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The Moyal product is a formal deformation quantization of a linear Poisson manifold, hence of a vector space $V$ equipped with a Poisson bivector $\pi \in V \wedge V$, regarded as a constant (translation invariant) bivector field.
Moyal quantization serves as an intermediate step in quantization of more general situations:
Given a symplectic manifold, then Moyal quantization applies in each fiber of the tangent bundle. The resulting fiber bundle of Moyal algebras admits a flat connection (non-uniquely) compatible with the algebra structure. The covariantly constant sections of this Moyal-algebra bundle constitute a formal deformation quantization of the symplectic manifold, see at Fedosov's deformation quantization.
With a little care, the Moyal construction applies also to infinite-dimensional Poisson vector spaces such as appear in local field theory. Here the Moyal quantization yields formal deformation quantization of free field theories to perturbative quantum field theories, the result are the Wick algebras of free field theory (Dito 90, Dütsch-Fredenhagen 01). Combining this this with Fedosov's deformation quantization as above yields interacting perturbative quantum field theories as constructed via causal perturbation theory (Collini 16), see at locally covariant perturbative quantum field theory for more on this.
The Moyal star product on smooth functions $C^\infty(V)$ is given on $f,g \in C^\infty(V)$ by
where in the exponent we regard $\pi$ as an endomorphism on the tensor product $C^\infty(V) \otimes C^\infty(V)$ by differentiation in each argument, where the exponential denotes the corresponding formal power series of iterated applications of this endomorphism, and where $prod \colon C^\infty(V) \otimes C^\infty(V) \to C^\infty(V)$ is the usual pointwise product of functions.
This means that given a choice of basis $\{x^i\}_i$ of $V$ such that $\pi$ has components $\{\pi^{i j}\}_{i j}$ in this basis, the resulting formal power series in the formal parameter $\hbar$ (“Planck's constant”) starts out as
(integral representation of star product)
If the functions $f,g$ admit Fourier analysis (are functions with rapidly decreasing partial derivatives), then their star product is equivalently given by the following integral expression:
(Baker 58, see at star product this prop).
The Moyal quantization of a Poisson vector space $(V,\pi)$ arises equivalently as the canonical geometric quantization of symplectic groupoids of the symplectic groupoid which is the Lie integration of the corresponding Poisson Lie algebroid (Weinstein 91, p. 446, Garcia-Bondia & Varilly 94, section V, Hawkins 06).
See at star product this prop. for the proof; and see at geometric quantization of symplectic groupoids – Examples – Moyal quantization for more.
The Moyal product was introduced independently in
Hilbrand Groenewold, On the Principles of elementary quantum mechanics, Physica,12 (1946) pp. 405-460.
José Moyal, Quantum mechanics as a statistical theory. Mathematical Proceedings of the Cambridge Philosophical Society 45: 99 (1949)
The integral expression (prop. ) is apparently due to
General accounts include
D. B. Fairlie, Moyal Brackets, Star Products and the Generalised Wigner Function (arXiv:hep-th/9806198)
Maciej Blaszak, Ziemowit Domanski, Maciej Blaszak, Ziemowit Domanski (arXiv:1009.0150)
The understanding of the Moyal product as the polarized groupoid convolution algebra of the corresponding symplectic groupoid, hence as an example of geometric quantization of symplectic groupoids had been suggested without proof in
and was proven in detail in
In a broader context this was reconsidered in
The observation that Moyal deformation quantization applied to the Peierls-Poisson bracket yields the Wick algebra quantization of free field theories is due to
and was amplified in the broader context of perturbative AQFT in
That moreover the corresponding Fedosov deformation quantization based on this free field theory star product yields the causal perturbation theory quantization of interacting field theories is due to
Last revised on April 18, 2020 at 01:43:38. See the history of this page for a list of all contributions to it.