nLab
spacetime support

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

In local field theory with fields on a given spacetime XX, the spacetime support of an observable AA is the maximal region in spacetime such that AA depends on (“observes”) the values of fields at the points in this region.

In a local field theory spacetime support of observables is typically required to be a compact subset of spacetime, which, under the Heine-Borel theorem, reflects the intuition that every experiment (every “observation” of physics) is necessarily bounded in spacetime.

For more see at geometry of physics – perturbative quantum field theory the chapter 7. Observables

Definition

Definition

(spacetime support)

Let EfbΣE \overset{fb}{\to} \Sigma be a field bundle over a spacetime Σ\Sigma (def. ), with induced jet bundle J Σ (E)J^\infty_\Sigma(E)

For every subset SΣS \subset \Sigma let

J Σ (E)| S ι S J Σ (E) (pb) S Σ \array{ J^\infty_\Sigma(E)\vert_S &\overset{\iota_S}{\hookrightarrow}& J^\infty_\Sigma(E) \\ \downarrow &(pb)& \downarrow \\ S &\hookrightarrow& \Sigma }

be the corresponding restriction of the jet bundle of EE.

The spacetime support supp Σ(A)supp_\Sigma(A) of a differential form AΩ (J Σ (E))A \in \Omega^\bullet(J^\infty_\Sigma(E)) on the jet bundle of EE is the topological closure of the maximal subset SΣS \subset \Sigma such that the restriction of AA to the jet bundle restrited to this subset does not vanishes:

supp Σ(A)Cl({xΣ|ι {x} *A0}) supp_\Sigma(A) \coloneqq Cl( \{ x \in \Sigma | \iota_{\{x\}}^\ast A \neq 0 \} )

We write

Ω Σ,cp r,s(E){AΩ Σ r,s(E)|supp Σ(A)is compact}Ω Σ r,s(E) \Omega^{r,s}_{\Sigma,cp}(E) \coloneqq \left\{ A \in \Omega^{r,s}_\Sigma(E) \;\vert\; supp_\Sigma(A) \, \text{is compact} \right\} \;\hookrightarrow\; \Omega^{r,s}_\Sigma(E)

for the subspace of differential forms on the jet bundle whose spacetime support is a compact subspace.

References

Created on November 9, 2018 at 02:51:37. See the history of this page for a list of all contributions to it.