Path Integral Formulation - Quantum Field Theory - Schwinger-Dyson Equations

Schwinger-Dyson Equations

Since this formulation of quantum mechanics is analogous to classical action principles, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case.

In the language of functional analysis, we can write the Euler-Lagrange equations as

(the left-hand side is a functional derivative; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the Schwinger-Dyson equations.

If the functional measure turns out to be translationally invariant (we'll assume this for the rest of this article, although this does not hold for, let's say nonlinear sigma models) and if we assume that after a Wick rotation

which now becomes

for some H, goes to zero faster than any reciprocal of any polynomial for large values of φ, we can integrate by parts (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger-Dyson equations for the expectation:

for any polynomially bounded functional F.

in the deWitt notation.

These equations are the analog of the on shell EL equations.

If J (called the source field) is an element of the dual space of the field configurations (which has at least an affine structure because of the assumption of the translational invariance for the functional measure), then, the generating functional Z of the source fields is defined to be:

Note that

or

where

Basically, if is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of QFT, unlike its Wick rotated statistical mechanics analogue, because we have time ordering complications here!), then are its moments and Z is its Fourier transform.

If F is a functional of φ, then for an operator K, F is defined to be the operator which substitutes K for φ. For example, if

and G is a functional of J, then

Then, from the properties of the functional integrals

we get the "master" Schwinger-Dyson equation:

or

If the functional measure is not translationally invariant, it might be possible to express it as the product where M is a functional and is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the target space is diffeomorphic to Rn. However, if the target manifold is some topologically nontrivial space, the concept of a translation does not even make any sense.

In that case, we would have to replace the in this equation by another functional

If we expand this equation as a Taylor series about J = 0, we get the entire set of Schwinger-Dyson equations.