Moyal product

In mathematics, the Moyal product (also called the star product or Weyl-Groenewold product) is perhaps the best-known example of a phase-space star product. It is an associative, non-commutative product, ★, on the functions on ℝ²ⁿ, equipped with its Poisson bracket (with a generalization to symplectic manifolds, described below).

Historical comments

The Moyal product is named after José Enrique Moyal, but is also sometimes called the Weyl-Groenewold product as it was introduced by H. J. Groenewold in his 1946 doctoral dissertation, in a trenchant appreciation^[1] of the Weyl correspondence. Moyal actually appears not to know about the product in his celebrated paper,^[2] and was crucially lacking it in his legendary correspondence with Dirac, as illustrated in his biography.^[3] The popular naming after Moyal appears to have emerged only in the 1970s, in homage to his flat phase-space quantization picture.^[4]

Definition

The product for smooth functions f and g on ℝ²ⁿ takes the form

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

where each C_n 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.

$2. \quad f\star g-g\star f = \mathrm i\hbar\{f,g\} + \mathcal O(\hbar^3) \equiv \mathrm i\hbar \{\{f,g\}\}$

(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 $\mathrm i$ in condition 2 and eliminates condition 4.

If one restricts to polynomial functions, the above algebra is isomorphic to the Weyl algebra A_n, and the two offer alternative realizations of the Weyl map 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 ∏ on ℝ²ⁿ:

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

where ∏^ij is 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 ħ 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 $C_n$ is

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

As indicated, often one eliminates all occurrences of $\mathrm i$ 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 torsion-free symplectic connection. This makes it a Fedosov manifold.

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 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 the Wigner–Weyl transform: Two Gaussians compose with this ★-product according to a hyperbolic tangent law,^[5]

$\exp \left (-a (x^2+p^2)\right ) ~ \star ~ \exp \left (-b (x^2+p^2)\right ) = {1\over 1+\hbar^2 ab} \exp \left (-{a+b\over 1+\hbar^2 ab} (x^2+p^2)\right ) .$

N.B. Note the classical limit, ħ → 0.

Every correspondence prescription between phase space and Hilbert space, however, induces its own proper ★-product.^[6][7]

References

1. H.J. Groenewold, "On the Principles of elementary quantum mechanics", Physica,12 (1946) pp. 405-460.
2. Moyal, J. E.; Bartlett, M. S. (1949). "Quantum mechanics as a statistical theory". Mathematical Proceedings of the Cambridge Philosophical Society 45: 99. doi:10.1017/S0305004100000487.
3. Ann Moyal, "Maverick Mathematician: The Life and Science of J.E. Moyal", ANU E-press, 2006. Online copy
4. Curtright, T. L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space". Asia Pacific Physics Newsletter 1: 37. arXiv:1104.5269. doi:10.1142/S2251158X12000069.
5. C. Zachos, D. Fairlie, and T. Curtright, "Quantum Mechanics in Phase Space" ( World Scientific, Singapore, 2005) ISBN 978-981-238-384-6 .
6. Cohen, L. (1995) Time-Frequency Analysis, Prentice-Hall, New York, 1995. ISBN 978-0135945322.
7. Lee, H. W. (1995). "Theory and application of the quantum phase-space distribution functions". Physics Reports 259 (3): 147. doi:10.1016/0370-1573(95)00007-4.