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

Understand

We investigate whether a 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.
• We 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 testfunctions.
• This class of smearing functions is smaller than the class of all non-negative testfunctions -- a fact which follows from Hilbert's Theorem.