constraint (18) ensures a perfect and faithful purification. Details of conditions for faithful purification and their satisfaction by $\hat{\rho}_{G}$ and $\hat{\rho}_{\infty}$ is discussed in Appendix F.

### 3.4.1 Physical interpretation of purified states

Using linearity of the map $\Lambda$ and the last equality in (70), which is the condition for faithfulness, it is straightforward to show that either $\Lambda$ maps all $\hat{\rho}_{G}$ to the same pure density matrix, or it cannot be a universal purifier for an ensemble of closely related mixed density matrices [33]$^7$. Thus, when it is a universal purifier, its outcome would be a pure quantum state independent of details of $\hat{\rho}_{G}$. For $\hat{\rho}_{G}$ in a given representation of G symmetry, amplitudes $A_{G_{\infty}}(k_G; \eta, \zeta, \cdots)$ would be independent of $A_G(k_G; \eta, \zeta, \cdots)$ up to a unitary transformation of $\hat{\rho}_G \times \hat{\rho}_{\infty}$ basis. Physically, this means that when subsystems with the same internal symmetry are approximately disentangled from the rest of the Universe, they see the same environment, and therefore are indistinguishable. This conclusion is also consistent with the global equivalence of all $\hat{\rho}_U$ states up to a physically irrelevant unitary transformation. By contrast, subsystems with different internal symmetries or different representations of the same symmetry see the rest of the Universe differently. Consequently, their purifying map $\Lambda$ would be different, because their projections into the extended Hilbert space would be different.

<footnote>$^7$Here by closeness we mean density matrices that can be possible states of a given quantum system or subsystem. If states $\hat{\rho}_1$ and $\hat{\rho}_2$ are in the set of possible states for a system, their sum can be also a plausible state. The proof of Theorem 1 in [33] uses this superposition property.</footnote>
In quantum information and quantum technologies purification of the states of quantum systems is necessary for compensating noise and decoherence induced by interaction with an uncontrollable environment. Here, the purification of states $\hat{\rho}_G$ and $\hat{\rho}_\infty$ has rather a conceptual purpose. It shows that in a Universe with infinite number of mutually commuting observables, pure quantum states can be associated to subsystems and their environment, at least intermittently. Indeed, the random action of global operators $\hat{O} \in SU(\infty)$, which can be considered to be part of the environment, affects both. On the other hand, similarity of the purified subsystem and environment states demonstrates that they are not really separable, and irrespective of what happens to them and the way their states are presented, their entanglement is retained.

### 3.5 Emergence of area/size parameter

In Sec. 3.4 we explicitly showed that in $SU(\infty)$-QGR subsystems of the Universe represent $G \times SU(\infty)$. This means that the Hilbert space $\mathcal{H}_{G\infty}$ of each subsystem is a representation of $SU(\infty)$. In [2, 1] we argued that in presence of multiple subsystems representing $SU(\infty)$, the area of their diffeo-surfaces becomes a relative observable and should be added to the list of parameters characterizing their states. Here we investigate this property further.

The Poisson algebra (37) of $ADiff(D_2)$, which is homomorphic to the algebra $SU(\infty)$ [6, 12] is invariant under scaling. Thus, (37) and associated algebras, that is the Poisson algebra of spherical harmonics in sphere basis [6], algebras (42) and (43) in torus basis, and their limit (39) are homomorphic to the Lie algebra $SU(\infty) + U(1)$ and represent the symmetry group $SU(\infty) \times \mathbb{R} \cong SU(\infty) \times U(1) = U(\infty)$. The scaling symmetry $\mathbb{R} \cong U(1)$ reflects the irrelevance of the area of compact diffeo-surfaces $D_2$ in the limit where these algebras are homomorphic to $ADiff(D_2)$.

It is easy to prove that $\otimes^n SU(\infty) \cong SU(\infty)$, $\forall n$ see e.g. appendices of [2]. This relation can be also expressed as:

$$ \otimes^n SU(\infty) \cong \bigcup_{i=1}^{n} ADiff(D_2^{(i)}) \cong ADiff(D_2), \quad D_2 \equiv \bigcup_{i=1}^{n} D_2^{(i)} \quad (19) $$

In isolation the area of each diffeo-surface $D_2^{(i)}$ is arbitrary and its variation does not affect the homomorphism of $ADiff(D_2^{(i)})$ with $SU(\infty)$. But, once diffeo-surfaces are stuck together, only the area of their ensemble $D_2$ can be arbitrarily scaled. Thus, scaling diffeo-surface of one subsystem must be necessarily compensated by scaling of others. The algebra becomes:

$$ \otimes^n (SU(\infty) + U(1)) \rightarrow \otimes^n SU(\infty) + U(1) \cong SU(\infty) + U(1) \quad (20) $$

The consequence of these properties is the dependence of quantum states of subsystems to an additional continuous parameter $r > 0$ that indicates relative area (or its square root) of compact diffeo-surfaces of subsystems. Notice that $r = 0$ is equivalent to trivial representation of $SU(\infty)$ and is excluded by axioms of the model. The following diagrams summarize the relationship of $SU(\infty)$ and $ADiff$ for single and multiple representations:

$$
\begin{tikzcd}
& & \text{Single subsystem} & & \\
SU(\infty) \arrow[r, "\cong", "\text{area irrel.}"] \arrow[d, "\cong"'] & SU(\infty) \arrow[r, "\cong"'] & SU(\infty) + U(1) \arrow[d, "\cong"'] & & \\
ADiff(D_2) \arrow[r, "\text{area irrel.}"] & ADiff(D_2) \times U(1) & & & \\
\arrow[d, "\cong"'] & \arrow[d, "\cong"'] & & & \\
& & & & (21)
\end{tikzcd}
$$
Multiple subsystems

$$
\begin{equation}
\begin{split}
SU(\infty) \times \dots \times SU(\infty) &\cong SU(\infty) \xrightarrow{\text{area irrel.}} SU(\infty) + U(1) \\
\xrightarrow{\text{area irrel.}} & \\
(ADiff(D_2^{(1)}) \times U(1)) \times \dots \times (ADiff(D_2^{(n)}) \times U(1)) &\xrightarrow[\text{symm. break}]{\text{area preserv.}} ADiff(D_2) \times U(1)
\end{split}
\tag{22}
\end{equation}
$$

3.5.1 Consistency of the two perceptions of the quantum Universe

Once the quantum state of the Universe is self-clustered to approximately isolated subsystems
and a clock parameter *t* is chosen, the purified state of the ensemble of subsystems $\hat{\rho}_{U_s}$ can be
written as:

$$
\begin{align}
\hat{\rho}_{U_s} &= \bigotimes_{i=1}^{\infty} \hat{\rho}_{G^{(i)}} = \bigotimes_{i=1}^{\infty} \sum_{\substack{\{k_{G^{(i)}}, k'_{G^{(i)}}\} \\ (x_i; \ell_i; x'_i; \ell'_i)}} A(k_{G^{(i)}}; x_i; \ell_i) A^*(k'_{G^{(i)}}; x'_i; \ell'_i) \left( \hat{\rho}_{G^{(i)}}(k_{G^{(i)}}, k'_{G^{(i)}}) \times \hat{\rho}_{\infty}(x_i; \ell_i; x'_i; \ell'_i) \right) \nonumber \\
&= \prod_i \left( \sum_{\substack{\{k_{G^{(i)}}, k'_{G^{(i)}}\} \\ (x_i; \ell_i; x'_i; \ell'_i)}} A(k_{G^{(i)}}; x_i; \ell_i) A^*(k'_{G^{(i)}}, x'_i; \ell'_i) \right) \bigotimes_{i=1}^{\infty} \hat{\rho}_{G^{(i)}}(k_{G^{(i)}}, k'_{G^{(i)}}) \times \bigotimes_{i=1}^{\infty} \hat{\rho}_{\infty}(x_i; \ell_i; x'_i; \ell'_i) \tag{23}
\end{align}
$$

Here we have assumed that each subsystem *i* may represent a different internal finite rank symmetry *G*⁽^i⁾. However, from cosmological and laboratory observations we know that many subsystems/particles share the same internal symmetry. Therefore, many of *G*⁽^i⁾’s and their representations in (23) are the same. In addition, the number of subsystems does not need to be countable.

In appendices of [2] we demonstrated that tensor products $\otimes_{n \to \infty} G \cong SU(\infty)$ for any $G$, including $G = SU(\infty)$ (see also [16]). Hence, both factors in the tensor product in the last equality of (23) are density matrices representing $SU(\infty)$. Even if the time parameter $t$ is fixed$^8$, despite apparent dependence of $\hat{\rho}_{U_s}$ on time it belongs to the same Hilbert space $\mathcal{H}_U$ as the state $\hat{\rho}_U$ in (6), namely the quantum state of the entire Universe without breaking it to subsystems. Specifically, due to $\otimes_{i=1}^\infty G_i \cong SU(\infty)$, the state $\hat{\rho}_{U_s}(t) \in \mathcal{H}_{U(t)} \subset \mathcal{H}_U$, and can be transformed to any other state of $\mathcal{H}_U$ by application of $SU(\infty)$ group, which according to axioms of $SU(\infty)$-QGR should not affect observables. The state of the Universe can be evolved only if the clock is approximately isolated from other subsystems. In this case, as explained in Sec. (3.2), it will remain entangled to the rest of the Universe and can evolve by both internal - self-interaction - and interaction with other subsystems through both shared internal symmetries and $SU(\infty)$ - gravity.

In conclusion, considering the Universe as the ensemble of its infinite number of subsystems is
not in contradiction with treating it as a single quantum system with infinite number of mutually
commuting observables. The difference of perception is on whether concentration or access is on
the local features of the Universe’s state or its global - topological - properties. In what concerns
our present observational capabilities, we can only detect its local aspects.

<footnote>$^8$Assuming a fixed value for the time parameter $t$, rather than integrating it, means a projective measurement of the clock observable associated to $t$. Nonetheless, the state of the clock for this outcome should be included in $\hat{\rho}_{U_s}$, because the clock may have other observables commuting with that used as time. For instance, if $t$ corresponds to the phase of an oscillating spin of an atom, its uncorrelated (commuting) degrees of freedom to spin, such as kinetic energy would be in general in coherent superposition and full quantum state of the clock would be similar to (16).</footnote>
# 4 Dynamics over a parameter space

Quantum state of the ensemble of subsystems $\hat{\rho}_U(t)$ calculated in (23) includes all information about them. However, it is not suitable for applications, because it does not show how subsystems interact with each others and evolve with respect to the clock. In quantum mechanics and QFT the evolution is usually formulated through definition of an action functional or a Hamiltonian in non-relativistic cases, which are usually taken or inspired from classical limit of the model. However, $SU(\infty)$-QGR is built as a quantum theory and it does not have a classical formulation, or even a true classical limit, see footnote 2 and [2]. Therefore, dynamics must be constructed solely based on the axioms of the model and properties concluded from them.

In [2] we introduced dynamics by constructing a Lagrangian functional over parameter space $\Xi$. It consists of symmetry invariant traces of $SU(\infty)$ generators, and those of internal symmetry $G$ when the Universe is divided to subsystems. After imposing invariance under reparameterization of $\Xi$, we showed that at lowest order in traces it has the form of a Yang-Mills theory for both symmetries. Traces of multiple generators of Lie groups are functions of their rank and structure coefficients. Moreover, in the Lagrangian functional such terms are proportional to higher orders of coupling constant. Therefore, trace terms with more than two generators can be considered as perturbative corrections to lowest order effective Lagrangian. Consequently, methodology and techniques of QFT's such as path integral and Feynman diagrams are applicable to $SU(\infty)$-QGR. Importantly, as Yang-Mills models are known to be renormalizable $SU(\infty)$-QGR does not have any issue in this regard.

In this section after a brief reminding of the $SU(\infty)$-QGR action we elaborate some of details which were not addressed in the previous works.

## 4.1 $SU(\infty)$-QGR as a Yang-Mills QFT on the parameter space of subsystems

The effective Lagrangian of $SU(\infty)$-QGR is analogous to the effective Lagrangian of QFT's, with the difference that it is defined on the parameter space $\Xi$, rather than an external spacetime. Reparameterization and symmetry invariance lead, at lowest order, to a Yang-Mills type Lagrangian functional on $\Xi$. Specifically, for the whole Universe, that is when the Universe is treated as a single isolated quantum system, the lowest order effective Lagrangian $\mathcal{L}_U$ is a 2D Yang-Mills QFT on the diffeo-surface $\Xi = D_2$ (see Fig. 1 for a schematic depiction):

$$ \mathcal{L}_U = \int d^2\Omega \left[ \frac{1}{2} \mathrm{tr}(F^{\mu\nu} F_{\mu\nu}) + \frac{1}{2} \mathrm{tr}(\mathcal{D}\hat{\rho}_U) \right], \quad d^2\Omega \equiv d^2x \sqrt{|\eta^{(2)}|} \quad (24) $$

$$ F_{\mu\nu} \equiv F_{\mu\nu}^a \hat{L}^a \equiv [D_\mu, D_\nu], \quad D_\mu = (\partial_\mu - \Gamma_\mu) + \sum_a i\lambda A_\mu^a(x) \hat{L}^a(x), \quad (25) $$

where $\hat{L}^a(x)$ are generators of $SU(\infty)$ symmetry, $\Gamma_\mu$ is a suitable connection for $\Xi$, and reparameterization invariant derivative $\mathcal{D}$ depends on how $\hat{\rho}_U$ is transformed under diffeomorphism of $\Xi$. See the next paragraph for definition of $\mathcal{D}$. Notice that both the gauge field $A_\mu^a$ and generators $\hat{L}^a$ depend on the coordinates of the parameter space - the diffeo-surface. For instance, for the sphere basis [6] (reviewed in [1]) $x \equiv (\theta, \phi)$ are angular coordinates on $\Xi$, index $a = (l, m)$, and $\hat{L}_{lm}(\theta, \phi)$ are generators of $SU(\infty)$. In torus basis reviewed in Appendix A, $x$ is the 2D Cartesian coordinates of a point on $\Xi$, index $a = \mathbf{m}$, and $\hat{L}_m$ is defined in (38). In $SU(\infty)$ Yang-Mills on a background spacetime of any dimension, in addition to the internal coordinates $x$, the gauge field depends on the spacetime, see Appendix G for a brief review of these models. The Lagrangian $\mathcal{L}_U$ is clearly static, because it does not include any time parameter. Moreover, it has been shown that it is topological [1, 2] and its value does not depend on the 2D metric $\eta_{\mu\nu}$, $\mu, \nu = 1, 2$.

When the Universe is considered as ensemble of its subsystems representing $G \times SU(\infty)$, as explained in Sec. 3 their purified states $\hat{\rho}_{G\infty}$ depends, in addition to coordinates of the diffeo-surface, on an area/size parameter and on time. The symmetry and reparameterization invariant
Lagrangian $\mathcal{L}_{U_s}$ is defined on the 4D parameter space $\Xi$ (depicted in Fig. 1), and has the following expression:

$$ \mathcal{L}_{U_s} = \int d^4 x \sqrt{|\eta|} \left[ \mathrm{tr}(F^{\mu\nu} F_{\mu\nu}) + \frac{1}{4} \mathrm{tr}(G^{\mu\nu} G_{\mu\nu}) + \frac{1}{2} \sum_s \mathrm{tr}(\mathcal{D} \hat{\rho}_{G_s}) \right] \quad (26) $$

$$ F_{\mu\nu}(x) \equiv [D_{\mu}, D_{\nu}], \quad D_{\mu} = (\partial_{\mu} - \Gamma_{\mu}) - i\lambda A_{\mu}, \quad A_{\mu} \equiv \sum_{a} A_{\mu}^{a}(x; \eta, \zeta) \hat{L}^{a}(\eta, \zeta) \quad (27) $$

$$ G_{\mu\nu}(x) \equiv [D'_{\mu}, D'_{\nu}], \quad D'_{\mu} = (\partial_{\mu} - \Gamma_{\mu} - i\lambda A_{\mu}) \mathbb{1}_{G} - i\lambda_{G} B_{\mu}, \quad B_{\mu} \equiv \sum_{b} B_{\mu}^{b}(x) \hat{T}^{a} \quad (28) $$

Definitions of quantities in this Lagrangian are as follows. The symmetric tensor $\eta_{\mu\nu}$ is the metric of the parameter space $\Xi$. It should not be confused with the metric $g_{\mu\nu}$ of the emergent classical spacetime $\Xi_{sp}$. According to the Proposition 2 of [2] $\eta_{\mu\nu}$ is arbitrary, because geometry of $\Xi$ is not a physical observable. We further discuss this property in Sec. 4.2 The first and second terms in (26) are the Lagrangian density for the $SU(\infty)$ and internal symmetry $G$ gauge fields $A_{\mu}^{lm}$ and $B_{\mu}^{a}$, respectively. The $SU(\infty)$ generators $\hat{L}^{lm}$ and the range of $(l, m)$ are defined in Appendix A. $\Gamma_{\mu}$ is the geometric connection of the parameter space $\Xi$. Operators $T^a$ are generators of the internal symmetry $G$ of subsystems. Their number is determined by the range of index $a$, which must be finite because the rank of $G$ is finite. The density matrix $\hat{\rho}_s$ is the quantum state of a subsystem. The covariant operator $\mathcal{D}$ is a differential operator and its exact definition depends on the representation of the symmetries of $\Xi$ by the states of subsystems. For instance, for spinors $\mathcal{D} \equiv \gamma^0 \gamma^i e_i^\mu (\partial_\mu - \Gamma_\mu) \mathbb{1} - \sum_{lm} i \lambda_s A_\mu^{lm} \hat{L}^{lm} - \sum_a i \lambda_G B_\mu^a \hat{T}^a$, where $\gamma^\mu, \mu = 0, \dots, 3$ are Dirac matrices and $e_i^\mu, i = 0, \dots, 3$ are tetrads, see [2] for more details. The reason for the presence of $SU(\infty)$ field in (28) and $\mathcal{D}$ is the fact that all subsystems represent $SU(\infty)$. In Sec. 4.2 we demonstrate that geometry connection terms in $D'_\mu$, $\mathcal{D}$ and field equations can be neutralized by a $SU(\infty)$ gauge transformation. Therefore, as mentioned above the Lagrangian (26) does not depend on $\eta_{\mu\nu}$. Finally, notice that the interaction of $G$ gauge field (amplitude) $B_\mu$ with $SU(\infty)$ is implicit, because it depends on the parameters $x \in \Xi$ of the $SU(\infty)$ symmetry of subsystems.

Aside of including terms for Yang-Mills fields representing $G$, the main difference between the $SU(\infty)$ gauge field in $\mathcal{L}_{U_s}$ and $\mathcal{L}_U$ is its dependence on additional parameters. In $\mathcal{L}_U$ the gauge field depends only on two continuous parameters characterizing $SU(\infty)$ algebra. By contrast, $A_\mu^a$ in $\mathcal{L}_{U_s}$ depends on 4D vectors $x \in \Xi$ and on $(\eta, \zeta) \in D_2$, where $D_2$ is the diffeo-surface of $SU(\infty)$ symmetry it represents. Although the $SU(\infty)$ gauge field (79) defined on a background spacetime has apparently a similar structure, there is an important difference. In contrast to coordinates of so called *internal space* in (79), the induced coordinates $(\eta, \zeta)$ of the diffeo-surface $D_2$ are functions of $x$. As the Fig. 1 shows, $D_2$ is immersed in $\Xi$, that is $D_2 \subset \Xi$. By contrast, in (79) the internal space is independent of the background spacetime. It is the reason for using the name *diffeo-surface* rather than *internal space* for a compact 2D surface that its $ADiff$ group is homomorphic to $SU(\infty)$.

### 4.1.1 Classical spacetime from quantum gravity action

Perturbation theory applied to $\mathcal{L}_{U_s}$ allows to calculate components of $\hat{\rho}_{G_s}$. Tracing out the contribution of $G$ symmetry parameters gives $\hat{\rho}_\infty$, which can be then used to determine the effective classical metric in (13). However, this procedure only determines $ds^2$ rather than the metric $g_{\mu\nu}$ of the emergent classical spacetime $\Xi_{sp}$. For determining components of $g_{\mu\nu}$ we have to use the classical limit of $\mathcal{L}_{U_s}$ as explained in [2]. The procedure consists of treating pure $SU(\infty)$ term in (26) as the classical scalar curvature, other terms as *matter*, and identifying the metric of parameter space $\eta_{\mu\nu}$ with the effective metric $g_{\mu\nu}$ defined in (13). However, the $SU(\infty)$ gauge field $A_\mu$ must be included in the connection $\Gamma_\mu$, because in classical gravity, it is the connection that convoys gravitational interaction. The effective energy-momentum tensor should be determined from effective Lagrangian for matter part of (26), which can include quantum corrections in a curved spacetime, see e.g. [44, 45, 46] for examples. Then, the Einstein equation obtained from
<figure><img src="image_2.png" /><figcaption>Figure 1: Schematic presentation of parameter spaces Ξ of $SU(\infty)$ symmetry. Left: A single 2D compact surface corresponding to diffeo-surface of $SU(\infty)$ symmetry of the Universe. Its area is arbitrary and irrelevant. Axes $(\eta, \zeta)$ are local coordinates. Right: The 4D parameter space generated by infinite number of 2D diffeo-surfaces of subsystems and time parameter of a quantum clock. Axes $x^i$, $i = 0, \dots, 3$ are coordinates at each point $x \in \Xi$ and axes $\eta(x), \zeta(x)$ are induced coordinates on one of the diffeo-surfaces passing through $x$.</figcaption></figure>

applying variational principle to this approximation can be solved to calculate components of the effective metric. Of course, the Lagrangian $\mathcal{L}_U$ can be used to calculate Green's functions and other measurables of both quantum matter and quantum gravity - the $SU(\infty)$ gauge field - without having to consider the classical limit.

## 4.2 Equivalence of diffeomorphism of parameter space and $SU(\infty)$ gauge transformation

The equivalence of reparameterization of the parameter space $\Xi$ and transformation of its 2D subspaces under the action of $SU(\infty)$ group is necessary for proving that geometry of $\Xi$ is irrelevant for physics (Proposition 2 in [2]). It is straightforward to verify that the connection form $\Gamma_\mu$ does not affect gauge invariant field intensity $F_{\mu\nu}$. But, it appears in its covariant derives as well as in the covariant derivative $\partial\!\!\!/\!$ of the density matrix. In Einstein gravity metric, connection, and curvature tensors of the background spacetime are related to the energy-momentum tensor of matter through Einstein-Hilbert action and Einstein equation. By contrast, in $SU(\infty)$-QGR geometry of parameter space $\Xi$ is arbitrary and should not affect observables. In [2] we used algebraic arguments to prove this property. Here we demonstrate it analytically.

Equivalence of the diffeomorphism of $\Xi$ and $SU(\infty)$ gauge transformation in the Lagrangian $\mathcal{L}_U$ of the whole Universe is trivial, because $\Xi$ is the same as the diffeo-surface $D_2$. Thus, area preserving diffeomorphisms of $\Xi$ are simply $SU(\infty)$ transformations, under which $\mathcal{L}_U$ is invariant. To see this, consider a change of $\Gamma_\mu \to \Gamma'_\mu$ that preserves the area of $\Xi = D_2$. We remind that vectors in $\mathcal{H}_U$ are 2D complex-valued functions<fnref n="9" />. Therefore, in analogy with $A_\mu$, components of $\Gamma_\mu$ can be considered as vectors of $\mathcal{H}_U$. Using, for instance, the sphere basis for $SU(\infty)$, both $\Gamma_\mu$ and $A_\mu$ can be expanded with respect to spherical harmonic functions $Y_{lm}(\theta, \phi)$. As $\hat{L}_{lm}Y_{l'm'} = -if^{l''m''}_{lm,l'm'}Y_{l''m''}$ (for $\hbar = 1$) where $f^{l''m''}_{lm,l'm'}$ are structure coefficients of $SU(\infty)$ [6], a $SU(\infty)$ gauge transformation of $A_\mu$ changes coefficients of its expansion with respect to $Y_{lm}(\theta, \phi)$.

<footnote n="9">⁹Nonetheless, in [2] we showed that phase of the gauge fields is irrelevant and without loss of generality these fields can be considered to be real.</footnote>
Thus, a gauge transformation such that:

$$ (\Gamma'_{\mu} - \Gamma_{\mu}) = \Omega(x) A_{\mu}(x) \Omega^{-1}(x) - i \Omega(x) \partial_{\mu} \Omega^{-1}(x) - A_{\mu}(x) \quad (29) $$

$$ \Omega(x) \equiv e^{i \int_{D_2} d^2 \Omega' \epsilon_{lm}(\theta, \phi, \eta', \zeta')} \hat{L}^{lm}(\eta', \zeta') \quad (30) $$

can compensate the change of $\Gamma_{\mu}$. A general diffeomorphism of any manifold $\mathcal{M}$ can be decomposed to:

$$ \mathrm{Diff}(\mathcal{M}) \cong \mathrm{ADiff}(\mathcal{M}) \times \Lambda(\mathcal{M}), \quad \forall \mathcal{M} \quad (31) $$

where $\Lambda(\mathcal{M})$ presents the operation of a global scaling of the manifold $\mathcal{M}$. Such transformation changes $\mathcal{L}_U \to \Lambda\mathcal{L}_U$. In other words, it changes the volume of $\Xi$ by rescaling its coordinates. But, due to the $U(1)$ symmetry discussed in Sec. 3.5 such a rescaling is not an observable.

Demonstration of the invariance of Lagrangian $\mathcal{L}_U$ of subsystems and the corresponding field equations under diffeomorphism of their 4D parameter space $\Xi$ is more complicated, but follows the same line of reasoning. As discussed above, the 2D diffeo-surface $\Sigma$ characterizing $SU(\infty)$ generators is immersed in $\Xi$, that is $\Sigma \subset \Xi$. Thus, the induced coordinates $(\eta, \zeta)$ of $\Sigma$ are functions of the 4D parameter space $\Xi$. To demonstrate the irrelevance of the geometry of $\Xi$ for physical observables, we show that gauge transformations involving all parameters can restore curvatures, geometrical connection, covariant derivatives of deformed or reparameterized $\Xi^{10}$. Specifically, Riemann and Ricci curvature tensors - hence the connection at a point $x$ of a Riemannian manifold - can be determined from sectional curvatures of 2D surfaces passing through that point [47, 48]. There are exactly 6 such pairs of axes at each point $x$ of the 4D parameter space $\Xi$ leading to the following expression for sectional curvatures:

$$ K(\Pi_{ij}(x)) \equiv K(\hat{x}^i, \hat{x}^j) = \frac{R_p(\hat{x}^i, \hat{x}^j, \hat{x}^i, \hat{x}^j)}{\langle \hat{x}^i, \hat{x}^i \rangle \langle \hat{x}^j, \hat{x}^j \rangle - \langle \hat{x}^i, \hat{x}^j \rangle^2}, \quad i, j = 0, \dots, 3, \quad i \neq j \quad (32) $$

The 4D unit vectors $\hat{x}^i \in T\Xi(x)$, where $T\Xi(x)$ is the tangent space of $\Xi$ at $x \in \Xi$, and the plane $\Pi_{ij}(x)$ passes through vectors $\hat{x}^i$ and $\hat{x}^j$. As the 2D compact surfaces used for the definition of $\hat{L}^a(\eta, \zeta)$ in (27) are arbitrary, we can locally identify them with surfaces generated by a pair of unit axes $(\hat{x}^i, \hat{x}^j) \in T\Xi(x)$ at $x$. Moreover, as $SU(\infty)^n \cong SU(\infty)$, the union of diffeo-surfaces is again a diffeo-surface associated to a representation of $SU(\infty)$. Thus, $SU(\infty)$ gauge transformations, such as:

$$ A_{\mu}(x; \eta, \zeta) \to A'_{\mu}(x; \eta, \zeta) = \Omega(x; \eta, \zeta) A_{\mu}(x; \eta, \zeta) \Omega^{-1}(x; \eta, \zeta) - i \Omega(x; \eta, \zeta) \partial_{\mu} \Omega^{-1}(x; \eta, \zeta) \quad (33) $$

$$ \Omega(x; \eta, \zeta) \equiv \exp(i\epsilon_{lm}(x; \eta, \zeta) \hat{L}^{lm}(\eta, \zeta)), \quad x \in \Xi \quad (34) $$

can be performed using $(\eta, \zeta) \in \mathrm{Span}(\hat{x}^i, \hat{x}^j)$, $i, j = 0, \dots, 3, i \neq j$, which means $(\eta, \zeta)$ are chosen such that the surface passing through them corresponds to $\Pi_{ij}(x)$ at $x$. Consequently, by properly choosing a set of $SU(\infty)$ gauge transformation (33), Riemann and Ricci curvatures of the deformed or reparameterized $\Xi$ can be restored. Hence, even in the case of Lagrangian $\mathcal{L}_U$ that depends on 4 parameters a $SU(\infty)$ gauge transformation is homomorphic to an $\mathrm{ADiff}$ of the parameter space and vis-versa. In other words, a change in the field space that preserves the Lagrangian $\mathcal{L}_U$ is equivalent to a diffeomorphism of its parameter space and vis-versa. This demonstration finalizes analytic proof of the Proposition 2 in [2] and shows that the geometry of $\Xi$ is irrelevant for physical observables of subsystems. Thus, without loss of generality we can consider a fixed geometry for $\Xi$ in both the Universe as a whole, and when it is perceived through the ensemble of its subsystem. This choice is equivalent to fixing $SU(\infty)$ gauge.

<footnote n="10">¹⁰Riemann and Ricci curvatures depend on the derivatives of Levi-Civita connection. For this reason, calculating connection as a function of curvatures amount to solving a partial differential equation, which is not a trivial task. Therefore we take the inverse approach and prove that curvatures are restored by $SU(\infty)$ gauge transformation.</footnote>
# 5 Perspectives for test and applications of SU($\infty$)-QGR

Currently there is no observed evidence of quantum behaviour of gravity. However, the existence of a quantum description for the universal interaction called gravity seems inevitable [49, 50, 51]. Therefore, the absence of evidence is most probably due to the insufficiency of precision and resolution of current experiments. Nonetheless, with the huge progress of quantum technologies in the past few decades there is strong hope that in the near future tests of the quantumness of gravity and eventually discrimination between various QGR proposals become achievable. Indeed, there are already some progress in this direction. Specifically, cosmological and astro-particle observations put stringent constraint on the modification of dispersion relation of high energy photons due to the micro-structure of the spacetime [52] predicted by some QGR models such as loop quantum gravity; gravitational waves constrain graviton mass [53] and Lorentz invariance violation [54] predicted by some string theories; large extra dimensions are constraint by various analyses and observations, for example [55, 56, 57].

In what concerns the test of SU($\infty$)-QGR, as mentioned earlier, its most discriminative signature is a spin-1 boson as mediator of QGR. However, observation of this attribute in laboratory experiments - usually based on the detection of a change in coherence [58, 59, 60] or entanglement [61, 62]<fnref n="11" /> of the state of a quantum system by gravitational interaction - is much harder. Meanwhile and until the limited capabilities of laboratory experiments become available, cosmological observations may be a faster road to detection of a QGR signal, in particular that of SU($\infty$)-QGR. Here we outline a proposal for the detection of QGR signature using coherent astronomical sources.

## 5.1 Difference in coherence of astronomical masers as a signature of QGR

Astronomical masers have short duration quantum coherence [63] and can be used as a coherent source for testing QGR. The change of their polarization may be also useful for estimate spin of particles which led to their decoherence [62]. On their path from the place of their formation - usually in molecular clouds of galaxies at cosmological distances - to Earth, maser photons may be affected by gravitational field of another galaxy or galaxy cluster and their dark matter halos and being lensed [64].

In a quantum view, the deflection of photons is the result of their interaction with gravitons. In general, such process changes the coherence of photons. Specifically, a quantum process (channel) affecting a quantum system can be described by a set of Krauss operators: $\hat{\rho}_f = \sum_i \hat{K}_i \hat{\rho}_i \hat{K}_i^\dagger$, where $\hat{\rho}_i$ and $\hat{\rho}_f$ are the initial and final states of the system, respectively, and Krauss operators $\hat{K}_i$, satisfying $\sum_i \hat{K}_i \hat{K}_i^\dagger = 1$ define the quantum process. Using the sum of off-diagonal components $C = \sum_{i \neq j} \hat{\rho}_{ij}$ as a measure of the coherence of a state $\hat{\rho}$ [65], it is evident that in general $C_f \neq C_i$.

In the framework of SU($\infty$)-QGR, the SU($\infty$) gauge-invariant Krauss operators for the process of photon deflection by a gravitational field can be written as: $\hat{K}_{\mu\nu}^a \equiv NF_{\mu\nu}^a \hat{L}^a$, where $N$ is a normalization constant, $a \equiv (l, m)$ defined in Appendix A is the SU($\infty$) graviton color, and $F_{\mu\nu}^a$ is the spin-1 graviton field defined in (27). Similarly, the U(1) gauge invariant photon state can be expanded with respect to $G_{\mu\nu}$ defined in (28): $\hat{\rho} = N' \hat{G}_{\mu\nu} \hat{G}^{\mu\nu}$, where $N'$ is a normalization constant. A hat is added to emphasize that the field should be considered as an operator in $B[H_{G=U(1)}]$.

Due to the deflection of maser's photons by the gravitational field of a galaxy or cluster playing the role of a gravitational lens, the optical paths of photons in different images are not the same. Consequently, the Krauss operators affecting them would be different. Thus, coherence of the

<footnote n="11">¹¹There are many proposals for test of QGR in the literature. The citations here are only examples and far from being exclusive.</footnote>
maser emission in the images would not be the same. Therefore, detection of coherence difference in lensed masers - after taking into account other sources of coherence distortion - would be a signature of QGR. Another measure of the quantum effect of gravity on the lensed photons is entanglement fidelity [66] (also called channel fidelity [67]) defined as $F_e = \sum_i \text{tr}(\hat{\rho}_{K_i})\text{tr}(\hat{\rho}_{K_i}^\dagger)$. This quantity measures how much a quantum channel changes the state of a signal. Assuming that before interacting with gravitational field of the lens the maser photons have the same single-photon state, the difference between channel fidelity of images would be due to the different Krauss operators of the channel (path) taken by photons in separate images<fnref n="12" />. In contrast to the measurement of coherence that only depends on the quantum state of detected photons, channel fidelity depends also on the properties of the channel, in particular the spin of gravity mediator. But, it needs modeling of the lens and is more sensitive to its uncertainties. Nonetheless, a priori it should make possible to discriminate between different QGR models, in particular to test the prediction of a spin-1 intermediate boson in $SU(\infty)$-QGR.

This test of QGR has the advantage to be intrinsically multi-messenger. Incoherent radiation from the galaxy or cluster can be used to model the lens and its gravitational field, and various baryonic effects on the maser photons, which would be the main source of uncertainty for the observation of a tiny QGR signal. The large volume and depth of future radio surveys should be able to detect with better resolution more lensed galaxies with maser sources.

## 5.2 Perspective for further investigations

Aside from seeking best methods for testing $SU(\infty)$-QGR, future works should investigate formulation of dynamics of subsystems as open quantum systems, because there is no truly isolated quantum system in the Universe. An open system approach would be particularly necessary for the design of experimental setups for the test of $SU(\infty)$-QGR and other QGR models, for example the experiment proposed in [68]. Other topics to be studied, both for verifying the capability of $SU(\infty)$-QGR to solve puzzles of QGR and for determining its predictions include: physics of quantum black holes, their evaporation, and puzzles of apparent information loss and singularity; QGR effects in the early Universe and inflation; nature of dark energy and whether it has a QGR origin; particle physics in the framework of $SU(\infty)$-QGR - in other words how the Universe was fragmented to subsystems, emergence of local symmetries which have ultimately led to the Standard Model at low energies; and potential role of QGR in matter-antimatter symmetry breaking.

## Acknowledgments

The author acknowledges the support of the Institut Henri Poincaré UAR 839 CNRS-Sorbonne Université), and the LabEx CARMIN (ANR-10-LABX-59-01).

## Funding statement

This work did not have dedicated funding.

<footnote n="12">¹²Photons of a maser beam are entangled by their phase. A priori the difference between entanglement entropy of images could be used as a complementary signature of QGR. However, different optical path of images, short coherence time of the maser photons, and different arrival time of the images means that their entanglement before getting lensed cannot be used.</footnote>
## Conflicts of Interest statement

This work does not have any known conflict of interest.

## Data Availability Statement

Results of this work can be freely used if the present work is cited.

## A A brief review of SU($\infty$) representations

The symmetry group SU($\infty$) and its relevance for physics were first noticed in the 1980's when J. Hoppe [6] demonstrated its homomorphism, in what is called sphere basis, with the group of area-preserving diffeomorphism ADiff of 2D sphere $S^{(2)}$, see e.g. appendices of [1] for a brief review of its algebra. In this basis the algebra does not allow a central charge, and thereby is an infinite dimensional algebra independent of Virasoro algebra. The latter is another infinite dimensional algebra extensively studied and employed in physics since 1970's for construction and quantization of string or more generally membrane theories and random matrices or M-theories. Indeed, the main subject of interest for [6] was field theories on 2D (pseudo)-Riemannian surfaces, assumed to be immersed in multi-dimensional Lorentzian manifolds. Follow-up investigations of SU($\infty$) were specially motivated by infinite dimensional matrices as symplectic approximation of 2D surfaces, SU($N \to \infty$) symmetry represented by such matrices, and their diffeomorphism as 2D gravity [7, 8, 15, 10, 12, 11].

Consider the area integral of a 2D compact surface $D_2$ immersed in a $(1, d-1)$ dimensional Minkowski space $M$. The induced metric on $D_2$ can be described with respect to coordinates $X_\alpha$, $\alpha = 0, \dots, d-1$ of $M$ [6, 7]. The coordinates $X_\alpha$ on $D_2$ are scalar functions of local coordinates $(\eta, \zeta)$ of $D_2$ and time. Thus, the area integral can be considered as action for dynamical variables $X_\alpha$ and their dynamics can be obtained by application of variational principle. The solution of field equations corresponds to minimizing or fixing the area of the membrane $D_2$, and it must be invariant under reparameterization of its coordinates. This implies that the Jacobian of transformation $(\eta, \zeta) \to (f(\eta, \zeta), g(\eta, \zeta))$ must be unity:

$$ \mathrm{Det}\left(\frac{\partial(f,g)}{\partial(\eta, \zeta)}\right) = \frac{\partial f}{\partial \eta} \frac{\partial g}{\partial \zeta} - \frac{\partial g}{\partial \eta} \frac{\partial f}{\partial \zeta} = \{f,g\} = 1 \quad (35) $$

where $\{f,g\}$ is the Poisson bracket of functions $f$ and $g$. It can be shown [6, 7] that infinitesimal area preserving coordinate transformations, that is ADiff($D_2$) can be generated by operators having the following form:

$$ \hat{L}_f = \frac{\partial f}{\partial \eta} \frac{\partial}{\partial \zeta} - \frac{\partial f}{\partial \zeta} \frac{\partial}{\partial \eta} , \quad \hat{L}_g = \{f,g\} \quad (36) $$

where $f$ is any $C^\infty$ scalar function on $D_2$. The Lie algebra of these operators is:

$$ [\hat{L}_f, \hat{L}_g] = \hat{L}_{\{f,g\}}, \quad (37) $$

The functions $f$ and $g$ can be in general complex valued. This is crucial when this algebra and its associated group is assumed to be the symmetry of the Hilbert space of a quantum system.

Functions on $D_2$ can be locally decomposed to any orthogonal function basis. However, area of a compact surface is a non-local property and topology of $D_2$ may dictate a globally preferred basis. For instance, for $D_2 = S^{(2)}$ a suitable basis for definition of generators $\hat{L}_f$ in (37) is the set of spherical harmonic functions $Y_{lm}(\theta, \phi)$, where $\theta$ and $\phi$ are angular coordinates of $S^{(2)}$. Thus, $(\eta, \zeta) \equiv (\cos \theta, \phi)$ and generators are indexed by $l = 0, 1, \dots, -l \le m \le l$. In [6] it
is demonstrated that the Lie algebra of generators $\hat{L}_{lm}(\theta, \phi)$ is homomorphic to the algebra $SU(N \to \infty)$. They have the expression (36) for $f = Y_{lm}(\theta, \phi)$. Properties of this representation of $SU(\infty)$ and its application to the Hilbert space of the Universe and its subsystems in the framework of $SU(\infty)$-QGR are studied in some details in [2, 1].

Another interesting example is $D_2 = T^{(2)}$, a 2D torus. In this case, the globally valid basis has the form of a 2D harmonic oscillator [7, 8, 10, 11, 9]:

$$f(\eta, \zeta) \equiv \exp[i(m_1\eta + m_2\zeta)] = \exp(i\mathbf{m}\cdot\mathbf{x}), \quad \mathbf{m} \equiv (m_1, m_2), \quad \mathbf{x} \equiv (\eta, \zeta). \qquad (38)$$

and the algebra (37) becomes:

$$[\hat{L}_{\mathbf{m}}, \hat{L}_{\mathbf{n}}] = (\mathbf{m} \times \mathbf{n}) \hat{L}_{\mathbf{m}+\mathbf{n}} \qquad (39)$$

$$\hat{L}_{\mathbf{m}} = -i \exp(i\mathbf{m}\cdot\mathbf{x})(\mathbf{m} \times \partial_{\mathbf{x}}), \quad \partial_{\mathbf{x}} = (\partial/\partial\eta, \partial/\partial\zeta). \qquad (40)$$

The symbol $\times$ means vector product in 2D. For instance, $\mathbf{m} \times \mathbf{n} = m_1 n_2 - m_2 n_1$. It is demonstrated [13] that $ADiff(T^{(2)})$ is also homomorphic to $SU(\infty)$. The difference between $SU(N \to \infty)$ related to $ADiff(S^{(2)})$ and that related $ADiff(T^{(2)})$ is in the way the number of roots in the $SU(N \to \infty)$ Dynkin diagram is extended to infinity [13]. The group $ADiff(S^{(2)})$ is homomorphic to $SU(\infty)_+$, corresponding to one side extension of Dynkin's diagram to infinity. By contrast, $ADiff(T^{(2)})$ is homomorphic to a $SU(N \to \infty)$ which its Dynkin's root diagram is extended to infinity from both sides. These results show that there are more than one $SU(\infty)$ group and they may not be homomorphic to each others.

The algebra (39) can have a central charge without violating its homomorphism with $ADiff(T^{(2)})$ [8]. This property demonstrates its relationship with Virasoro algebra, when the components of indices $\mathbf{m}$ and $\mathbf{n}$ are integer [7, 8]. Moreover, in [10] it is shown that this algebra is a special case of a more general class of algebras of the form:

$$[\hat{K}_{\mathbf{m}}, \hat{K}_{\mathbf{n}}] = c(\mathbf{m}, \mathbf{n}) \hat{K}_{\mathbf{m}+\mathbf{n}} \qquad (41)$$

The Jacobi identity of structure coefficients $c(\mathbf{m}, \mathbf{n})$ is solved by $c(\mathbf{m}, \mathbf{n})\alpha(m-n)$, corresponding to Virasoro algebra, and by $c(\mathbf{m}, \mathbf{n})\alpha\sin(k\mathbf{m} \times \mathbf{n})$:

$$[\hat{K}_{\mathbf{m}}, \hat{K}_{\mathbf{n}}] = r \sin(k(\mathbf{m} \times \mathbf{n})) \hat{K}_{\mathbf{m}+\mathbf{n}} \qquad (42)$$

It is straightforward to see that (39) is a special case of (42) in which $r = 1/k$ and $k \to 0$. It is demonstrated that these algebras are homomorphic to $SU(\infty)$ [12, 14]. Here we review the proof for the case of $k = 2\pi/N$, $N \in \mathbb{Z}$, $(m_1, m_2)$, $(n_1, n_2) \in (\mathbb{Z}, \mathbb{Z})$:

$$[\hat{K}_{\mathbf{m}}, \hat{K}_{\mathbf{n}}] = \left(\frac{N}{2\pi}\right) \sin\left(\frac{2\pi}{N}(\mathbf{m} \times \mathbf{n})\right) \hat{K}_{\mathbf{m}+\mathbf{n}} \qquad (43)$$

There is a generalization of Pauli matrices for generators of $SU(N)$ [69, 70]:

$$g \equiv \mathrm{diag}\{1, \omega, \omega^2, \dots, \omega^{N-1}\}, \quad \omega \equiv 2\pi i/N, \qquad (44)$$

$$\mathbf{h} \equiv \begin{bmatrix} 0 & 1 & 0 & \dots \\ 0 & 0 & 1 & 0 \dots \\ \vdots & & & \\ 0 & 0 & 0 & \dots 1 \\ 1 & 0 & 0 & 0 \dots \end{bmatrix} \qquad (45)$$

$$\mathbf{h}g = \omega\mathbf{g}\mathbf{h}, \quad \mathbf{g}^N = \mathbf{h}^N = \mathbf{1}. \qquad (46)$$

In this basis unitary unimodular $N \times N$ matrices, that is unitary matrices with determinant $\pm 1$ representing $SU(N)$ group can be factorized to $g$ and $\mathbf{h}$. In particular, matrices (operators) $\hat{K}_{\mathbf{m}}$ satisfying the algebra (43) with integer or more generally fractional $N$ can be written as:

$$\hat{K}_{\mathbf{m}} = \omega^{m_1 m_2} \mathbf{g}^{m_1} \mathbf{h}^{m_2} \qquad (47)$$
Using (46) it is straightforward to show that these operators satisfy (43). Hence, this algebra is isomorphic to $SU(\infty)$ when $N \to \infty$, or equivalently $k = 2\pi/N \to 0$. In this case the algebra (43) asymptotically becomes equal to (39), which is homomorphic to $ADiff(T^{(2)})$. We will later explain what we mean by asymptotic equality.

It is clear that in (43) there is only $N-1$ independent structure coefficients, because they are invariant under shifting of indices:

$$ (m_1, m_2) \to (m_1 + lN, m_2 + qN), \quad l, q \in \mathbb{Z} \qquad (48) $$

This symmetry is in addition to modular transformation $\mathbf{m} \to \mathbf{m}' = U\mathbf{m}$, where $U$ is a $2 \times 2$ unimodular matrix with integer elements. Owing to these symmetries, the space of indices that characterize the algebra (43) can be considered as a 2D lattice of cell size $N$ in which boundaries are pairwise identified, that is $m_1 + lN = m_1 + l'N$ and $m_2 + qN = m_2 + q'N$. Here, the symbol $=$ means periodic identification. This structure corresponds to flat space representation of high genus compact Riemann 2D surfaces. Therefore, when $N \to \infty$ the algebra (43), which owing to (47) is associated to $SU(\infty)$, is also homomorphic to $ADiff$ of compact orientable Riemann surfaces of any genus. Moreover, indices $(m_1 + lN, m_2 + qN)$ enumerate independent cocycles of these surfaces. A detailed proof of the homomorphism of $ADiff$ of any compact orientable 2D Riemann surface with $SU(N \to \infty)$ for $N$ countable is given in [16]. We call *diffeo-surface* those surfaces which their $ADiff$ is at least asymptotically homomorphic to $SU(\infty)$.

The algebra (43) does not include any information about diffeo-surfaces to which the corresponding $SU(N \to \infty)$ is homomorphic for a given $N$, because the structure coefficients of this algebra are independent of cocycle indices $(l, q)$ in (48). On the other hand, diffeomorphism cannot transfer Riemann surfaces with different genera to each others. Therefore, diffeo-surfaces with different genus should be considered as different representations of $SU(\infty)$ symmetry.

## B No exact description of sine algebra on a 2D surface

In [12] it is claimed that the following operators generate the algebra (42):

$$ \hat{K}_{\mathbf{m}} \equiv \left(\frac{ir}{2}\right) \exp(i\mathbf{m}\cdot\mathbf{x}) \hat{G}_{\mathbf{m}}(k) \quad , \quad \hat{G}_{\mathbf{m}}(k) \equiv \exp(k \mathbf{m} \times \partial_{\mathbf{x}}) \qquad (49) $$

It is straightforward to show that:

$$ [\exp(i\mathbf{m}\cdot\mathbf{x}), \hat{G}_{\mathbf{n}}(k)] = -\exp(i\mathbf{m}\cdot\mathbf{x}) \exp(ik \mathbf{m} \times \mathbf{n}) \qquad (50) $$

$$ [\hat{K}_{\mathbf{m}}, \hat{K}_{\mathbf{n}}] = \left(\frac{ir}{2}\right)^2 \exp(i(\mathbf{m} + \mathbf{n})) \left( \exp(ik \mathbf{m} \times \mathbf{n}) \hat{G}_{\mathbf{n}}(k) - \exp(-ik \mathbf{m} \times \mathbf{n}) \hat{G}_{\mathbf{m}}(k) \right) \qquad (51) $$

Only when $k \to 0$ and $|\mathbf{m}|, |\mathbf{n}| \gg 0$, the r.h.s. of (51) approximately corresponds to the r.h.s. of (42), and for all other values of $k$, the r.h.s. of (51) and (42) are different. Thus, question arises whether another expression for $\hat{K}_{\mathbf{m}}$ as a functional of $\mathbf{x}$ and $\partial_{\mathbf{x}}$ can be found such that their commutations exactly corresponds to (42). Here we prove that there is no such expression.

Consider $\hat{K}_{\mathbf{m}} = f(\mathbf{x}, \mathbf{m} \times \partial_{\mathbf{x}})$. As $\mathbf{x}$ and $\partial_{\mathbf{x}}$ do not commute, an analytical expression for $f$ is ambiguous and needs an ordering. Giving the fact that at this stage there is no constraint on $f$, without loss of generality we can assume $\hat{K}_{\mathbf{m}} = f_1(\mathbf{x})f_2(\mathbf{m} \times \partial_{\mathbf{x}})$. The functions $f_1$ and $f_2$ can be separately Fourier expanded:

$$ f_1(\mathbf{x}) = \sum_{\mathbf{s}} F_1(\mathbf{s}) \exp(i\mathbf{s}\cdot\mathbf{x}), \quad \sum_{\mathbf{q}} F_2(\mathbf{q}) \exp(i\mathbf{q}\cdot\mathbf{m} \times \partial_{\mathbf{x}}) \qquad (52) $$

$$ \hat{K}_{\mathbf{m}} = \sum_{\mathbf{s},\mathbf{q}} F_1(\mathbf{s}) F_2(\mathbf{q}) \exp(i\mathbf{s}\cdot\mathbf{x}) \exp(i\mathbf{q}\cdot\mathbf{m} \times \partial_{\mathbf{x}}) \qquad (53) $$
Notice that $\mathbf{s}$ and $\mathbf{q}$ are in general continuous variables and can be also complex. Thus, sums in (52-53) can be rather integrals. Using these expansions, we find:

$$ 
\begin{aligned}
[\hat{K}_{\mathbf{m}}, \hat{K}_{\mathbf{n}}] &= \sum_{\mathbf{s}, \mathbf{q}, \mathbf{s}', \mathbf{q}'} F_1(\mathbf{s}) F_2(\mathbf{q}) F_1(\mathbf{s}') F_2(\mathbf{q}') \exp(i(\mathbf{s} + \mathbf{s}')) \\
&\quad \left( \exp(-\mathbf{q} \mathbf{m} \times \mathbf{s}') \exp(i\mathbf{q}' \mathbf{n} \times \partial_{\mathbf{x}}) - \exp(-\mathbf{q}' \mathbf{n} \times \mathbf{s}) \exp(i\mathbf{q} \mathbf{m} \times \partial_{\mathbf{x}}) \right) \\
&= \sum_{\mathbf{s}, \mathbf{q}, \mathbf{s}', \mathbf{q}'} F_1(\mathbf{s}) F_2(\mathbf{q}) F_1(\mathbf{s}') F_2(\mathbf{q}') \exp(i(\mathbf{s} + \mathbf{s}')) \\
&\quad \left( \exp(-\mathbf{q} \mathbf{m} \times \mathbf{s}') \hat{G}_{\mathbf{n}}(i\mathbf{q}') - \exp(-\mathbf{q}' \mathbf{n} \times \mathbf{s}) \hat{G}_{\mathbf{m}}(i\mathbf{q}') \right)
\end{aligned} 
\quad (54) $$

Comparison of (54) with (51) shows that the latter corresponds to a special case where $F_2 \neq 0$ only for one mode $\mathbf{q} = -i\mathbf{k}$ and $\mathbf{s} = \mathbf{m}$. Therefore, giving the independence of Fourier modes, the more general case of the algebra presented in (54) cannot become homomorphic to (42) for any choice of the functions $f_1$ and $f_2$.

### C Equivalence of amplitude factorization and algebraic criteria for compositeness

Product states satisfy the three criteria for division of a quantum system to multiple subsystems according to [19]:

**Commuting operator blocks:** We assume that $A_{k_1 k_2 \cdots k_n}$ in (2) is factorized to $m$ blocks of dimension $r_j, j = 1, \cdots, m$ and:

$$ \sum_{j=1, \cdots, m} r_j = n \quad (55) $$

Each factor is associated to a subspace $\mathcal{H}_j \subset \mathcal{H}$ of the many-body system Hilbert space $\mathcal{H}$. The set of parameters $\{k_{j,i}, i = 1, \cdots, r_j\}$ characterize vectors of $\mathcal{H}_j$. Using the expansion (2), a factorized state $|\Psi_f\rangle$ can be written as:

$$ 
\begin{aligned}
|\Psi_f\rangle &= \sum_{\substack{j=1,\cdots,m, \\ \{k_{j,i}, i=1,\cdots,r_j\}}} A_{k_1,1}^{(1)} k_{1,2} \cdots k_{1,r_1} \cdots A_{k_m,1}^{(m)} k_{m,2} \cdots k_{m,r_m} \\
&\quad |\Psi_1(k_{1,1}, \cdots, k_{1,r_1})\rangle \times \cdots \times |\Psi_m(k_{m,1}, \cdots, k_{m,r_m})\rangle \\
&= \bigotimes_{j=1,\cdots,m} \left( \sum_{\{k_{j,i}, i=1,\cdots,r_j\}} A_{k_{j,1},1}^{(j)} k_{j,2} \cdots k_{j,r_j} |\Psi_j(k_{j,1}, \cdots, k_{j,r_j})\rangle \right)
\end{aligned} 
\quad (56) $$

Vectors $|\Psi_j\rangle$ may be in turn expandable with respect to a tensor basis depending on $k_{j,i}$:

$$ |\Psi_j(k_{j,1}, \cdots, k_{j,r_j})\rangle = |\psi_{k_{j,1}}\rangle \times |\psi_{k_{j,2}}\rangle \times \cdots \quad (57) $$

and the set of $\{|\psi_{k_{j,i}}\rangle\}$ constitute a tensor product basis for $\mathcal{H}$. But, by assumption amplitudes $A_{k_{j,1} k_{j,2} \cdots k_{j,r_j}}^{(j)}$ would not be further factorizable. Moreover, (57) shows that for different $j$'s states $|\Psi_j\rangle$ do not share characteristic parameters $k_{j,i}$. Thus, subspaces $\mathcal{H}_j$ are disjoint. This is the reason for the last equality in (56) and shows that the factorization of amplitudes of a state with a tensor product basis leads to symmetry breaking and factorization of the Hilbert space:

$$ \mathcal{H} \supseteq \mathcal{H}_1 \times \cdots \mathcal{H}_m \quad (58) $$

Therefore, $\forall \hat{A} \in \mathcal{B}[\mathcal{H}_i]$ and $\hat{B} \in \mathcal{B}[\mathcal{H}_j], i \neq j, [\hat{A}, \hat{B}] = 0$.
**Locality:** When the classical spacetime coordinates $x$ are included in the set of parameters characterizing state of a quantum system, locality means that components of the density matrix become negligibly small for $|x - y| \to \infty$, $x, y \in \mathbb{R}^{(3+1)}$ for $^{13}$. More generally, $x$ can be any characterizing parameter of the quantum state. Lack of locality means impossibility of tensor decomposition, even as an approximation. Locality condition is fulfilled if the factorized amplitudes are local. Endomorphism between characterizing parameters of factorized and unfactorized amplitudes ensures that if it is satisfied by the full basis, it is also satisfied by the factorized one.

**Complementarity:** A map $\phi: \mathcal{H} \to \mathcal{H}$ can be defined such that each factorized state $|\Psi_f\rangle$ is projected to an unfactorized state. The endomorphism condition is satisfied if:

$$ \phi(|\Psi_f\rangle) = |\Psi\rangle, \quad A_{k_1,1}^{(1)} A_{k_1,2}^{(1)} \cdots A_{k_1,r_1}^{(1)} \cdots A_{k_m,1}^{(m)} A_{k_m,2}^{(m)} \cdots A_{k_m,r_m}^{(m)} = A_{k_1 k_2 \cdots k_n} \quad (59) $$

where $\{k_l\}$ characterizes a basis for the full Hilbert space $\mathcal{H}$. Considering (55), there is an endomorphism between parameter sets $\{k_{i,j}, i = 1, \cdots, m; j = 1, \cdots, r_i\}$ and $k_l, l = 1, \cdots; n$. Therefore, complementary condition is fulfilled by states with factorized amplitudes.

In $SU(\infty)$-QGR the state of the Universe is not truly factorizable, because the components in a tensor product basis similar to (1) constitute the assumed global $SU(\infty)$ symmetry of the Hilbert space $\mathcal{H}_U$. Hence, any factorization of amplitudes can be reverted by a $SU(\infty)$ transformation under which observables remain unchanged. Nonetheless, approximate factorization, symmetry breaking, and division to subsystem arise due to quantum fluctuations and locality of dynamics.

## D Homomorphism of $SU(\infty)$ and $G \times SU(\infty)$

An explicit homomorphism between generators of $SU(\infty)$ and $G \times SU(\infty)$ can be established as the following. First consider the simplest case of $G = SU(2)$. Using torus basis (40) for the definition of $SU(\infty)$ generators $\hat{L}_m$, the following bijection is an example of possible maps between $\hat{\sigma}_i \times \hat{L}_m$, where $\hat{\sigma}_i, i = 1, 2, 3$ are Pauli matrices, and $\hat{L}_m$:

$$ \Phi_{\text{tor}} : \{\hat{\sigma}_i \times \hat{L}_m\} \to \{\hat{L}_m\} \mid \Phi_{\text{tor}}(\hat{\sigma}_i \times \hat{L}_m) = \hat{L}_{3m+i}, \quad i = (i-1, i-1) \quad (60) $$

In the sphere basis [6] (reviewed in Appendix D of [1]) we can use formal description of $SU(\infty)$ generators $\hat{L}(\theta, \phi)$ with respect to angular coordinates on sphere to define a bijection:

$$ \Phi_{\text{sph}} : \{\hat{\sigma}_i \times \hat{L}(\theta, \phi)\} \to \{\hat{L}(\theta, \phi)\} \mid \Phi_{\text{sph}}(\hat{\sigma}_i \times \hat{L}(\theta, \phi)) = \hat{L}(\theta, \frac{\phi + 2\pi(i-1)}{3}), \\ \theta = [0, \pi], \quad \phi = [0, 2\pi] \quad (61) $$

This example can be easily extended to other finite rank $G$. Specifically, (60) can be generalized to:

$$ \Phi_{\text{tor}} : \{\hat{\lambda}_i \times \hat{L}_m\} \to \{\hat{L}_m\} \mid \Phi_{\text{tor}}(\hat{\lambda}_i \times \hat{L}_m) = \hat{L}_{d_G m+i}, \quad i = (i-1, i-1), \quad i = 1, \cdots, n_G \quad (62) $$

where $n_G$ is the dimension of $G$, that is the number of generators $\hat{\lambda}_i$.

<footnote n="13">¹³This is a naive definition of locality for a quantum system, but sufficient for our purpose here. A more rigorous definition should consider von Neumann classification of the system, see e.g. [71] for a review.</footnote>
## E Entanglement measures

Entanglement of quantum systems can be quantified and several entanglement measures are proposed in the literature [72, 35, 73, 74]. Here we calculate entanglement entropy and negativity for unspecified subsystems of the Universe. These estimations would be useful for future application of SU(∞)-QGR to physical systems and processes, such as black holes and their information puzzle, inflation, particle production during reheating of the Universe, and decoherence of quantum systems by gravitational interaction.

### E.1 Entanglement entropy

The amount of entanglement is reflected in mutual information $I(G : SU(∞)) = S(\hat{\rho}_G) + S(\hat{\rho}_∞) - S(\hat{\rho}_U)$, where $S(\hat{\rho}) = -\text{tr}(\hat{\rho} \ln \hat{\rho})$ is the von Neumann entropy. As $\hat{\rho}_U$ is pure, $S(\hat{\rho}_U) = 0$. In this case $S(\hat{\rho}_G) = S(\hat{\rho}_∞)$, see e.g. [72]. This can be confirmed by explicit calculation using (9) and (12). We find:

$$I(\hat{\rho}_G : \hat{\rho}_\infty) = 2S(\hat{\rho}_G) = 2S(\hat{\rho}_\infty) = -4 \sum_{\{k_G; (\eta, \zeta, \dots)\}} |A(k_G; \eta, \zeta, \dots)|^2 \ln |A(k_G; \eta, \zeta, \dots)| \quad (63)$$

This calculation also demonstrates that the reason for $S(\hat{\rho}_G) = S(\hat{\rho}_\infty)$ is the complementarity of $H_G$ and $H_\infty$ as subspaces of $H_U$. Using normalization and unitarity constraint (8), we find that mutual information $I(\hat{\rho}_G : \hat{\rho}_\infty) \ge 0$, which demonstrates the entanglement of $\hat{\rho}_G$ and $\hat{\rho}_\infty$ states and corresponding subsystems. Moreover, as (63) shows, the entropy cannot be divided to geometrical - SU(∞) related - and internal. Indeed, this is what the Einstein equation tells us. Gravitational effects represented by the curvature of spacetime is directly related to the state of matter - subsystem of the Universe - represented by their energy-momentum tensor, and vis-versa.

It is useful to compare the entanglement entropy (63) which depends on the degrees of freedom of both internal symmetry of the subsystem and its quantum gravity, with the entanglement entropy in QGR proposals based on the holography principle, such as AdS/CFT [75, 76, 77] and related models that interpret variation of entanglement as gravity [30, 31, 32]. First of all, equality of the Bekenstein entropy in a bulk spacetime with entropy of a dual Conformal Field Theory (CFT) on a boundary is demonstrated only for (2+1)D bulk with Anti-de Sitter (AdS) geometry. Evidently, this setup does not correspond to the (3+1)D physical spacetime with a positive cosmological constant and FLRW geometry. Even accepting the conjecture of the existence of a dual CFT for any geometry, the spacetime must have at least a virtual boundary on which the dual CFT could be defined and used for calculation of the geometrical entropy. AdS includes such a boundary, but FLRW does not. Although a priori horizon can play the role of a null surface, as requested in covariant formulation of holography [78, 79], it does not solve the issue of indivisibility of QFT's due to their type III von Neumann algebra, see e.g [71] for a review. None of such restrictions exists in SU(∞)-QGR.

### E.2 Entanglement negativity

Another entanglement measure frequently advocated in the literature is entanglement negativity or the related quantity called logarithmic negativity [73, 74]. The latter is a monotone but non-convex measure of entanglement [74]. For a density matrix $\hat{\rho}$ they are defined as:

$$N(\hat{\rho}_g) = \frac{\|\hat{\rho}_g^T\|_1 - 1}{2}, \quad LN(\hat{\rho}_g) = \log \|\hat{\rho}_g^T\|_1 \quad (64)$$

where for any operator $\hat{O}$, $\|\hat{O}\|_1 = \text{tr}|\hat{O}| = \text{tr}\sqrt{\hat{O}^\dagger\hat{O}}$, and $\hat{\rho}_g^T$ is the partial transpose of $\hat{\rho}$ over its subspace g. We remind that partial transpose preserves the trace. Therefore, $\hat{\rho}_g^T$ is also a
density matrix. Entanglement negativity has the advantage of being purely quantum, in contrast to mutual information which is also defined for classical systems. In particular, it better reflects evolution of entanglement, see e.g. [80].

The partial transpose of $\hat{\rho}_U$ in (6) with respect to $G$ subspace, that is the subspace generated by $\hat{\rho}_G$ is:

$$ \hat{\rho}_{UG}^T = \sum_{\substack{k_G, k'_G \\ (\eta, \zeta, \eta', \zeta', \dots)}} A(k_G; \eta, \zeta, \dots) A^*(k'_G; \eta', \zeta', \dots) \hat{\rho}_G(k'_G, k_G) \times \hat{\rho}_{\infty}(\eta, \zeta, \eta', \zeta', \dots) \quad (65) $$

It is straightforward to verify that $\hat{\rho}_{UG}^{T\dagger} = \hat{\rho}_{UG}^T$. Therefore, $\hat{\rho}_{UG}^T$ is also a density operator and $\text{tr}\hat{\rho}_{UG}^T = 1$. Thus:

$$ ||\hat{\rho}_{UG}^T||_1 = \text{tr}\sqrt{\hat{\rho}_{UG}^{T\dagger}\hat{\rho}_{UG}^T} = \text{tr}|\hat{\rho}_{UG}^T| \quad (66) $$

The necessary - but not sufficient - condition for separability of $\hat{\rho}_U$ is positivity of $\hat{\rho}_{UG}^T$ eigen values [35]. In this case $\text{tr}|\hat{\rho}_{UG}^T| = \text{tr}\hat{\rho}_{UG}^T = 1$ and negativity of $\hat{\rho}_U$ would be zero. However, explicit calculation shows that $(\hat{\rho}_{UG}^T)^2 \neq \hat{\rho}_{UG}^T$:

$$ (\hat{\rho}_{UG}^T)2 = \frac{1}{2} \left\{ \sum_{\Omega} \left( A(k_1; \eta_1, \zeta_1, \dots) A^*(k'_1; \eta'_1, \zeta'_1, \dots) A(k_2; \eta'_2, \zeta'_2, \dots) A^*(k_1; \eta'_2, \zeta'_2, \dots) \right. \right. \\ \left. \left. \hat{\rho}_G(k'_1, k_2) \times \hat{\rho}_{\infty}(\eta'_1, \zeta'_1; \eta'_2, \zeta'_2, \dots) \right) + \right. \\ \left. \sum_{\Omega'} \left( A(k_2; \eta'_2, \zeta'_2, \dots) A^*(k'_2; \eta_1, \zeta_1, \dots) A(k_1; \eta_1, \zeta_1, \dots) A^*(k_2; \eta'_1, \zeta'_1, \dots) \right. \right. \\ \left. \left. \hat{\rho}_G(k'_2, k_1) \times \hat{\rho}_{\infty}(\eta_2, \zeta_2, (\eta'_1, \zeta'_1, \dots)) \right) \right\} \quad (67) $$

$$ \Omega \equiv \{k_1, k'_1, k_2\}; \{(\eta_1, \zeta_1, \dots)\}; \{(\eta'_1, \zeta'_1, \dots)\}; \{(\eta'_2, \zeta'_2, \dots)\}; \quad (68) $$

$$ \Omega' \equiv \{k_1, k_2, k'_2\}; \{(\eta_1, \zeta_1, \dots)\}; \{(\eta'_1, \zeta'_1, \dots)\}; \{(\eta_2, \zeta_2, \dots)\}; \quad (69) $$

It is important to notice that tensor product basis $\hat{\rho}_G$ and $\hat{\rho}_{\infty}$ depend only on two sets of parameters $\{k\}$ and $\{\eta, \zeta, \dots\}$. There are, however, three sets of parameters in the summations in (67). They are indicated as $\Omega$ and $\Omega'$ in (68) and (69), respectively. For this reason, amplitudes in (67) cannot be factorized, unless $k$ and $(\eta, \zeta, \dots)$ dependence of $A(k; \eta, \zeta, \dots)$ can be factorized. However, the assumed $SU(\infty)$ symmetry of the Universe mixes the two sectors and in general dependency on $k$ and $(\eta, \zeta, \dots)$ of the amplitudes $A(k; \eta, \zeta, \dots)$ are not separable. Thus, $(\hat{\rho}_{UG}^T)^2 \neq \hat{\rho}_{UG}$, the density matrix $\hat{\rho}_{UG}$ is a mixed state, and is not separable. This means that $\hat{\rho}_{UG}$ may have negative eigen values [35], and $N(\hat{\rho}_U) > 0$, $LN(\hat{\rho}_U) > 0$. These results confirm the conclusion made from entanglement entropy and Proposition 1 in [2] about entanglement of every subsystem to the rest of the Universe.

## F Conditions for perfect purification

Imposing perfect purification conditions (18) does not guarantee that they would be satisfied in the framework of $SU(\infty)$-QGR and construction of subsystems as explained in Sec. 3.2 Necessary conditions for purification of a set of quantum states and faithfulness of such process are investigated in [33]. We apply their approach to mixed states such as (9) and (12) in $SU(\infty)$-QGR.

The process of purifying a mixed state $\hat{\rho} \in \mathcal{B}[\mathcal{H}]$ can be considered as a map:

$$ \Lambda: \mathcal{B}[\mathcal{H}] \to \mathcal{B}[\mathcal{H} \times \mathcal{H}'] \quad | \quad \Lambda(\hat{\rho}) = \hat{\rho}', \quad \hat{\rho} \in \mathcal{B}[\mathcal{H}], \quad \hat{\rho}' \in \mathcal{B}[\mathcal{H} \times \mathcal{H}'], \quad \hat{\rho}'^2 = \hat{\rho}'. \quad (70) $$
In order to project a density matrix to a density matrix, the map $\Lambda$ must be completely positive and trace preserving. Moreover, for a faithful purification $\Lambda$ must be linear<fnref n="14" />. Such maps can be obtained from appending a pure ancilla state $|\chi\rangle\langle\chi|$ to a mixed state $\hat{\sigma}$ and performing a unitary rotation $U$ on the extended state. On the other hand, purifying process needs extending the state and its Hilbert space by an auxiliary state $\hat{\omega}_{aux}$. A set of states are called essentially pure if the result of the above sets of operations are equal:

$$ \hat{\rho} \times \omega_{aux} = U(|\chi\rangle\langle\chi| \times \hat{\sigma})U^{\dagger} \quad (71) $$

It is demonstrated [33] that an essentially pure set of states, and more generally orthogonal union of such sets have a perfect purifier.

Here we show that the set of states $\hat{\rho}_G$ calculated in (9) are essentially pure and have a perfect purifier. Irrespective of the value of amplitudes $A_G$ in (9), the set of $\hat{\rho}_G$ states have the same general structure. Thus, it is enough to demonstrate essentially pureness for one of them. As we had traced out the $SU(\infty)$ component of the tensor product basis $\hat{\rho}_G \times \hat{\rho}_\infty$ in (9) to obtain $\hat{\rho}_G$, the natural candidate for the auxiliary Hilbert space to append to $\mathcal{H}_G$ is a space generated by the basis $\hat{\rho}_\infty(\eta, \zeta, \eta', \zeta', \dots)$. We choose the pure state $|\chi\rangle$ and the mixed state $\hat{\sigma}$ belonging to a representation of $SU(\infty)$ as the followings:

$$ |\chi\rangle \equiv \sum_{k_G} \chi(k_G) |k_G\rangle, \qquad \hat{\chi}_G = |\chi\rangle\langle\chi| \quad (72) $$

$$ \hat{\sigma} = \sum_{\{(\eta, \zeta, \dots)\}, \{(\eta', \zeta', \dots)\}} \sigma(\eta, \zeta; \eta', \zeta'; \dots) \hat{\rho}_{\infty}(\eta, \zeta; \eta', \zeta'; \dots), \quad (73) $$

In Appendix D we proved that $G \times SU(\infty) \cong SU(\infty)$. Thus, $\hat{\chi}_G \times \hat{\sigma} \in \mathcal{B}[\mathcal{H}_{G\infty}] \cong \mathcal{B}[\mathcal{H}_U]$ and the unitary operator $U$, which must perform a global rotation is also in $\mathcal{B}[\mathcal{H}_{G\infty}]$. The result of this rotation can be expanded as:

$$ U(\hat{\chi}_G \times \hat{\sigma})U^{\dagger} = \sum_{\{(k_G, k'_G) \\ (\eta, \zeta, \eta', \zeta', \dots)\}} C(k_G, k'_G; \eta, \zeta; \eta', \zeta'; \dots) \hat{\rho}_G(k_G, k'_G) \times \hat{\rho}_{\infty}(\eta, \zeta, \eta', \zeta'; \dots), \quad (74) $$

Thus the r.h.s. of (71) for $\hat{\rho}_G$ is also in $\mathcal{B}[\mathcal{H}_{G\infty}]$, and a minimal choice for $\hat{\omega}_{aux}$ would be an operator in $\mathcal{B}[\mathcal{H}_{\infty}]$ expanded by the basis $\hat{\rho}_{\infty}$:

$$ \hat{\omega}_{aux} = \sum_{\{(\eta, \zeta, \dots)\}, \{(\eta', \zeta', \dots)\}} \omega(\eta, \zeta; \eta', \zeta'; \dots) \hat{\rho}_{\infty}(\eta, \zeta; \eta', \zeta'; \dots), \quad (75) $$

The essentially pure condition (71) is satisfied for $\hat{\rho}_G$ and above choices of auxiliary Hilbert space and associated operators if:

$$ A_G(k_G; \eta, \zeta, \dots) A_G^*(k'_G; \eta, \zeta, \dots) \omega(\eta, \zeta; \eta', \zeta'; \dots) = C(k_G, k'_G; \eta, \zeta; \eta', \zeta'; \dots), \quad \forall k_G, k'_G; \eta, \zeta; \eta', \zeta'; \dots \quad (76) $$

and the condition for perfect purification is satisfied if:

$$ \sum_{\{(\eta', \zeta', \dots)\}} C(k_G, k'_G; \eta, \zeta; \eta', \zeta'; \dots) = A_G(k_G; \eta, \zeta, \dots) A_G^*(k'_G; \eta, \zeta, \dots) \quad (77) $$

After summing (76) over $\{(\eta', \zeta', \dots)\}$, we find the following constraint for $\omega$:

$$ \sum_{\{(\eta', \zeta', \dots)\}} \omega(\eta, \zeta; \eta', \zeta'; \dots) = 1 \quad (78) $$

This constraint on the auxiliary state $\omega$ is independent of how $A_G$ amplitudes in $\hat{\rho}_G$ depend on the external parameters $(\eta, \zeta, \dots)$. Therefore, this purification procedure is universal, and $\hat{\omega}_{aux}$ can be chosen such that $\hat{\rho}_G \times \hat{\omega}_{aux}$ be a faithful purification of $\hat{\rho}_G$.

<footnote n="14">¹⁴Faithfulness means tracing over the auxiliary space used for the purification should recover the original density matrix. Thus, if $\hat{\rho}_i$, $i = 1, 2$ can be faithfully purified by $\Lambda$, so does $\hat{\rho}_1 + \hat{\rho}_2$. Moreover, following equalities are satisfied: $\text{tr}_{aux}\Lambda(\hat{\rho}_i) = \hat{\rho}_i$ and $\text{tr}_{aux}\Lambda(\hat{\rho}_1) + \text{tr}_{aux}\Lambda(\hat{\rho}_2) = \text{tr}_{aux}\Lambda(\hat{\rho}_1 + \hat{\rho}_2)$. The latter equality would be valid for any density matrix only if $\Lambda(\hat{\rho}_1 + \hat{\rho}_2) = \Lambda(\hat{\rho}_1) + \Lambda(\hat{\rho}_2)$. Thus, $\Lambda$ must be linear.</footnote>
# G SU(∞) Yang-Mills QFT on a background spacetime

The SU(∞)-Yang-Mills theory on a background spacetime is first studied in [81]. For a finite rank non-Abelian symmetry G the gauge field is defined as:

$$A_{\mu}(x) = \sum_{a} A_{\mu}^{a} T^{a} \qquad (79)$$

where x is a point in the background spacetime and a is an index - usually an integer - enumerating symmetry generators $T^a$ of G in a given representation. When the symmetry group is the infinite rank SU(∞), the definition (79) still applies with the difference that index a consists of two independent continuous variables, because SU(∞) $\cong ADiff(D_2)$. For instance, in sphere basis $a = (\theta, \phi) \in S^{(2)}$ are angular coordinates on a sphere. Generators of SU(∞) algebra can be formally described with respect to these parameters, see appendices of [1]. Alternatively, they can be decomposed to spherical harmonic functions $Y_{lm}(\theta, \phi)$, which satisfy Poisson bracket (algebra) in the l.h.s. of (36). In turn, Poisson algebra is homomorphic to SU(∞) [6]. Therefore, $a = \{(\theta, \phi); (l \ge 0, |m| \le l)\}$. As we reviewed in Appendix A, in torus basis generators are expressed as 2D harmonic oscillators depending on two continuous parameters $(x_1, x_2) = \mathbf{x}$ and a 2D integer vector $(m_1, m_2) = \mathbf{m}$. Thus, $a = \{\mathbf{x}; \mathbf{m}\}$ [12].

Using sphere basis, which is used by [81], the SU(∞) gauge field defined on a background spacetime is expressed as:

$$A_{\mu}(x; \theta, \phi) = \sum_{lm} A_{\mu}^{lm}(x) \hat{L}_{lm}(\theta, \phi) \qquad (80)$$

The authors of [81] call the 2D surface generated by $(\theta, \phi)$ the internal space to distinguish it from the background spacetime. It corresponds to what we have called diffeo-surface of SU(∞) representation in the context of SU(∞)-QGR.

# H Quantum fields and density matrices

In QFT's a field operator $\hat{\phi}$ acts on a multi-particle Fock space and depends on the parameters constituting a space $\Xi$. The latter is usually - but not necessarily - the spacetime or its conjugate mode space. The field is often decomposed to creation and annihilation operators:

$$\hat{\phi}(y) = \hat{a}(y) + \hat{a}^{\dagger}(y), \quad y \in \Xi \qquad (81)$$

$$\hat{a}^{\dagger}(y)|\emptyset\rangle = |y\rangle, \quad \langle\emptyset|\hat{a}^{\dagger} = 0, \quad \hat{a}(y)|\emptyset\rangle = 0, \quad \langle\emptyset|\hat{a} = \langle y|, \quad [\hat{a}(y), \hat{a}^{\dagger}(y')] = 1. \qquad (82)$$

The third equality in (82) defines the vacuum state $|\emptyset\rangle$ and the first equality defines a single particle state at point y of the parameter space. The commutation (or anti-commutation for fermions) is the quantization condition of the field. Notice that although creation and annihilation operators $\hat{a}^\dagger$ and $\hat{a}$, respectively, act on the multi-particle Fock space, their action is restricted to a single particle/subsystem. Therefore, $\hat{a}(y), \hat{a}^\dagger(y), \hat{\phi}(y) \in \mathcal{B}[\mathcal{H}]$, where $\mathcal{H}$ is the single particle/subsystem Hilbert space parameterized by $\Xi$.

In a quantum mechanical view, the vacuum $|\emptyset\rangle$ can be identified with the null vector of the Hilbert space $\mathcal{H}$, and $|y\rangle, y \in \Xi$ with eigen vector of an operator, which with eigen value $y \in \Xi$. Therefore, a general 1-particle state $|\psi\rangle \in \mathcal{H}$ and its density matrix $\hat{\rho} \in \mathcal{B}[\mathcal{H}]$ can be written as:

$$|\psi\rangle = \sum_{y} \psi(y)|y\rangle = \sum_{y} \psi(y)a^{\dagger}(y)|\emptyset\rangle, \quad \sum_{y} |\psi(y)|^2 = 1, \quad \langle y|y'\rangle = \delta(y - y') \qquad (83)$$

$$\hat{\rho} = |\psi\rangle\langle\psi| = \frac{1}{2} \sum_{y,y'} [\psi(y)\psi^*(y')\hat{a}^\dagger(y)|\emptyset\rangle\langle\emptyset|\hat{a}(y') + C.C.] \qquad (84)$$

The description of density matrix with respect to creation and annihilation operators demonstrates that fields and density matrices are different way of presenting state of a quantum particle/system, and one can be easily deduced from the other. The formulation of dynamics using a
path integral defined on the configuration space with fields and their derivatives as generalized coordinates has its analogous with density matrices defined on the Hilbert space, see e.g. [82] (Chapter 6) for details.

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