Essential self-adjointness of Wick squares in quasi-free Hadamard representations on curved spacetimes

We investigate whether a symmetric, second order Wick polynomial T of a free scalar field, including derivatives, is essentially self-adjoint on the natural (Wightman) domain in a quasi-free (i.e., Fock space) Hadamard representation. Our results apply to arbitrary spacetime dimensions d >= 2, but we do restrict our attention to the case where T is smeared with a test-function from a particular class S, namely the class of sums of squares of test-functions. (This class of smearing functions is smaller than the class of all non-negative test-functions - a fact which follows Hilbert's theorem.) Combining techniques from microlocal and functional analysis we prove that T is essentially self-adjoint if it is a Wick square (without derivatives). In the presence of derivatives we prove the weaker result that T is essentially self-adjoint if its compression to the one-particle Hilbert space is essentially self-adjoint. For the latter result, we use Wust's theorem and an application of Konrady's trick in Fock space. In the presence of derivatives, we also prove that one has some control over the spectral projections of T, by describing it as the strong graph limit of a sequence of essentially self-adjoint operators. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3703516]