Lorentzian bordisms in algebraic quantum field theory

TL;DR

This paper demonstrates that algebraic quantum field theories inherently possess a functorial structure on Lorentzian bordisms, linking algebraic and geometric perspectives and illustrating this with a free scalar field example.

Contribution

It establishes a functorial framework for algebraic quantum field theories on Lorentzian bordisms, connecting algebraic and geometric formulations.

Findings

Every AQFT has an underlying functorial field theory on Lorentzian bordisms.

The functor encodes time evolution but not spatial locality.

Comparison of algebraic and functorial descriptions of a free scalar field.

Abstract

It is shown that every algebraic quantum field theory has an underlying functorial field theory which is defined on a suitable globally hyperbolic Lorentzian bordism pseudo-category. This means that globally hyperbolic Lorentzian bordisms between Cauchy surfaces arise naturally in the context of algebraic quantum field theory. The underlying functorial field theory encodes the time evolution of the original theory, but not its spatially local structure. As an illustrative application of these results, the algebraic and functorial descriptions of a free scalar quantum field are compared in detail.