Consider a certain quantum mechanical system with action $S[\phi]$, and let $$ G(1,\dots,n)\equiv\langle\phi_1\cdots\phi_n\rangle $$ be the $n$-point function. It is well-known that these functions satisfy a certain set of recurrence relations (cf. the Schwinger-Dyson equations) that allow us to write any $n$-point function as a combination of propagators and $n'$-point functions, for $n'=n,n+1,n+2,\dots$

The path integral of $S$ is the exponential generating functional of the sequence $\{G(1,\dots,n)\}$.

Let $G_t(1,\dots,n)$ be the tree-level contribution to the connected $n$-point function. The generating functional of these functions is the stationary-phase approximation to the path integral of $S$, and as such, I expect that there should exist a set of recurrence relations among them. In other words, we should be able to write any function $G_t(1,\dots,n)$ in terms of propagators and other functions $G_t(1,\dots,n')$. In fact, this is easily confirmed by calculating the first few tree-level $n$-point functions, which can always be written in terms of a finite series that involves propagators and vertices.

Question: What is the exact form of these recurrence relations? What is the analogous to the Schwinger-Dyson equations, but in terms of tree-level $n$ point functions instead of standard $n$-point functions?

  • $\begingroup$ What have you done to try to resolve this yourself (eg computing tree level amplitudes and relating them)? $\endgroup$
    – JamalS
    Oct 16 '17 at 17:06
  • $\begingroup$ @JamalS I computed the first few (up to $n=4$) correlation functions, and checked that any of them could be written as linear combinations of the correlation functions of lower order. I couldn't come up with a general formula though. $\endgroup$ Oct 16 '17 at 17:09
  • $\begingroup$ @AccidentalFourierTransform you could try substituting $\hbar \rightarrow 0$ in the ordinary $n$-point functions. $\endgroup$ Oct 16 '17 at 17:16
  • $\begingroup$ @SolenodonParadoxus I tried to do that. The result is the same equations as in the Schwinger-Dyson equations but without the contact terms (those with Dirac deltas). That seems to work because, after all, contact terms are usually thought of as quantum corrections, so they should be absent in tree amplitudes. But it doesn't quite work: the resulting set of equations couple $n$-point functions to other functions with higher $n$ instead of lower $n$, which is what I expected (cf. my comment above). ¯\_(ツ)_/¯ $\endgroup$ Oct 16 '17 at 17:20
  • $\begingroup$ @SolenodonParadoxus yes, that is precisely my point. The normal SD equation involve terms of higher order. The equations I am looking for involve terms of lower order. Thus, you cannot really derive the latter from the former. $\endgroup$ Oct 16 '17 at 17:35

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