nLab
local observable

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

Φ Σf(Φ(x),Φ(x), 2Φ(x),)b(x)dvol Σ(x) \Phi \;\mapsto\; \int_\Sigma f(\Phi(x), \nabla \Phi(x), \nabla^2 \Phi(x), \cdots) b(x) dvol_\Sigma(x)

where ff is some smooth function of its arguments and bb 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 xyΣx \neq y \in \Sigma are two distinct spacetime points, then

ΦΦ(x)Φ(y) \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

E fb Σ \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 Γ Σ(E)\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(Σ)dim(\Sigma) on the jet bundle J Σ (E)J^\infty_\Sigma(E) of EE.

Products of local observables are called multilocal observables.

types of observables in perturbative quantum field theory:

local field linear microcausal polynomial general regular \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)PolyObs(E) cm 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 November 9, 2018 at 02:52:46. See the history of this page for a list of all contributions to it.