Wick rotation and the positivity of energy in quantum field theory

TL;DR

The paper proposes a novel axiomatic framework for unitary quantum field theories on curved spacetimes by embedding complex-valued metrics into a domain Met_C(M) and positing holomorphic extendability of partition functions and correlators. It develops a category-theoretic, holomorphic formulation of QFT where theories are functors from a complex-metric cobordism category to nuclear Fréchet spaces, with unitarity tied to time-symmetric boundary data and antilinear dualities. A detailed analysis of allowable complex metrics, their Shilov boundaries, and holomorphic convexity underpins the Wick-rotation philosophy without fully complexifying spacetime. The work draws deep analogies to representation theory, showing boundary-unitary phenomena via oscillator semigroups and boundary representations, and proves that globally hyperbolic Lorentzian cobordisms provide a robust setting for unitary evolution within this framework. Finally, it connects these constructions to causality, locality of observables, and the holomorphic structure of vacuum correlators, while outlining conjectures about the maximal holomorphic envelope of Minkowski-space correlators.

Abstract

We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics. Ordinary Riemannian metrics are contained in the allowable domain, while Lorentzian metrics lie on its boundary.