Quantum oscillator in a non-self-interacting radiation field: Exact calculation of the partition function

With the use of functional integration, an exact analytic expression valid for the whole temperature range is derived for the partition function of a charged quantum oscillator coupled to a non-self-interacting (with the A2 term neglected) radiation field. To derive this result, special care is taken with renormalization which is carried out at two stages. The temperature-dependent free-energy shift due to blackbody radiation is studied and compared with recent work. It is shown that the ground state is the only stable state and that the higher states have a finite lifetime.