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Fedosov's deformation quantization

Context

Symplectic geometry

Background

Basic concepts

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Classical mechanics and quantization

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Physics

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Contents

Idea

Fedosov’s method (Fedosov 94) is a prescription for constructing formal deformation quantizations of the algebra of smooth functions on any symplectic manifold (X,ω)(X,\omega).

In particular this establishes the existence of formal deformation quantizations of all symplectic manifolds. While Kontsevich 97 more generally proves the existence of deformation quantization of all Poisson manifolds of finite dimension, Fedosov’s method, in a variant applicable to almost Kähler structures (Karabegov-Schlichenmaier 01) generalizes to infinite-dimensional symplectic manifolds as they appear in local field theory, where it yields the quantization to perturbative quantum field theory equivalent to the method of causal perturbation theory (Collini 16). (Kontsevich’s deformation quantization is however not compatible with field theory (Hawkins-Rejzner 16, section 5.3.2).) For more on this see at locally covariant perturbative quantum field theory.

Fedosov’s method proceeds by the following steps:

  1. For each point xXx \in X regard the tangent space T xXT_x X as a symplectic vector space (T xX,ω x)(T_x X, \omega_x) and consider its Moyal deformation quantization 𝒲 xA Moy(T xX,ω x)\mathcal{W}_x \coloneqq A_{Moy}(T_x X, \omega_x). In the context of local field theory these Moyal algebras are the Wick algebras of the underlying free field theory.

    The union of these yelds an associative algebra-fiber bundle 𝒲X\mathcal{W} \to X. From the fiber-wise product in A Moy(T xX,ω x)A_{Moy}(T_x X, \omega_x) the space of sections Γ X(𝒲)\Gamma_X(\mathcal{W}) inherits itself the structure of an associative algebra.

  2. Find a flat connection Fed\nabla_{Fed} on 𝒲\mathcal{W} which respects the associative algebra structure, and such that there is a linear isomorphism

    Γ X(𝒲)ker( Fed)C (X)[[]] \Gamma_X(\mathcal{W}) \supset ker(\nabla_{Fed}) \simeq C^\infty(X) [ [ \hbar ] ]

    between the covariantly constant sections of the algebra bundle and the algebra of smooth functions on XX with a formal variable \hbar adjoined.

    Such Fed\nabla_{Fed} exists, (non-uniquely), it is called a Fedosov connection.

  3. By this linear isomorphism the algebra structure on Γ X(𝒲)\Gamma_X(\mathcal{W}) induces an associative algebra structure on C (X)[[]]C^\infty(X)[ [ \hbar ] ], and this turns out to be a formal deformation quantization of (X,ω)(X,\omega).

    In the context of local field theory, this deformation quantization is equivalent to the result of quantization via causal perturbation theory (Collini 16).

References

The method is due to

  • Boris Fedosov, A simple geometrical construction of deformation quantization Journal of Differential Geometry, 40(2):213–238, 1994 (Euclid)

Its generalization to almost Kähler structures is due to

  • V. Karabegov, M. Schlichenmaier, Almost-Kähler deformation quantization, Letters in Mathematical Physics, 57(2):135–148, 2001 (proceedings pdf)

The observation that the construction of perturbative quantum field theory via causal perturbation theory is equivalent to Fedosov quantization is due to

Discussion showing that this generalization to field theory is not given by Kontsevich deformation quantization is in

Last revised on August 10, 2017 at 14:11:16. See the history of this page for a list of all contributions to it.