spacetime support

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

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

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*

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.

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