I am currently studying "Critical Dynamics - A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior", and came across an issue I can't solve.
If you know about functional derivatives and/or Martin-Siggia-Rose formalism, please skip the rather long paragraph meant to bring some context. I hope the notations are standard (I'm new to the field).
The context
If you have access to the book (no publicly available version unfortunately), starting from chapter 4, the Martin-Siggia-Rose formalism is set up for Langevin dynamics, with in this case the example :
Generalizing the Landau–Ginzburg–Wilson Hamiltonians for the Ising and Heisenberg models,[...], to n-component isotropic systems undergoing a second-order phase transition, we consider \begin{equation} H[S]=\int d^{d}x \sum_{\alpha} \left(\frac{r}{2}[S^{\alpha}(x)]^{2}+\frac{1}{2}[\nabla S^{\alpha}(x)]^{2}+\frac{u}{4!} \sum_{\beta}[S^{\alpha}(x)]^{2}[S^{\beta}(x)]^{2}-h^{\alpha}(x)S^{\alpha}(x) \right) \end{equation} With $S$ the n-component order parameter and h the conjugate external field.
The associated evolution is a Langevin equation, given by :
\begin{equation} \frac{\partial S^{\alpha}(x,t)}{\partial t}=-D(i\nabla)^{a} \frac{\delta H[S]}{\delta S^{\alpha}(x,t)}+\zeta^{\alpha}(x,t) \end{equation}
Where fluctuations are taken into account via $\zeta$, a zero mean diagonal white noise $\left< \zeta^{\alpha}(x,t) \zeta^{\beta}(x',t') \right> = 2 L\delta(x-x')\delta(t-t')\delta^{\alpha\beta}$, and $L=D(i\nabla)^{a}$.
From here is derived the generating functional for correlation functions, \begin{equation} Z[\tilde{j},j]=\left< exp \int d^{d}xdt \sum_{\alpha} \tilde{j^{\alpha}}(x,t) \tilde{S^{\alpha}}(x,t)+j^{\alpha}(x,t)S^{\alpha}(x,t)\right> \end{equation}
The $j$'s being the source currents and the tilde quantities the auxiliary MSR fields/source currents
By means of functional derivatives w.r.t. the currents, taken at vanishing currents, one obtains the correlation functions.
Taking the logarithm of Z gives you the generating functional of the connected correlation functions, and now performing a Legendre transform of the logarithm with the new variables,
\begin{align} \tilde{\Phi^{\alpha}}(x,t)=\frac{\delta \log Z[\tilde{j},j]}{\delta \tilde{j^{\alpha}}(x,t)} && \Phi^{\alpha}(x,t)=\frac{\delta \log Z[\tilde{j},j]}{\delta j^{\alpha}(x,t)} \end{align}
One ends up with the generating functional for vertex functions, \begin{equation} \Gamma[\tilde{\Phi^{\alpha}},\Phi^{\alpha}]=-\log Z[\tilde{j},j]+ \int d^{d}xdt \sum_{\alpha} \left( \tilde{j^{\alpha}}(x,t)\tilde{\Phi^{\alpha}}(x,t)+j^{\alpha}(x,t)\Phi^{\alpha}(x,t)\right) \end{equation}
All of this to map a perturbation expansion of the correlation functions in terms of Feynman diagrams. The connected correlation functions will correspond in perturbation theory to the contribution from connected graph, but as often in field theory one is interested in the "real" building blocks of the diagrammatic expansion, namely the 1-particle irreducible graphs whose contribution come from the functional derivatives of $\Gamma$, the vertex functions,
\begin{equation} \Gamma^{(\tilde{N},N)}_{\{\alpha_{i}\};\{\alpha_{k}\}}(\{x_{i},t_{i}\};\{x_{k},t_{k}\})=\prod^{\tilde{N}}_{i} \frac{\delta}{\delta \tilde{\Phi^{\alpha_{i}}}(x_{i},t_{i})} \prod^{N}_{k} \frac{\delta}{\delta \Phi^{\alpha_{k}}(x_{k},t_{k})} \Gamma[\tilde{\Phi^{\alpha}},\Phi^{\alpha}] \biggr\rvert_{j=\tilde{j}=0} \end{equation}
It can be shown, that $\Gamma^{(1,1)}_{\alpha;\beta}(x,t;x',t')$ is of the form $\Gamma^{(1,1)}(x-x',t-t')\delta^{\alpha\beta}$. In turn this form is interesting because we have, after performing a Fourier transform,
\begin{equation} \Gamma^{(1,1)}(q,\omega)=G(-q,-\omega)^{-1} \end{equation}
Thus, up to a sign in the variables, the loop diagrams contributing to $\Gamma^{(1,1)}(q,\omega)$ are nothing but the one-particle irreducible self-energy graphs.
Therefore, one can compute a perturbative expansion of the propagator by computing only the contribution from these graphs.
The issue itself
To compute the perturbative expansion of the propagator, it seems that you need to establish the Feynman rules for your theory first, and then make use of the results above to assert that you only need to compute the contributions from the 1-particle irreducible graphs.
But we also saw an explicit form of the propagator in terms of the vertex functions. Let's compute directly $\Gamma^{(1,1)}_{\alpha;\beta}(x,t;x',t')$ then !
\begin{equation} \frac{\delta \Gamma[\tilde{\Phi},\Phi]}{\delta \tilde{\Phi}^{\alpha}(x,t)} =\tilde{j}^{\alpha}(x,t) \end{equation}
and
\begin{equation} \frac{\delta^{2} \Gamma[\tilde{\Phi},\Phi]}{\delta \tilde{\Phi}^{\alpha}(x,t) \delta\Phi^{\beta}(x',t')} =\frac{\delta \tilde{j}^{\alpha}(x,t)}{\delta\Phi^{\beta}(x',t')} = ? \end{equation}
The first identity is rather easy to obtain. Though, I am not able to compute the derivative of the current at all. I tried to make use of the definition of $\tilde{\Phi}$ to make a change in variable but there it seems to be a functional change of variables. I tried the usual method, ie to take the multivariable case and let the number of variables go to infinity, changing sums into integrals, but nothing came out of it.
Is there any way to proceed like this ? Is this "functional change of variable" even a thing ? And if not, is it possible to compute the vertex functions in another way ?
Many thanks for reading up to here. I don't really expect a complete answer (unless Mr. Tauber is on this site and sees my post), but any reference, either to related content in QFT or SFT, or to a math text about this idea of functional change of variable will be greatly appreciated.
Please do not hesitate to ask any precision you'd like to see on the matter, I may have been too quick in the context (even though it is already too long).
Note: I could not attend functional analysis courses during my studies, that is why I had troubles finding anything from mathSE/math articles. Nevertheless, I'm ready to dive in if necessary.
