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

TL;DR

This paper proves that certain second order Wick polynomials of a free scalar field are essentially self-adjoint in curved spacetime quantum field theory, using microlocal analysis and operator theory techniques.

Contribution

It establishes essential self-adjointness of Wick squares with derivatives in curved spacetimes under specific conditions, extending previous results.

Findings

Wick squares without derivatives are essentially self-adjoint.

Self-adjointness of the operator's compression to the one-particle space implies full self-adjointness.

Spectral projections can be controlled via strong graph limits.

Abstract

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. Exploiting the microlocal spectrum condition we prove that T is essentially self-adjoint if it is a Wick square (without derivatives). More generally we show 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 W\"ust's Theorem and an application of Konrady's trick in Fock space. In this more general case 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.