Positivity and structure of even-power \u03c0_a operators in the open EFToI

Determine whether there exist combinations of local operators involving even powers of the advanced field \u03c0_a that are neither positive-definite nor total derivatives within the open Effective Field Theory of single-clock inflation, subject to the non-equilibrium constraints and locality. If such operators exist, classify them and establish conditions under which the influence functional remains positive; otherwise, provide a proof of their absence at and beyond leading-derivative order.

Background

Ensuring positivity of the influence functional is a core non-equilibrium constraint in the Schwinger–Keldysh construction of open systems. For operators containing odd powers of \u03c0_a, positivity is comparatively straightforward under real Wilsonian coefficients.

The authors highlight a subtlety for even powers of \u03c0_a: while one expects positivity or reduction to total derivatives, they have not identified counterexamples or a general proof, even after considering higher-derivative structures. Resolving this question would clarify the allowed operator basis and the robustness of positivity in the open EFToI.