Quantization of a charged scalar field on a charged black hole background

Abstract

We study the canonical quantization of a massless charged scalar field on a Reissner-Nordstr\"om black hole background. Our aim is to construct analogs of the standard Boulware, Unruh and Hartle-Hawking quantum states which can be defined for a neutral scalar field, and to explore their physical properties by computing differences in expectation values of the scalar field condensate, current and stress-energy tensor operators between two quantum states. Each of these three states has a non-time-reversal-invariant past and future charged field generalization, whose properties are similar to those of the corresponding past and future states for a neutral scalar field on a Kerr black hole. In addition, we present some tentative, time-reversal-invariant, equilibrium states. The first is a ``Boulware''-like state which is as empty as possible at both future and past null infinity. Second, we posit a ``Hartle-Hawking''-like state which may correspond to a thermal distribution of particles. The construction of both these latter states relies on the use of nonstandard commutation relations for the creation and annihilation operators pertaining to superradiant modes.