  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.
