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Quantum field theory on the Bertotti-Robinson space-time

We consider the problem of quantum field theory on the Bertotti-Robinson space-time, which arises naturally as the near horizon geometry of an extremal Reissner-Nordstrom black hole but can also arise in certain near-horizon limits of non-extremal Reissner Nordstrom space-time. The various vacuum states have been considered in the context of $AdS_{2}$ black holes by Spradlin and Strominger who showed that the Poincare vacuum, the Global vacuum and the Hartle-Hawking vacuum are all equivalent, while the Boulware vacuum and the Schwarzschild vacuum are equivalent. We verify this by explicitly computing the Green's functions in closed form for a massless scalar field corresponding to each of these vacua. Obtaining a closed form for the Green's function corresponding to the Boulware vacuum is non-trivial, we present it here for the first time by deriving a new summation formula for associated Legendre functions that allows us to perform the mode-sum. Having obtained the propagator for the Boulware vacuum, which is a zero-temperature Green's function, we can then consider the case of a scalar field at an arbitrary temperature by an infinite image imaginary-time sum, which yields the Hartle-Hawking propagator upon setting the temperature to the Hawking temperature. Finally, we compute the renormalized stress-energy tensor for a massless scalar field in the various quantum vacua.