Path integrals vs. Diagrammatics

Question

The question is about the approximation techniques available in the path integral formulation and their equivalents in the context of the traditional Feynman-Dyson expansion (aka diagrammatic techniques). Of course, the Feynman-Dyson expansion can be also done in terms of path integrals, but the point here is the techniques that are specific to this approach or where it presents significant advantages.

Here is what comes to my mind in terms of the techniques:

• Quasi-classical approximation, i.e. calculating fluctuations around the extremum trajectory.
• Instanton techniques - these seem to be of limited practical utility, a few known state-of-the-art solutions.
• Renormalization group seems to work particularly well in the path integral formulation.
• ... On the other hand, they seem needlessly complicate the Keldysh approach.

In terms of the equivalence: I have recently encountered a claim that the mean phase approximation for a path integral is equivalent to the random phase approximation, that is to summing the bubble diagrams. This raises a question of what could be the equivalent of summing the ladder diagrams? The maximally crossed diagrams?

Remark: I am mostly interested in the condensed matter applications, but other backgrounds are welcome.

Answer 1

The connecting thread between mean field theory, ladder diagrams, instantons, and renormalization group, is that these are techniques for describing non-perturbative phenomena. In some rare cases we have exact non-perturbative solutions, but usually non-perturbative physics proceeds with effective theories that are not rigorously connected to microscopic models; but more is different, so don't let that bother you.

An interesting and often overlooked formalism that connects these techniques is n-particle irreducible (nPI) effective actions. I like the writing of Jürgen Berges on this subject, especially this long pedagogical introduction https://arxiv.org/abs/hep-ph/0409233. In this paper you will find connections to mean field theory, ladder diagrams, and renormalization.

In mean field theory we posit that a field (a one point function) takes non-zero expectation and then derive the results, this is a 1PI effective action. In 2PI theories we posit that a two point function takes non-zero expectation, e.g., the electron Green's function, Cooper pair propagator, or magnetic susceptibility. The framework of nPI effective actions gives us a formal way to convert the idea "I believe this n point function has non-zero expectation value" into an effective field theory that we can calculate with.

For a concrete, down to earth example of nPI actions in condensed matter, check out Sec III of the supporting information of this paper https://arxiv.org/abs/1205.4780. First they apply mean field theory to a chiral magnet model; then they apply 2PI theory to derive the magnetic susceptibility. At the mean field level, the system has a continuous phase transition. When non-linear effects are including using 2PI theory, it is seen that the phase transition is first order! The result agrees with neutron scattering experiments.

Edit: This is a key source for 1PI and 2PI effective action. It mentions connections between Hartree-Fock MFT (1PI) and the Bethe-Saltpeter ladder diagrams. https://doi.org/10.1103/PhysRevD.10.2428

Answer 2

To carry out a non-perturbative calculation from QED it is usual to use the Foldy-Wouthuysen transformation. This is necessary to ensure that the time evolution of states matches the time evolution of field operators, without which constraint the phase differences corrupt the definition of the momentum operator. It is possible to simplify the Foldy-Wouthuysen transformation (which incorporates spin) and define the field picture $$|f_F(t)\rangle = e^{-iH_It} |f(t)\rangle = e^{-iH_0t} |f(0)\rangle $$ In the field picture, kets evolve as in the Schrödinger picture for non-interacting particles. The momentum operator in the field picture is $$P_F^a= e^{-iH_It}i\partial^ae^{iH_It} $$ In the semi-classical correspondence, evolution may be treated for small $t$ as a perturbation to the evolution of a non-interacting particle, by replacing the interaction Hamiltonian with its expectation (in effect summing diagrams for the non-perturbative case). For a classical particle with position $x$ and velocity $\dot x$, the classical current is $$J=-e\dot x$$ The expectation of the interaction Hamiltonian is $$\langle H_I\rangle=J \cdot\langle A \rangle = -e \dot x \cdot\langle A \rangle $$ Replacing the interaction Hamiltonian with its expectation gives a semiclassical model in which the electron is quantum but the field is classical. In this semi-classical model, the momentum operator in the field picture is $$ P_F^a = e^{ie \dot x \cdot\langle A \rangle}i\partial^a e^{-ie \dot x \cdot\langle A \rangle} = i\partial^a-e\langle A^a\rangle $$ Thus, the expectation, $\langle A^a\rangle$, of the operator which creates and annihilates photons acts in the manner of a classical vector field, modifying energy and momentum. This is the standard formula for generalised momentum in the presence of a classical field, often assumed on phenomenological grounds, but here seen from the emission and absorption of photons in interaction. Replacing momentum in the Dirac equation with generalised momentum gives the interacting Dirac equation (covered in many textbooks).

Again working in the field picture we have, from Ehrenfest’s theorem,

$$ {d \over dt}\langle P^a_F\rangle= \langle {d \over dt} P^a_F\rangle + i\langle[H,P^a_F]\rangle $$ Replacing the interaction in the Hamiltonian with the expectation as before $$H=H_0 + H_I \approx H_0 + \langle H_I\rangle =H_0 -e\dot x\cdot\langle A\rangle $$

Substituting, using generalised momentum, and dropping the subscript F (since expectations are the same in any picture)

$$ {d \over dt}\langle P^a\rangle= e {d \over dt}\langle A^a\rangle +i\langle [ H_0 -e\dot x\cdot\langle A\rangle, i\partial^a-e\langle A^a\rangle]\rangle $$ $$ {d \over dt}\langle P^a\rangle= e {d \over dt}\langle A^a\rangle -e\partial^a \dot x\cdot\langle A\rangle $$

To intepret this, write it in the rest frame of the particle (so that we have proper time) $$ \partial^0 \langle P^a\rangle= e \partial^0\langle A^a\rangle -e\partial^a \langle A^0\rangle $$ Then we only have to do a Lorentz transformation to find the Lorentz force law in terms of the Faraday tensor.

The derivation of Maxwell's equations is more straightforward, working from the Gupta-Bleuler gauge condition which yields Lorenz gauge, because it is not necessary to use the Field picture. I have given a full treatment in A Construction of Full QED Using Finite Dimensional Hilbert Space and in The Mathematics of Gravity and Quanta