interacting field algebra of observables in nLab

Contents

Idea

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).

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Properties

Causal locality of interacting field quantum observables

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

For proof see this prop. at S-matrix.

• Bogoliubov's formula

• quantum Møller operator

References

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

• 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, doi:10.1007/s002200050004]

and made more explicit in

• 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)

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

• Giovanni Collini, Fedosov Quantization and Perturbative Quantum Field Theory (arXiv:1603.09626)

• Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory (arXiv:1612.09157)