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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 ℝ<sup>2n</sup>,  equipped with its [[Poisson bracket]] (with a generalization to [[symplectic manifold]]s below).
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This particular star product is also sometimes called [[Hermann Weyl|Weyl]]-Groenewold product, as it was introduced by [[Hilbrand J. Groenewold | H. J. Groenewold]] in 1946, in a trenchant appreciation<ref>H.J. Groenewold, "On the Principles of elementary quantum mechanics", ''Physica'','''12''' (1946) pp. 405-460.</ref> of [[Weyl quantization]] &mdash;Moyal actually appears not to know about it in his celebrated paper,<ref>{{cite  doi|10.1017/S0305004100000487|noedit}}</ref> and in his legendary correspondence with Dirac, as adduced in his biography.<ref>Ann Moyal, "Maverick Mathematician: The Life and Science of J.E. Moyal", ANU E-press, 2006. [http://epress.anu.edu.au/maverick_citation.html Online copy]</ref> (The paradoxical popular naming after Moyal, utilized in this stub,  appears to have emerged only in the 1970s, in homage to his flat [[Weyl quantization|phase-space quantization]] picture.)<ref>
{{cite journal
|last1=Curtright |first1=T. L.
|last2=Zachos |first2=C. K.
|year=2012
|title=Quantum Mechanics in Phase Space
|journal=[[Asia Pacific Physics Newsletter]]
|volume=1 |issue= |pages=37
|arxiv=1104.5269
|bibcode=
|doi=10.1142/S2251158X12000069
}}</ref>
==Definition==
The product for [[smooth function]]s ''f'' and ''g'' on ℝ<sup>2n</sup>  takes the form


:<math>f\star g = fg + \sum_{n=1}^{\infty} \hbar^{n} C_{n}(f,g)</math>
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where each ''C''<sub>n</sub>  is a certain bi[[differential operator]] of order ''n''  with the following properties. (See below for an explicit formula).
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:<math>1.\quad f\star g = fg + \mathcal O(\hbar)</math>
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(Deformation of the pointwise product) &mdash; implicit in the definition.
 
:<math>2. \quad f\star g-g\star f = \mathrm i\hbar\{f,g\} + \mathcal O(\hbar^2) \equiv \mathrm i\hbar \{\{f,g\}\}</math>
(Deformation of the Poisson bracket, called  [[Moyal bracket]].)
 
:<math>3. \quad f\star 1=1\star f=f</math>
(The 1 of the undeformed algebra is also the identity in the new algebra.)
 
:<math>4. \quad \overline{f\star g} = \overline{g}\star \overline{f}</math>
(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 <math>\mathrm i</math> in condition 2 and eliminates condition 4.
 
If one restricts to polynomial functions, the above algebra is isomorphic to the [[Weyl algebra]] ''A''<sub>n</sub>,  
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 2''n'').
 
To provide an explicit formula, consider a constant [[Poisson bivector]] ∏ on ℝ<sup>2n</sup>:  
::<math>\Pi=\sum_{i,j} \Pi^{ij} \partial_i \wedge \partial_j,</math>
where ∏<sup>''ij''</sup> is a complex number for each ''i,j''.
The star product of two functions <math>f</math> and <math>g</math> can then be defined as
:<math>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</math>
 
where  ħ  is the [[reduced Planck constant]], treated as a formal parameter here. 
 
A closed form can be obtained by using the [[Matrix exponential|exponential]],  
::<math>f\star g = m \circ e^{\frac{i\hbar}{2} \Pi}(f \otimes g),</math>
where  <math>m</math>  is the multiplication map, <math>m(a\otimes b) = ab</math>, and the exponential is treated as a power series, <math>\textstyle e^A := 1 + \sum_{n=1}^{\infty} \frac{1}{n!} A^n</math>. 
 
That is, the formula for <math>C_n</math> is
::<math>C_n = \frac{i^n}{2^n n!} m \circ \Pi^n.</math>
 
As indicated, often one eliminates all occurrences of <math>\mathrm i</math> 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 (mathematics)|flat]] symplectic [[Connection (mathematics)|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 of the construction and utility of the <small>★</small>-product (for the simplest case of a two-dimensional euclidean [[phase space]]) is given in the article on [[Weyl quantization]]: Two Gaussians compose with this <small>★</small>-product according to a hyperbolic tangent law,<ref>[[Cosmas Zachos|C. Zachos]], [[David Fairlie|D. Fairlie]], and [[Thomas Curtright|T. Curtright]], "Quantum Mechanics in Phase Space" ( World Scientific, Singapore, 2005) ISBN 978-981-238-384-6 .</ref>
: <math>
\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 ) .
</math>
N.B. Note the classical limit, {{math|''ħ'' → 0}}.
 
Every correspondence prescription between phase space and Hilbert space, however, induces ''its own'' [[Transformation between distributions in time–frequency analysis|proper]]  <small>★</small>-product.<ref>Cohen, L. (1995) ''Time-Frequency Analysis'', Prentice-Hall, New York, 1995. ISBN 978-0135945322. </ref>
 
==References==
<references/>
 
{{DEFAULTSORT:Moyal Product}}
[[Category:Mathematical quantization]]
[[Category:Mathematical physics]]

Latest revision as of 00:32, 15 December 2014

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