Lorentzian bordisms in algebraic quantum field theory
Severin Bunk, James MacManus, Alexander Schenkel
TL;DR
The paper establishes a precise relationship between algebraic quantum field theory (AQFT) and functorial field theory (FFT) in globally hyperbolic Lorentzian spacetimes by modeling Lorentzian bordisms as a globally hyperbolic bordism pseudo-category $\mathcal{LB}\mathsf{ord}_m$. It proves that every AQFT has an underlying FFT that captures time evolution but generally forgets spatially local structure, via a faithful functor $\mathfrak{F}_{(-)}: \mathbf{AQFT}_m\to \mathbf{FFT}_m^{\mathrm{t.s.}}$ and shows an equivalence when restricting to spatially global AQFTs (and in 1D, $m=1$). The paper further grounds these abstract results by comparing a free scalar quantum field in both AQFT and FFT formalisms, demonstrating compatibility through CCR quantization and the time-slice axiom. Overall, it clarifies how Lorentzian bordism geometry underpins AQFT time evolution and outlines a path to incorporating spatial locality within extended FFT frameworks.
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.