Feynman’s iϵ prescription, almost real spacetimes, and acceptable complex spacetimes

Abstract

Feynman’s iϵ prescription for quantum field theoretic propagators has a quite natural reinterpretation in terms of a slight complex deformation of the Minkowski space-time metric. Though originally a strictly flat-space result, once reinterpreted in this way, these ideas can be naturally extended first to semi-classical curved-spacetime QFT on a fixed background geometry and then, (with more work), to fluctuating spacetime geometries. There are intimate connections with variants of the weak energy condition. We shall take the Lorentzian signature metric as primary, but note that allowing the complex deformation to become large leads to a variant of Wick rotation, and more importantly leads to physically motivated constraints on the configuration space of acceptable off-shell geometries to include in Feynman’s functional integral when attempting to quantize gravity. Ultimately this observation allows one to connect the discussion back to recent ideas on “acceptable” complex metrics, in the Louko-Sorkin and Kontsevich-Segal-Witten sense, with Lorentzian signature spacetimes occurring exactly on the boundary of the set of “acceptable” complex metrics. By adopting the tetrad formalism we explicitly construct the most general set of acceptable complex metrics satisfying the 0-form, 1-form, and 2-form acceptability conditions.