Axiomatic quantum field theory in curved spacetime

TL;DR

This paper proposes a fully local and covariant framework for quantum field theory in curved spacetime by elevating the operator product expansion ($OPE$) to a fundamental axiom, replacing the Minkowski-specific Wightman requirements of Poincaré invariance and a preferred vacuum with a microlocal spectrum condition and $OPE$-driven structure. The theory is specified by a background ${\bf M}=(M,g,T,e)$, a set of quantum fields $\mathcal{I}$ with star and Bose/Fermi data, and a collection of $OPE$ coefficients $\mathcal{C}({\bf M})$ that satisfy nine axioms (covariance, identity, star-compatibility, locality, scaling, asymptotic positivity, spectrum, associativity, and metric-analytic dependence). From the $OPE$ data, one constructs a *-algebra $\mathcal{A}({\bf M})$ and a state space $\mathcal{S}({\bf M})$ that together define the quantum field theory, with normal (anti-)commutation relations and curvature-dependent spin-statistics and $PCT$ theorems proven in this curved setting. The framework preserves locality and covariance across spacetimes, supports functoriality under spacetime embeddings, and offers a path toward perturbative construction of interacting theories via consistency conditions on the $OPE$. It thereby provides a robust, background-independent approach to QFT in curved spacetime that parallels classical field theory in focusing on local equations (encoded in the $OPE$) rather than global state selection. The results have potential implications for quantum gravity and the analytic behavior of parameters, illustrating that the local $OPE$ coefficients can remain well-behaved even when state-dependent objects (like vacua) do not exist or are non-analytic in parameters.

Abstract

The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features--such as Poincare invariance and the existence of a preferred vacuum state--that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is locally and covariantly constructed from the spacetime metric), a microlocal spectrum condition, an "associativity" condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.

Theorems & Definitions (7)

• Lemma 1
• Lemma 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5