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.

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.

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 in 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 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.

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.