nLab spacetime support

Idea

In local field theory with fields on a given spacetime $X$, the spacetime support of an observable $A$ is the maximal region in spacetime such that $A$ 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 $E \overset{fb}{\to} \Sigma$ be a field bundle over a spacetime $\Sigma$ (def. ), with induced jet bundle $J^\infty_\Sigma(E)$

For every subset $S \subset \Sigma$ let

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

be the corresponding restriction of the jet bundle of $E$.

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

$$supp_\Sigma(A) \coloneqq Cl( \{ x \in \Sigma | \iota_{\{x\}}^\ast A \neq 0 \} )$$

We write

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

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

Related concepts

• support, support of a distribution

• compact support

• local observable

References

• geometry of physics – perturbative quantum field theory the chapter 7. Observables