Contents

# Contents

## Idea

In physics (field theory) a local observable is an observable which is an average of a function of the values of the fields and their derivatives at each fixed spacetime point.

If $\Phi$ is a field configuration over some spacetime $\Sigma$ then a local observable is a function of $\Phi$ which, assigns values of the form

$\Phi \;\mapsto\; \int_\Sigma f(\Phi(x), \nabla \Phi(x), \nabla^2 \Phi(x), \cdots) b(x) dvol_\Sigma(x)$

where $f$ is some smooth function of its arguments and $b$ is some bump function on spacetime.

This is in contrast to an observable which combines the values of fields at different spacetime points other than by forming spacetime averages. For instance if $x \neq y \in \Sigma$ are two distinct spacetime points, then

$\Phi \mapsto \Phi(x) \Phi(y)$

is a smooth function on the space of fields, hence an observable, but it is not a local observables.

More in detail, if

$\array{ E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }$

is a fiber bundle regarded as the field bundle that defines the given field theory, then a local observables is a function on the the space of field configurations, hence the space of sections $\Gamma_\Sigma(E)$ (or else on the subspace of those which solve the equations of motion, the shell) which arises as the transgression of a horizontal differential form of degree $dim(\Sigma)$ on the jet bundle $J^\infty_\Sigma(E)$ of $E$.

Products of local observables are called multilocal observables.

$\array{ && \text{local} \\ && & \searrow \\ \text{field} &\longrightarrow& \text{linear} &\longrightarrow& \text{microcausal} &\longrightarrow& \text{polynomial} &\longrightarrow& \text{general} \\ && & \nearrow \\ && \text{regular} }$

## Definition

For the moment see at geometry of physics – perturbative quantum field theory this def.

## Properties

###### Example

(polynomial local observables are polynomial observables)

A local observable which comes form a horizontal differential form which is a polynomial in the fields and their jets times the volume form on spacetime is a polynomial observable.

These happen to be also microcausal observables (this example).

The resultiing inclusion

$LocPolyObs(E) \hookrightarrow PolyObs(E)_{cm}$

of the local polynomial observables into the microcausal polynomial observables is a dense subspace-inclusion. (Fredenhagen-Rejzner 12, p. 23)

## References

Last revised on February 8, 2020 at 06:06:41. See the history of this page for a list of all contributions to it.