 Moyal bracket

In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phasespace star product.
The Moyal Bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with Dirac ^{[1]} ^{[2]}. In the meantime this idea was independently introduced in 1946 by Hip Groenewold^{[3]}.
The Moyal bracket is a way of describing the commutator of observables in quantum mechanics when these observables are described as functions on phase space. It relies on schemes for identifying functions on phase space with quantum observables, the most famous of these schemes being Weyl quantization. It underlies Moyal’s dynamical equation, an equivalent formulation of Heisenberg’s quantum equation of motion, thereby providing the quantum generalization of Hamilton’s equations.
Mathematically, it is a deformation of the phasespace Poisson bracket, the deformation parameter being the reduced Planck constant ħ.
Up to formal equivalence, the Moyal Bracket is the unique oneparameter Liealgebraic deformation of the Poisson bracket. Its algebraic isomorphism to the algebra of commutators bypasses the negative result of the Groenewold–van Hove theorem, which precludes such an isomorphism for the Poisson bracket, a question implicitly raised by Paul Dirac in his 1926 doctoral thesis: the "method of classical analogy" for quantization.^{[4]}
For instance, in a twodimensional flat phase space, and for the Weylmap correspondence (cf. Weyl quantization), the Moyal bracket reads,
where ∗ is the starproduct operator in phase space (cf. Moyal product), while f and g are differentiable phasespace functions, and {f,g} is their Poisson bracket.
More specifically, this equals
Sometimes the Moyal bracket is referred to as the Sine bracket. E.g., a popular (Fourier) integral representation for it, introduced by George Baker^{[5]} is
Each correspondence map from phase space to Hilbert space induces a characteristic "Moyal" bracket (such as the one illustrated here for the Weyl map). All such Moyal brackets are formally equivalent among themselves, in accordance with a systematic theory.^{[6]}
The Moyal bracket specifies the eponymous infinitedimensional Lie algebra—it is antisymmetric in its arguments f and g, and satisfies the Jacobi identity. The corresponding abstract Lie algebra is realized by T_{f} ≡ f ∗ , so that
On a 2torus phase space, T^{2}, with periodic coordinates x and p, each in [0,2π], and integer mode indices m_{i} , for basis functions exp(i (m_{1}x+m_{2}p)), this Lie algebra reads,^{[7]}
which reduces to SU(N) for integer N ≡ 4π/ħ. SU(N) then emerges as a deformation of SU(∞), with deformation parameter 1/N.
Generalization of the Moyal bracket for quantum systems with secondclass constraints involves an operation on equivalence classes of functions in phase space,^{[8]} which might be considered as a quantum deformation of the Dirac bracket.
See also
References
 ^ J.E. Moyal, “Quantum mechanics as a statistical theory,” Proceedings of the Cambridge Philosophical Society, 45 (1949) pp. 99–124. doi:10.1017/S0305004100000487
 ^ "Maverick Mathematician: The Life and Science of J.E. Moyal (Chap. 3: Battle With A Legend)". http://epress.anu.edu.au/maverick/mobile_devices/ch03.html. Retrieved 201052.
 ^ H.J. Groenewold, “On the Principles of elementary quantum mechanics,” Physica,12 (1946) pp. 405–460. doi:10.1016/S00318914(46)800594
 ^ P.A.M. Dirac, "The Principles of Quantum Mechanics" (Clarendon Press Oxford, 1958) ISBN 9780198520115
 ^ G. Baker, “Formulation of Quantum Mechanics Based on the Quasiprobability Distribution Induced on Phase Space,” Physical Review, 109 (1958) pp.2198–2206. doi:10.1103/PhysRev.109.2198
 ^ C.Zachos, D. Fairlie, and T. Curtright, “Quantum Mechanics in Phase Space” (World Scientific, Singapore, 2005) ISBN 9789812383846 .
 ^ D. Fairlie and C. Zachos, "InfiniteDimensional Algebras, Sine Brackets and SU(∞)," Physics Letters, B224 (1989) pp. 101–107 doi:10.1016/03702693(89)910575
 ^ M. I. Krivoruchenko, A. A. Raduta, Amand Faessler, Quantum deformation of the Dirac bracket, Phys. Rev. D73 (2006) 025008.
Categories: Quantum mechanics
 Mathematical quantization
 Mathematical physics
Wikimedia Foundation. 2010.