An inverse problem in quantum field theory and canonical correlation functions

In this paper, we treat a quantum harmonic oscillator in thermal equilibrium with any systems in certain classes of bosons with infinitely many degrees of freedom. We describe the following results: (i) when a canonical correlation function is given, we so reconstruct a Hamiltonian by the rotating wave approximation from it that the Hamiltonian restores it. Namely, we solve an inverse problem in the quantum field theory at finite temperature in a finite volume. (ii) Taking an infinite volume limit for the result in (i), we consider long-time behavior of the canonical correlation function in the finite volume limit.