Positivity of G(+) for special Klein–Gordon operators

Establish whether, for any Lorentzian manifold M on which the Klein–Gordon operator −□ + Y(x) is special in the sense of Definition 3.6 (i.e., the sum of the operator-theoretic Feynman and anti-Feynman propagators has causal support), the bisolution defined by G(+) := −i(G_F^op − G_ret) = i(G_F^op − G_adv) always satisfies the positivity requirement needed to be a two-point function of a quantum state; equivalently, determine if the sesquilinear form ∫∫ f*(x) G(+)(x,y) f(y) dx dy is nonnegative for all test functions f.

Background

The paper defines operator-theoretic Feynman and anti-Feynman propagators as boundary values of the resolvent of the essentially self-adjoint Klein–Gordon operator on L2(M). A Klein–Gordon operator is called special if the sum of these two propagators has causal support. In such cases, the authors propose reconstructing classical propagators by splitting this sum according to causal support and then defining a candidate positive-frequency bisolution by G(+) := −i(G_F^op − G_ret) = i(G_F^op − G_adv).

While this construction works and yields positive two-point functions in all worked-out examples in the paper (e.g., certain stationary and de Sitter cases), the general validity of the positivity property for G(+) under the specialty condition is not established. Positivity is essential for G(+) to serve as the two-point function of a quasifree (Fock) state in the Klein–Gordon quantum field theory on curved spacetimes.