For a quantum mechanics system, assume that we know its partition function calculated using the path integral method. $$Z = \int [d\Phi]\cdots e^{iS[\Phi,\ldots]}$$

Can we derive the operator (for example, Hamiltonian operator) and the eigenvalues from the partition function?

Most text books take a different approach. There, we assume that we know the Hamiltonian of the system and try to calculate path integral. Here, we study a reverse problem.

Can anyone provide some help on this?

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    $\begingroup$ What, exactly, do you mean by "knowing" its path integral? The path integral is a technique to compute expectation values, not a single object with a single value. $\endgroup$ – ACuriousMind Sep 30 '16 at 0:26
  • $\begingroup$ I think he meant that writing down the partition function $$Z=\int [D\Phi]\dots e^{i S[\Phi,\dots]}$$, for example, can you define other physical quantities/observables from here. $\endgroup$ – wonderich Sep 30 '16 at 1:05
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    $\begingroup$ If $S$ is replaced by $S + \int J \Phi$, that is $Z$ is replaced by $Z[J]$, you can do many things...Much depends on what you mean for "$\ldots$" in $S[\Phi, \ldots]$ $\endgroup$ – Valter Moretti Sep 30 '16 at 7:38

The path integral contains the action (in the exponent), which contains the Lagrangian of the theory. Being equipped with the Lagrangian for a theory, one's options are almost limitless. One can use the standard transformation that converts Lagrangians into Hamiltonians and visa versa. Or, if the theory is shift invariant in time and space, one can compute the energy-momentum tensor (stress-energy tensor) which is the Noether current associated with shift invariance. The time-time component would then be the Hamiltonian for the theory.

  • $\begingroup$ The question is, assume we know the partition function ( path integral ) of a system, how to recover the Lagrangian of the system ? $\endgroup$ – david Sep 30 '16 at 18:25

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