nLab Moyal deformation quantization

Idea

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.

Definition

The Moyal star product on smooth functions $C^\infty(V)$ is given on $f,g \in C^\infty(V)$ by

$$f \star g \coloneqq prod \circ \exp(\hbar \pi)(f , g) \,,$$

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

$$(f \star g) = f \cdot g + \hbar \sum_{i,j} \pi^{i j} \frac{\partial f}{\partial x^i}\cdot \frac{\partial g}{\partial x^j} + \frac{1}{2} \hbar^2 \sum_{i,j,k, l} \pi^{k l} \pi^{i j} \frac{\partial^2 f}{\partial x^k\partial x^i}\cdot \frac{\partial^2 g}{\partial x^l \partial x^j} + \cdots$$

Properties

Integral representation

Proposition

(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:

$$\begin{aligned} \left(f \star_\omega g\right)(x) &= \frac{(det(\omega)^{2n})}{(2 \pi \hbar)^{2n} } \int e^{ - \tfrac{1}{i \hbar} \omega((x - \tilde y),(x-y))} f(y) g(\tilde y) \, d^{2 n} y \, d^{2 n} \tilde y \end{aligned}$$

(Baker 58, see at star product this prop).

Via geometric quantization

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.

References

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

• George A. Baker, Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space, Jr. Phys. Rev. 109, 2198 – Published 15 March 1958 (doi:10.1103/PhysRev.109.2198)

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

• Alan Weinstein, p. 446 in P. Donato et al. (eds.) Symplectic Geometry and Mathematical Physics, (Birkhäuser, Basel, 1991);

and was proven in detail in

• José Gracia-Bondia, Joseph Varilly, From geometric quantization to Moyal quantization, J. Math. Phys. 36 (1995) 2691-2701 (arXiv:hep-th/9406170)

In a broader context this was reconsidered in

• Eli Hawkins, example 6.2 of A groupoid approach to quantization, J. Symplectic Geom. Volume 6, Number 1 (2008), 61-125. (arXiv:math.SG/0612363)

The observation that Moyal deformation quantization applied to the Peierls-Poisson bracket yields the Wick algebra quantization of free field theories is due to

• J. Dito, Star-product approach to quantum field theory: The free scalar field. Letters in Mathematical Physics, 20(2):125–134, 1990 (spire)

and was amplified in the broader context of perturbative AQFT in

• Michael Dütsch, Klaus Fredenhagen, Perturbative algebraic field theory, and deformation quantization, in Roberto Longo (ed.), Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001

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

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