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  Schwinger-Dyson equation from the Heisenberg formalism?

+ 4 like - 0 dislike

All the derivations of the Schwinger-Dyson equation I can find are done using either the path integral formalism, or for the oldest papers, Schwinger's own quantum action principle formalism, which, while it resembles the Heisenberg formalism, assumes that the operator derived in the process is an operator version of the action.

Does there exist any derivation of the Schwinger-Dyson equations derived purely from regular matrix mechanics, using the Hamiltonian as its basis? I assume the trick might be to simply show that the Lagrangian operator used in the quantum action principle is $\approx \hat \pi {\partial_t \hat \phi} - \hat H$, but I have been unable to find any such derivation.

This post imported from StackExchange Physics at 2016-09-03 19:17 (UTC), posted by SE-user Slereah

asked Oct 9, 2015 in Theoretical Physics by Slereah (540 points) [ revision history ]
edited Sep 3, 2016 by Dilaton
Combining section 7.1 and 14.7.2 of Schwartz' book on QFT yields a derivation of the S-D differential equation (is that what you're looking for?), which is (there) also used to show that the canonical and path integral approach are equivalent. I could reproduce it for you, if you insists.

This post imported from StackExchange Physics at 2016-09-03 19:17 (UTC), posted by SE-user Danu
The consistent cohabitation of the Schwinger-Dyson equations and the Heisenberg eom's are shown for quadratic Hamiltonians in my Phys.SE answer here.

This post imported from StackExchange Physics at 2016-09-03 19:17 (UTC), posted by SE-user Qmechanic

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