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I am wondering, whether the Wick theorem for free particles (in the form: $n$-point correlators split into $2$-point correlators) also hold for particles in a finite volume or for particles on a circle (periodic boundary conditions)? Or are there controlled extra terms?

I was curious to derive a set of Feynman rules for a quantum mechanical particle (not yet a field) in some potential etc, in either of the two settings.

PS: I might add I would be also interested to see any finite or periodic analogon of the harmonic oscillator, for which again Wicks theorem holds, so I can use it as the underlying free theory for deriving Feynman rules.

I am very sure that many people have thought about this before, but I have a hard time finding literature on it. Any hint would be much appreciated :-)

Greetings, Simon

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  • $\begingroup$ also posted at physicsoverflow.org/43194 (it is best practice to disclose cross-postings, in order to avoid duplication of efforts) $\endgroup$
    – Carlo Beenakker
    Commented Aug 29, 2020 at 13:06
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    $\begingroup$ I thought that Wick's theorem holds for any quadratic Hamiltonian; a finite volume or periodic boundary conditions would be realized by a potential term, without spoiling the quadratic nature, so it should not invalidate Wick's theorem. $\endgroup$
    – Carlo Beenakker
    Commented Aug 29, 2020 at 13:10
  • $\begingroup$ Carlo, that would be great - do you have a reference? I would somehow expect that, but the proofs I found where too involved for myself to easily include additional potential terms (e.g. the binomial formula for V(x+deltax) does not hold if I include a cut-off)? $\endgroup$
    – Simon Lentner
    Commented Aug 29, 2020 at 13:34
  • $\begingroup$ see, for example, page 170 of these notes --- the only assumption is a bilinear Hamiltonian in the fermionic (or bosonic) fields. $\endgroup$
    – Carlo Beenakker
    Commented Aug 29, 2020 at 13:55
  • $\begingroup$ The algebraic steps involved in a Wick decomposition are fairly simple: Introducing sources with respect to which you can take derivatives, and completing the square. Off the top of my head, I don't see how spatial boundary conditions would interact nontrivially with these operations - the boundary conditions will influence the propagator, not the combinatorics of Gaussian integrals. Of course, in practice, I'd quickly run through these steps for any particular model in question to make sure. $\endgroup$
    – Michael Engelhardt
    Commented Aug 29, 2020 at 15:48

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