Contents

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

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.

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

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