nLab interacting field algebra of observables



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



In perturbative quantum field theory the algebra of observables of an interacting field theory constructed as a perturbation of the Wick algebra of observables of a free field theory is called, for emphasis, the interacting field algebra of observables, often just “interacting field algebra”, for short.

In terms of causal perturbation theory, the interacting field algebra is obtained from the free field Wick algebra of observables and the perturbative S-matrix by differentiating Bogoliubov's formula, yielding a Møller operator.

More abstractly, the algebra of observables is the formal deformation quantization (specifically Fedosov deformation quantization) of the interacting field theory (Collini 16, Hawkins-Rejzner 16).



Causal locality of interacting field quantum observables


(causal locality)

As the spacetime support varies, the algebras of interacting field quantum observables spanned via the Bogoliubov formula consistitute a causally local net of observables, hence an instance of perturbative AQFT.

(Dütsch-Fredenhagen 00, section 3, following Brunetti-Fredenhagen 99, section 8, Il’in-Slavnov 78)

For proof see this prop. at S-matrix.

product in perturbative QFT\,\, induces
normal-ordered productWick algebra (free field quantum observables)
time-ordered productS-matrix (scattering amplitudes)
retarded productinteracting quantum observables

quantum probability theoryobservables and states


The observation that the pertruabtive interacting field quantum observables form a causally local net of quantum observables is due to

  • V. A. Il’in and D. S. Slavnov, Observable algebras in the S-matrix approach, Theor. Math. Phys. 36 (1978) 32. (spire, doi)

then rediscovered in

and made more explicit in

The observation that these algebras are the formal deformation quantization of the interacting field theory is due to

Last revised on June 10, 2023 at 10:03:56. See the history of this page for a list of all contributions to it.