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.

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    $\begingroup$ 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. $\endgroup$ – Danu Oct 9 '15 at 22:23
  • $\begingroup$ The consistent cohabitation of the Schwinger-Dyson equations and the Heisenberg eom's are shown for quadratic Hamiltonians in my Phys.SE answer here. $\endgroup$ – Qmechanic Oct 10 '15 at 16:42
  • $\begingroup$ @Danu: ''I could reproduce it for you" - maybe also for me? it would be worth an answer! $\endgroup$ – Arnold Neumaier Jan 7 '18 at 14:05
  • $\begingroup$ @ArnoldNeumaier unfortunately, that offer has expired ;) I still gave a rather precise reference though! $\endgroup$ – Danu Jan 8 '18 at 6:52
  • $\begingroup$ @Danu: But I (like many readers here, I guess) do not have Schwartz's book! $\endgroup$ – Arnold Neumaier Jan 8 '18 at 10:56

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