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In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space 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 phase-space Poisson bracket, the deformation parameter being the reduced Planck constant ħ. Thus, its group contraction ħ→0 yields the Poisson bracket Lie algebra.

Up to formal equivalence, the Moyal Bracket is the unique one-parameter Lie-algebraic 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 two-dimensional flat phase space, and for the Weyl-map correspondence (cf. Weyl quantization), the Moyal bracket reads,

\( \begin{align} \{\{f,g\}\} & \stackrel{\mathrm{def}}{=}\ \frac{1}{i\hbar}(f\star g-g\star f) \\ & = \{f,g\} + O(\hbar^2), \\ \end{align} \)

where ★ is the star-product operator in phase space (cf. Moyal product), while f and g are differentiable phase-space functions, and {f,g} is their Poisson bracket.

More specifically, this equals

\( \{\{f,g\}\}\ = \frac{2}{\hbar} ~ f(x,p)\ \sin \left ( {{\frac{\hbar }{2}}(\stackrel{\leftarrow }{\partial }_x \stackrel{\rightarrow }{\partial }_{p}-\stackrel{\leftarrow }{\partial }_{p}\stackrel{\rightarrow }{\partial }_{x})} \right ) \ g(x,p). \)

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

\( \{ \{ f,g \} \}(x,p) = {2 \over \hbar^3 \pi^2 } \int dp' \, dp'' \, dx' \, dx'' f(x+x',p+p') g(x+x'',p+p'')\sin \left( {2\over \hbar} (x'p''-x''p')\right). \)

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 infinite-dimensional Lie algebra—it is antisymmetric in its arguments f and g, and satisfies the Jacobi identity. The corresponding abstract Lie algebra is realized by Tf ≡ f ∗ , so that

\( [ T_f ~, T_g ] = T_{i\hbar \{ \{ f,g \} \} }. \)

On a 2-torus phase space, T2, with periodic coordinates x and p, each in [0,2π], and integer mode indices mi , for basis functions exp(i (m1x+m2p)), this Lie algebra reads,[7]

\( [ T_{m_1,m_2} ~ , T_{n_1,n_2} ] = 2i \sin \left ({\hbar\over 2}(n_1 m_2 - n_2 m_1 )\right ) ~ T_{m_1+n_1,m_2+ n_2}, ~ \)

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 second-class 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

Weyl quantization
Wigner quasi-probability distribution
Poisson bracket

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)". Retrieved 2010-05-02.
^ H.J. Groenewold, “On the Principles of elementary quantum mechanics,” Physica,12 (1946) pp. 405–460. doi:10.1016/S0031-8914(46)80059-4
^ P.A.M. Dirac, "The Principles of Quantum Mechanics" (Clarendon Press Oxford, 1958) ISBN 978-0-19-852011-5
^ G. Baker, “Formulation of Quantum Mechanics Based on the Quasi-probability 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 978-981-238-384-6 .
^ D. Fairlie and C. Zachos, "Infinite-Dimensional Algebras, Sine Brackets and SU(∞)," Physics Letters, B224 (1989) pp. 101–107 doi:10.1016/0370-2693(89)91057-5
^ M. I. Krivoruchenko, A. A. Raduta, Amand Faessler, Quantum deformation of the Dirac bracket, Phys. Rev. D73 (2006) 025008.

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