Question

In standard lectures on advanced QFT one learnt that performing Legendre transformation leads to effective actions generating one-particle-irreducible (1PI) diagrams, which is encoded by Schwinger-Dyson equation. From the book by Altland and Simons, Hubbard-Stratonovich transformation provides an alternative approach to the diagrammatic perturbation exploiting Schwinger-Dyson equation or Bethe-Salpeter equation. I feel confused trying to put the two techniques in the same picture and would like to ask how these two are related if there is any? Or is there a criterion on when to apply one of the two techniques?

Answer 1

The Legendre transformation and the Hubbard-Stratonovich transformation are two different methods used in advanced QFT, with a connection through the Schwinger-Dyson equation.

Step-by-step explanation:

The Legendre transformation and the Hubbard-Stratonovich transformation are two different methods used in advanced quantum field theory (QFT) to simplify calculations and analyze particle interactions. While they serve different purposes, there is a connection between them in the context of 1PI (one-particle irreducible) diagrams and the Schwinger-Dyson equation.

The Legendre transformation is used to relate the classical action to the generating functional in QFT. It allows us to express the theory in terms of the expectation values of fields and their conjugate momenta. The effective action generated through the Legendre transformation captures the physics of the theory at a certain scale.

The Hubbard-Stratonovich transformation, on the other hand, is a technique used to introduce auxiliary fields into the theory. By doing this, it simplifies the calculation of certain integrals associated with Feynman diagrams. This transformation is particularly useful when dealing with interactions described by the Schwinger-Dyson or Bethe-Salpeter equations.

The connection between these two techniques lies in the fact that the effective action obtained through the Legendre transformation can be expressed as a sum of connected diagrams. These connected diagrams are related to the 1PI diagrams through a series of relations known as the Schwinger-Dyson equations. The Hubbard-Stratonovich transformation provides an alternative approach to calculating these connected diagrams by introducing auxiliary fields and exploiting these equations.

The choice of which technique to apply depends on the specific problem at hand. The Legendre transformation is generally used to obtain the effective action and study the behavior of the theory at different scales. The Hubbard-Stratonovich transformation is useful when calculating certain integrals associated with Feynman diagrams and can be seen as a way to simplify the perturbative expansion of the theory.