# Moyal product

Moyal product

In mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket (with a generalization to symplectic manifolds below). This particular star product is also sometimes called Weyl-Groenewold product, as it was introduced by H. J. Groenewold in 1946, in a trenchant appreciation[1] of Weyl quantization —Moyal actually appears to not know about it in his celebrated paper,[2] and in his legendary correspondence with Dirac, as adduced in his biography.[3] (The paradoxical popular naming after Moyal, utilized in this stub, appears to have emerged only in the 1970s, in homage to his flat phase-space quantization picture.)

## Definition

The product (for smooth functions f and g on $\mathbb R^{2n}$ takes the form

$f\star g = fg + \sum_{n=1}^{\infty} \hbar^{n} C_{n}(f,g)$

where each Cn is a certain bidifferential operator of order n with the following properties. (See below for an explicit formula).

$1.\quad f\star g = fg + \mathcal O(\hbar)$

(Deformation of the pointwise product) — implicit in the definition.

$2. \quad f\star g-g\star f = \mathrm i\hbar\{f,g\} + \mathcal O(\hbar^2) \equiv \mathrm i\hbar \{\{f,g\}\}$

(Deformation of the Poisson bracket, called Moyal bracket.)

$3. \quad f\star 1=1\star f=f$

(The 1 of the undeformed algebra is also the identity in the new algebra.)

$4. \quad \overline{f\star g} = \overline{g}\star \overline{f}$

(The complex conjugate is an antilinear antiautomorphism.)

Note that, if one wishes to take functions valued in the real numbers, then an alternative version eliminates the i in condition 2 and eliminates condition 4.

If one restricts to polynomial functions, the above algebra is isomorphic to the Weyl algebra An , and the two offer alternative realizations of Weyl quantization of the space of polynomials in n variables (or, the symmetric algebra of a vector space of dimension 2n).

To provide an explicit formula, consider a constant Poisson bivector Π on $\mathbb R^{2n}$ :

$\Pi=\sum_{i,j} \Pi^{ij} \partial_i \wedge \partial_j,$

where Πij is just a complex number for each i,j.

The star product of two functions f and g can then be defined as

$f\star g = fg + \frac{i\hbar}{2} \sum_{i,j} \Pi^{ij} (\partial_i f) (\partial_j g) - \frac{\hbar^2}{8} \sum_{i,j,k,m} \Pi^{ij} \Pi^{km} (\partial_i \partial_k f) (\partial_j \partial_m g) + \ldots$

where $\hbar$ is the reduced Planck constant, treated as a formal parameter here.

A closed form can be obtained by using the exponential,

$f\star g = m \circ e^{\frac{i\hbar}{2} \Pi}(f \otimes g),$

where m is the multiplication map, $m(a\otimes b) = ab$, and the exponential is treated as a power series, $\textstyle e^A := 1 + \sum_{n=1}^{\infty} \frac{1}{n!} A^n$.

That is, the formula for Cn is

$C_n = \frac{i^n}{2^n n!} m \circ \Pi^n.$

As mentioned, often one eliminates all occurrences of i above, and the formulas then restrict naturally to real numbers.

Note that if the functions f and g are polynomials, the above infinite sums become finite (reducing to the ordinary Weyl algebra case).

## On manifolds

On any symplectic manifold, one can, at least locally, choose coordinates so as make the symplectic structure constant, by Darboux's theorem; and, using the associated Poisson bivector, one may consider the above formula. For it to work globally, as a function on the whole manifold (and not just a local formula), one must equip the symplectic manifold with a flat symplectic connection.

More general results for arbitrary Poisson manifolds (where the Darboux theorem does not apply) are given by the Kontsevich quantization formula.

## Examples

A simple explicit example[4] of the construction and utility of the ∗-product (for the simplest case of a two-dimensional euclidean phase space) is given in the article on Weyl quantization. Every correspondence prescription between phase space and Hilbert space, however, induces its own proper ∗-product.

## References

1. ^ H.J. Groenewold, "On the Principles of elementary quantum mechanics", Physica,12 (1946) pp. 405-460.
2. ^ J.E. Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society, 45 (1949) pp. 99-124.
3. ^ Ann Moyal, "Maverick Mathematician: The Life and Science of J.E. Moyal", ANU E-press, 2006, http://epress.anu.edu.au/maverick_citation.html
4. ^ C. Zachos, D. Fairlie, and T. Curtright, "Quantum Mechanics in Phase Space" ( World Scientific, Singapore, 2005) ISBN 978-981-238-384-6 .

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