Moyal product

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In mathematics, the Moyal product is an example for an associative, non-commutative product * on the functions on \mathbb{R}^{2n}, equipped with its Poisson bracket (with a generalization to symplectic manifolds below). This star product is also sometimes called Weyl-Groenewold product, as it was introduced by H. J. Groenewold in 1946, in an appreciation of Weyl quantization---Moyal appeared to not know about it even in his celebrated paper.

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[edit] Definition

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

f*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. f*g = fg + O(\hbar) (Deformation of the pointwise product) - clear from the definition but important!
2. f*g-g*f = i\hbar\{f,g\} + O(\hbar^2) (Deformation in the direction of the Poisson bracket)
3. f * 1 = 1 * f = f (The 1 of the undeformed algebra is the 1 in the new algebra)
4. \overline{f*g} = \overline{g}*\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 points of view on the 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}^{n}:

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

where πij is just a complex number for each i,j. (Do not confuse π here with the constant pi, which is completely different.) The Moyal product of two functions f and g can then be defined as

f*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 (which is usually treated as a formal parameter here). A closed form can be obtained by using the exponential,

f*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, e^A := 1 + \sum_{n=1}^{\infty} \frac{1}{n!} A^n. That is, going back to the first paragraph, the formula for Cn is

C_n = \frac{i^n \hbar^n}{2^n n!} m \circ \pi^n.

As mentioned in the introduction, oftentimes one eliminates all occurrences of i above, and the formulas then make sense using only real numbers.

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

[edit] On manifolds

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

[edit] Examples

A simple example of the construction of the Moyal product for the case of a two-dimensional euclidean phase space is given in the article on Weyl quantization.

[edit] Historical References

  • J.E. Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society, 45 (1949) pp. 99-124.
  • H.J. Groenewold, "On the Principles of elementary quantum mechanics", Physica,12 (1946) pp. 405-460.
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