algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
quantum mechanical system, quantum probability
interacting field quantization
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
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
is a smooth function on the space of fields, hence an observable, but it is not a local observables.
More in detail, if
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.
types of observables in perturbative quantum field theory:
For the moment see at geometry of physics -- A first idea of quantum field theory this def.
(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
of the local polynomial observables into the microcausal polynomial observables is a dense subspace-inclusion. (Fredenhagen-Rejzner 12, p. 23)
Romeo Brunetti, Klaus Fredenhagen, section 5.4.1 of Quantum Field Theory on Curved Backgrounds, chapter 5 in Christian Bär, Klaus Fredenhagen (eds.), Quantum Field Theory on Curved Spacetime, Springer 2009
Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, section 3.2 of_Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)
Klaus Fredenhagen, Katarzyna Rejzner, Perturbative algebraic quantum field theory, In Mathematical Aspects of Quantum Field Theories, Springer 2016 (arXiv:1208.1428)
Katarzyna Rejzner, def. 3.14 in Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (web)
Michael Dütsch, def. 1.2 in From classical field theory to perturbative quantum field theory, 2018