Functional Equation & Feynman Path Integral Solution

[Karlisbad]

Is there any Functional equation In functional derivatives so the Feynman Path integral is its solution?.. i mean given:

$$A[\Phi]=\int \bold D[\Phi]e^{iS/\hbar}$$

Then A (functional) satisfies:

$$G( \delta , \delta ^{2} , B[\phi] )A[\Phi]=0$$

where B is a known functional and "delta" here is the functional derivative.

 

[StatMechGuy]

I think there would be some difficulty in defining such a thing since the path integral isn't technically an integral at all, since it's defined over a space with no clearly defineable measure.

 

[hellfire]

If you take this A to be the vacuum to vacuum transition amplitude then this equation exists and as far as I know it is known as Dyson-Schwinger equation. You can find the derivation in section 6.4 of Ryder's "Quantum Field Theory".

 

[Karlisbad]

"Hellfire" is the Dyson-Schwinger equation a method to evaluate propagators (Non-perturbative) without recalling to Path Integrals??