Diagram versus gradient expansion

Question

Suppose one starts from the Dyson equation $\int dy\left[G^{-1}\left(x_{1},y\right)\cdot G\left(y,x_{2}\right)\right]=\delta\left(x_{1}-x_{2}\right)$ with $G$ some Green's function. One may usually cut $G^{-1}=G_{0}^{-1}+G_{\text{int}}^{-1}$ into two parts, one of them being exactly solvable in term of some propagator $G_{0}$ of (for instance) a free theory. Then one expands the solution of the full problem in term of the free propagator using some Dyson's series. This process is at the base of the so-called diagrammatics expansion and is well documented in both high-energy and condensed matter situations. In this approach, one can group some diagrams to establish some effective theory, or to solve some dedicated problems, like the dilute particle gas for instance, to all orders of some sub-class of diagrams, ...

What is usually less documented (at least to my taste), is the following approach : one first transforms the Dyson's equation toward the so-called Wigner-Weyl representation transforming it to something like $G^{-1}\left(p,x\right)\star G\left(p,x\right)=1$ with the $G\left(p,x\right)$ the Wigner transform of $G\left(x_{1},x_{2}\right)$, and where the $\star$-/star-product can be defined rigorously as $\star\approx 1+i\hbar\left(\overleftarrow{\partial}_{x}\overrightarrow{\partial}_{p}-\overleftarrow{\partial}_{p}\overrightarrow{\partial}_{x}\right)/2+\cdots$ up to higher orders I don't care much here. The way to truncate the expansion at a given order is called a gradient expansion (I don't know any good reference for pedagogical review of this expansion, so please add one if you're aware of such a thing :-)

Clearly starting from the same equation, and just transforming via Fourier and/or Wigner transform should not change much the problem, but after these few steps, either diagrammatics or gradient expansion develop on their own. I wonder whether there are some relations between the two approaches, as e.g. complementarity (the two methods give results in differents regimes for instance), equivalence (the two methods in fact give the same result up to [what ?] ...), ... Any kind of resource (documented answer, citation to (reachable) reference, comment, ...) is warm welcome.

Answer 1

Your question is too open ended, and I would leave practical issues aside. But the translations of the Wigner-Weyl map are completely injective and unambiguous, summarized in a crib-Chapter 12 in the booklet referenced below, (Ch 0.18 in the linked update notes below.)

A (gothic) operator ${\mathfrak G}$ has equivalent representations of coordinate-space matrix elements on the one hand, $$ \langle x | {\mathfrak G} | y \rangle , $$ and, equivalently, Wigner transforms (${\mathfrak G}\mapsto g(x,p) $) in phase space, which are apparently perverse half-Fourier transforms of these,
$$ g(x,p) = \hbar \int dy~ e^{-iyp} \left \langle x +\frac{\hbar}{2}y \right | {\mathfrak G}({\mathfrak x},{\mathfrak p}) \left | x-\frac{\hbar}{2}y \right \rangle\\ = \frac{\hbar}{2\pi} \int d\tau d\sigma ~ e^{i(\tau p + \sigma x)} \operatorname{Tr}\left ( e^{-i(\tau {\mathfrak p} + \sigma {\mathfrak x})} {\mathfrak G} \right ) , $$ until the genius of the twist comes into evidence--it took two grad students a decade in the 1940s to convince the world...

Given this Wigner transform, you may reconstruct both the operator via the Weyl map, \begin{equation} {\mathfrak G}({\mathfrak x},{\mathfrak p}) =\frac{1}{(2\pi)^2}\int d\tau d\sigma dx dp ~g(x,p) \exp \Bigl ( i\tau ({\mathfrak p}-p)+i\sigma ({\mathfrak x}-x) \Bigr ) , \end{equation} and your coordinate-space kernels through a simple Fourier inversion, $$\langle x | {\mathfrak G} | y \rangle = \int\! \frac{dp}{2\pi\hbar} ~ \exp\left( i p {(x-y)\over \hbar} \right ) ~ g\left( {x+y\over 2}, p\right) . $$

Further note the traces and multiplication rules of these objects, $$ h~ \hbox{Tr}{\mathfrak G} = \int\! dx dp~ g =h\! \int\! dx ~ \langle x|{\mathfrak G} |x\rangle $$ $$h \hbox{Tr}({\mathfrak G} {\mathfrak F})= h\! \int\! dx dy ~ \langle x|{\mathfrak G} |y\rangle \langle y|{\mathfrak F} |x\rangle =\int\! dx dp~ g(x,p)f(x,p) ~.$$

Now, this last equation is deceptively simple, as the Wigner image of a product actually amounts to $$ {\mathfrak G} {\mathfrak F} \qquad \mapsto \qquad f\star g . $$ However, inside a phase-space integral, just one $\star$-product integrates out by parts, so $\int dx dp ~g\star f= \int dx dp~ gf $ above.

(It's subtle. See how it is evaded in the above correspondence I wrote, Groenewold's fundamental Theorem of the formulation, \begin{equation} {\mathfrak F}~{\mathfrak G}= \frac{1}{(2\pi)^2}\int d\tau d\sigma dx dp ~\exp i \Bigl (\tau ( {\mathfrak p}-p)+\sigma ( {\mathfrak x}-x)\Bigr ) ~(f\star g)(x,p) ~; \end{equation} it is crucial here to appreciate the parenthesis around $f\star g$, that is, that the star is not acting on the exponential to the left, and, as a result, it cannot be integrated out. Removing stars is not child's play.)

The identity operators are, of course, $$ \mathbb{1} \qquad \mapsto \qquad 1,\\ \langle x|{\mathbb 1} |y\rangle=\delta(x-y) . $$

Given this full correspondence, the combinatorics of your Dyson expansion should be identical... but the simplification techniques, of course, remarkably different.

Your starting non-relativistic Fredholm equation, then, $$ {\mathfrak G}={\mathfrak G}_0 -{\mathfrak G}_0 {\mathfrak G}_{int}^{-1} {\mathfrak G} $$ Wigner maps to $$ g(x,p)= g_0 - g_0 \star V \star g ~, $$ where, recall, the $\star$ is associative and V(x) is the potential, or, if the interaction is messier, just the Wigner image of your ${\mathfrak G}_{int}^{-1}$.

• The coupling Neumann series expansion of this in phase space, $ g(x,p)= g_0 - g_0 \star V \star g_0 + g_0 \star V \star g_0 \star V \star g_0 +...$, is not the $\hbar$-gradient expansion, of course, rarely recommended. So you should not expect to compare Feynman loops and *-product orders and obtain related results. (See, e.g. Ch 0.17 in the pdf linked below.) What I have outlined here is just how to transcribe generic recursive expansions in phase space.

References:

For documentation, you might consider our booklet, ISBN: 978-981-4520-43-0, Imperial press. W. Schleich's book is also tasteful.