In the ordinary path integral, the action is an integration over the time your interested in. In quantum statistical mechanics the integration is over an imaginary time with the limit $\frac{\beta}{\hbar}$. Does this imply this is the longest time relevant to the system is $\frac{\beta}{\hbar}$

I'm motivated to think about this because I'm trying to understand the analytic continuation used to go from imaginary time green functions to the retarded real time greens function used to study response. My reasoning is vaguely this:

If you only have information about the response function on the imaginary axis at discrete frequency intervals, then this lack of information would some how show up in the analytic continuation to the real axis.


The answer to your question is no: there is no relation between the integration limit of the path integral in imaginary time (${\beta}$), and a time scale relevant for the study of the system. Imaginary time is called time but has no physical relation to time, as considered in the time evolution of the system.

Concerning your motivation, and the problem of analytic continuation, you need to consider that the function you wish to continue is not known on a finite interval on the "imaginary axis". It is known for an infinite set of Matsubara frequencies. Those indeed form a discrete set, but knowledge of the function to be continued on this infinite discrete frequency set is, mathematically speaking, enough to determine the values of this very analytic function on the complete real frequency axis.

  • $\begingroup$ Thanks, I was wondering if you know anywhere that this "enough" is explained formally? $\endgroup$ Feb 21 '17 at 17:03
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    $\begingroup$ For a formal derivation, I would not know: my "enough" is that of a physicist, and a mathematician might be a bit unhappy with it. In the second answer to this question , an essential ingredient is introduced: the identity theorem for holomorphic functions. Nevertheless, I would not vouch for certain that the discrete set of Masubara frequencies respects its hypothesis. In the situations of practical interest in numerical approaches in physics, it remains an ill-posed problem. $\endgroup$ Feb 21 '17 at 21:29

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