We discuss the procedure of canonical quantization of free scalar, spinor and electromagnetic fields above the horizon of an eternal Schwarzschild black hole focusing on the case of electromagnetic field. In the resulting quantum theories, the canonical commutation relations are satisfied, and the Hamiltonians have the standard diagonal form without pathologies. The only difference from the case of Minkowski spacetime is the doubling of the number of quantum states of infinite motion, which is a consequence of the different topological structures of the Schwarzschild and Minkowski spacetimes.
Egorov et al. (Mon,) studied this question.
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