## 2 Disorder in Field Theories  
Let us begin by considering \(N\) interacting bosonic particles of mass \(m\) enclosed in a \(L^{d}\) volume and subject to an external _disordered_ potential \(U_{\rm D}({\bf r})\). Thus, the Hamiltonian is simply given by  
\[\hat{H}=\sum_{i=1}^{N}\bigg{[}-\frac{\hbar^{2}\
abla_{i}^{2}}{2m}+\hat{U}_{\rm D }({\bf r}_{i})\bigg{]}+\frac{1}{2}\sum_{i\
eq j}\hat{V}\big{(}|{\bf r}_{i}-{\bf r }_{j}|\big{)}\, \tag{1}\]  
with \(\hat{V}\big{(}|{\bf r}_{i}-{\bf r}_{j}|\big{)}\) the two-body central interaction potential. The thermodynamic description of the system can be derived from the partition function. Within the canonical ensemble, it is defined as  
\[Z_{N}=\sum_{\{n\}}\exp\big{(}-\beta E_{n}\big{)}, \tag{2}\]  
where \(\beta\equiv(k_{B}T)^{-1}\) and \(\{n\}\) denoting the bosonic states (i.e., totally symmetric with respect to particle exchange) with eigenenergy \(E_{n}\) of the operator \(\hat{H}\) in Equation (1). In order to adopt the functional integration formalism [44], Equation (1) has to be expressed in second quantization. By adding \(-\mu\hat{N}\), with \(\mu\) the chemical potential and \(\hat{N}\) the number operator, we can also move to the grand canonical ensemble. Here, a basis of coherent bosonic states can be chosen, leading to the imaginary-time representation of the (grand canonical) partition function  
\[\mathcal{Z}=\int\mathcal{D}[\psi,\psi^{*}]\,e^{-S[\psi,\psi^{*}]/\hbar}. \tag{3}\]  
In the equation above, the Bose statistics is encoded in the periodic boundary condition for the fields, namely \(\psi(0)=\psi(\beta\hbar)\) and \(\psi^{*}(0)=\psi^{*}(\beta\hbar)\). The Euclidean action appearing in Equation (3) is given by  
\[ \begin{split} S[\psi,\psi^{*}]=\int_{0}^{\beta\hbar}d\tau\int d^ {d}{\bf r}\,\bigg{\{}\psi^{*}({\bf r},\tau)\bigg{[}\hbar\partial_{\tau}-\frac {\hbar^{2}\
abla^{2}}{2m}-\mu-U_{\rm D}({\bf r},\tau)\bigg{]}\psi({\bf r},\tau) \\ +\frac{1}{2}\int d^{d}{\bf r}^{\prime}|\psi({\bf r},\tau)|^{2}V(| {\bf r}-{\bf r}^{\prime}|)|\psi({\bf r}^{\prime},\tau)|^{2}\bigg{\}}\.\end{split}  \tag{4}\]  
We have now to clarify what is meant by a (quantum) field theory for the disordered Bose gas. Also in the simplest case of non-interacting bosons, given a certain realization of the disordered confinement, an exact diagonalization of Equation (1) does not appear as a feasible task. A statistical approach is certainly more reasonable, since it allows, in principle, to characterize the system through a limited range of variables as, for instance, the strength of the impurity potential leading to the disorder, or the scale of space-time potential fluctuations. This obviously implies that it is possible to average (in a way to define) over different microscopic realization of the disorder potential. This outline may seem very vague: how can we implement a theoretical description in statistical terms? Apparently, it is simple to imagine a set of \(N\) impurities at \(\{{\bf r}_{i}\}_{i=1,\dots,N}\) positions. When they are assumed equivalent, the resulting _disorder_ (i.e., impurity) potential takes the obvious form \(U_{\rm D}({\bf r})=\sum_{i}U({\bf r}-{\bf r}_{i})\). Within this recipe, the disorder average is equivalent to the integration over the whole configuration space of impurities, namely  
\[\left\langle\dots\right\rangle_{\rm dis}\equiv L^{-Nd}\prod_{i=1}\int d^{d}{\bf r }_{i}\big{(}\dots\big{)}\, \tag{5}\]  
a technical task not easy to implement in a functional formalism. As pointed out throughout a vast literature ([35; 36; 37; 45; 46; 47; 48; 49], just to mention a few), a more fruitful approach relies upon the statistical features of the disorder potential or, in other words, a description in terms of probability density function (PDF) \(\mathsf{P}(U_{\mathrm{D}})\).