# If we know path integral of a quantum system, can we recover operators and eigenvalues from the system?

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?

• 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. – ACuriousMind Sep 30 '16 at 0:26
• 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. – wonderich Sep 30 '16 at 1:05
• 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 $S[\Phi, \ldots]$ – Valter Moretti Sep 30 '16 at 7:38