nLab Fedosov's deformation quantization

Idea

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

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 $x \in X$ regard the tangent space $T_x X$ as a symplectic vector space $(T_x X, \omega_x)$ and consider its Moyal deformation quantization $\mathcal{W}_x \coloneqq A_{Moy}(T_x X, \omega_x)$. 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 $\mathcal{W} \to X$. From the fiber-wise product in $A_{Moy}(T_x X, \omega_x)$ the space of sections $\Gamma_X(\mathcal{W})$ inherits itself the structure of an associative algebra.

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

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

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

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

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

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

Related concepts

• almost Kähler geometric quantization

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)

with a monograph account in

• Boris Fedosov, Deformation Quantization and Index Theory, John Wiley & Sons (1995) [ISBN:9783055017162]

Its generalization to almost Kähler structures is due to

• Alexander Karabegov, Martin Schlichenmaier, Almost-Kähler deformation quantization, Letters in Mathematical Physics, 57 2 (2001) 135–148 [doi:10.1023/A:1017993513935, arXiv:0102169]

with a survey in

• Martin Schlichenmaier, Deformation quantization for almost-Kähler manifolds, J. of Nonlinear Math. Phys.

  11, Supplement (2004) 49–54 [doi:10.2991/jnmp.2004.11.s1.6, pdf]

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

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

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

• Eli Hawkins, Kasia Rejzner, section 5.3.2 of The Star Product in Interacting Quantum Field Theory (arXiv:1612.09157)