nLab field observable

Idea

Given Lagrangian field theory with field bundle $E \overset{fb}{\to} \Sigma$ over some spacetime $\Sigma$, so that a field history is a smooth section $\Phi \in \Gamma_\Sigma(E)$, then for every point $x \in \Sigma$ (every event) and every coordinate function $\phi^a \in C^\infty(E\vert_{U_x}$) on the fibers of $E$ over a neighbourhood of $X$ there is the observable

$$\mathbf{\Phi}^a(x) \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C}$$

which reads out the value of that component of the field history $\Phi$ at that point (event)

$$\mathbf{\Phi}^a(x) \;\colon\; \Phi \mapsto \Phi^a(x) \,.$$

Hence in the case that $E$ is a vector bundle, then under the identification of linear observables with distributions (this prop.), these field observable are the Dirac delta distributions.

types of observables in perturbative quantum field theory:

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

In general such $\mathbf{\Phi}^a(x)$ itself is far from being on-shell gauge invariant and hence far from being an element in the BV-BRST cohomology of the theory. But since most observables that are may be expressed as algebraic combinations of these “point evaluation observables” they are of particular interest. Since they manifestly reflect the value of field histories, it is common to terminologically conflate the field observables $\mathbf{\Phi}^a(x)$ with “the fields” of the field theory.

Related entries

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