+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Algebraic Quantum Field Theory +--{: .hide} [[!include AQFT and operator algebra contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea In [[perturbative quantum field theory]] the [[algebra of observables]] of an [[interacting field theory]] constructed as a [[perturbation]] of the [[Wick algebra]] [[algebra of observables|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 [[differentiation|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](#Collini16), [Hawkins-Rejzner 16](#HawkinsRejzner16)). $\,$ ## Properties ### Causal locality of interacting field quantum observables +-- {: .num_prop #CausalLocalityOfBogoliubovFormula} ###### Proposition **([[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](#DuetschFredenhagen00), following [Brunetti-Fredenhagen 99, section 8](#BrunettiFredenhagen99), [Il'in-Slavnov 78](#IlinSlavnov78)) For **[[proof]]** see [this prop.](S-matrix#PerturbativeQuantumObservablesIsLocalnet) at _[[S-matrix]]_. ## Related concepts * [[Bogoliubov's formula]] * [[quantum Møller operator]] [[!include products in pQFT -- table]] [[!include states and observables -- content]] ## References The observation that the pertruabtive interacting field quantum observables form a [[causally local net of quantum observables]] is due to * {#IlinSlavnov78} V. A. Il'in and D. S. Slavnov, _Observable algebras in the S-matrix approach_, Theor. Math. Phys. 36 (1978) 32. ([spire](http://inspirehep.net/record/135575), [doi](http://dx.doi.org/10.1007/BF01035870)) then rediscovered in * {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], *Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds*, Commun. Math. Phys. **208** (2000) 623-661 [[math-ph/9903028](https://arxiv.org/abs/math-ph/9903028), [doi:10.1007/s002200050004](https://doi.org/10.1007/s002200050004)] and made more explicit in * {#DuetschFredenhagen00} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129)) The observation that these algebras are the [[formal deformation quantization]] of the interacting field theory is due to * {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626)) * {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157)) [[!redirects interacting field algebras of observables]] [[!redirects quantum observable on interacting fields]] [[!redirects quantum observables on interacting fields]] [[!redirects interacting field algebra of quantum observables]] [[!redirects interacting field algebras of quantum observables]] [[!redirects interacting quantum observable]] [[!redirects interacting quantum observables]] [[!redirects interacting field quantum observable]] [[!redirects interacting field quantum observables]] [[!redirects interacting field observable]] [[!redirects interacting field observables]] [[!redirects interacting field algebra]] [[!redirects interacting field algebras]] [[!redirects interacting observable algebra]] [[!redirects interacting observable algebras]] [[!redirects interacting quantum field observable]] [[!redirects interacting quantum field observables]] [[!redirects local interacting field observable]] [[!redirects local interacting field observables]] [[!redirects interacting local observable]] [[!redirects interacting local observables]]