nLab field observable



Algebraic Quantum Field Theory

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



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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

Φ a(x):Γ Σ(E) \mathbf{\Phi}^a(x) \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C}

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

Φ a(x):ΦΦ a(x). \mathbf{\Phi}^a(x) \;\colon\; \Phi \mapsto \Phi^a(x) \,.

Hence in the case that EE 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:

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} }

In general such Φ a(x)\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 Φ a(x)\mathbf{\Phi}^a(x) with “the fields” of the field theory.

quantum probability theoryobservables and states

Last revised on February 8, 2020 at 11:07:33. See the history of this page for a list of all contributions to it.