A Deformation Quantization for Non-Flat Spacetimes and Applications to QFT

TL;DR

This paper develops a deformation quantization approach for all globally hyperbolic spacetimes with Poisson structures, enabling the extension of quantum field theory states to non-commutative geometries while preserving key singularity properties.

Contribution

It introduces a Rieffel-type deformation quantization applicable to all such spacetimes, ensuring the Hadamard condition persists in the deformed quantum states.

Findings

Deformation quantization is applicable to all globally hyperbolic spacetimes with suitable Poisson structures.

Deformed quantum states retain the Hadamard singularity structure.

The Hadamard condition is preserved under the deformation, supporting the equivalence principle in non-commutative spacetimes.

Abstract

We provide a deformation quantization, in the sense of Rieffel, for \textit{all} globally hyperbolic spacetimes with a Poisson structure. The Poisson structures have to satisfy Fedosov type requirements in order for the deformed product to be associative. We apply the novel deformation to quantum field theories and their respective states and we prove that the deformed state (i.e.\ a state in non-commutative spacetime) has a singularity structure resembling Minkowski, i.e.\ is \textit{Hadamard}, if the undeformed state is Hadamard. This proves that the Hadamard condition, and hence the quantum field theoretical implementation of the equivalence principle is a general concept that holds in spacetimes with quantum features (i.e. a non-commutative spacetime).