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 Paul 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 the phase space formulation of 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. Thus, its group contraction ħ→0 yields the Poisson bracket Lie algebra.
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 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. WignerWeyl transform), the Moyal bracket reads,

\begin{align} \{\{f,g\}\} & \stackrel{\mathrm{def}}{=}\ \frac{1}{i\hbar}(f\star gg\star f) \\ & = \{f,g\} + O(\hbar^2), \\ \end{align}
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.^{[5]}
More specifically, this equals

\{\{f,g\}\}\ = \frac{2}{\hbar} ~ f(x,p)\ \sin \left ( (\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.
A popular (Fourier) integral representation for it, introduced by George Baker^{[6]} 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( \tfrac{2}{\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.^{[7]}
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

[ T_f ~, T_g ] = T_{i\hbar \{ \{ f,g \} \} }.
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,^{[8]}

[ T_{m_1,m_2} ~ , T_{n_1,n_2} ] = 2i \sin \left (\tfrac{\hbar}{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 secondclass constraints involves an operation on equivalence classes of functions in phase space,^{[9]} which can be considered as a quantum deformation of the Dirac bracket.
Sine bracket and Cosine bracket
Next to the sine bracket discussed, Groenewold further introduced^{[3]} the cosine bracket, elaborated by Baker,^{[6]}^{[10]}

\begin{align} \{ \{ \{f ,g\} \} \} & \stackrel{\mathrm{def}}{=}\ \tfrac{1}{2}(f\star g+g\star f) = f g + O(\hbar^2). \\ \end{align}
Here, again, ★ is the starproduct operator in phase space, f and g are differentiable phasespace functions, and f g is the ordinary product.
The sine and cosine brackets are, respectively, the results of antisymmetrizing and symmetrizing the star product. Thus, as the sine bracket is the Wigner map of the commutator, the cosine bracket is the Wigner image of the anticommutator in standard quantum mechanics. Similarly, as the Moyal bracket equals the Poisson bracket up to higher orders of ħ, the cosine bracket equals the ordinary product up to higher orders of ħ. In the classical limit, the Moyal bracket helps reduction to the Liouville equation (formulated in terms of the Poisson bracket), as the cosine bracket leads to the classical Hamilton–Jacobi equation.^{[11]}
The sine and cosine bracket also stand in relation to equations of a purely algebraic description of quantum mechanics.^{[11]}^{[12]}
References

^ Moyal, J. E.; Bartlett, M. S. (1949). "Quantum mechanics as a statistical theory". Mathematical Proceedings of the Cambridge Philosophical Society 45: 99.

^ "Maverick Mathematician: The Life and Science of J.E. Moyal (Chap. 3: Battle With A Legend)". Retrieved 20100502.

^ ^{a} ^{b} Groenewold, H. J. (1946). "On the principles of elementary quantum mechanics". Physica 12 (7): 405–460.

^ P.A.M. Dirac, "The Principles of Quantum Mechanics" (Clarendon Press Oxford, 1958) ISBN 9780198520115

^ Conversely, the Poisson bracket is formally expressible in terms of the star product, iħ{f, g} = 2f (log★) g.

^ ^{a} ^{b} 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

^

^ Fairlie, D. B.; Zachos, C. K. (1989). "Infinitedimensional algebras, sine brackets, and SU(∞)". Physics Letters B 224: 101.

^ M. I. Krivoruchenko, A. A. Raduta, Amand Faessler, Quantum deformation of the Dirac bracket, Phys. Rev. D73 (2006) 025008.

^ See also the citation of Baker (1958) in: Curtright, T.; Fairlie, D.; Zachos, C. (1998). "Features of timeindependent Wigner functions". Physical Review D 58 (2). arXiv:hepth/9711183v3

^ ^{}a ^{b} B. J. Hiley: Phase space descriptions of quantum phenomena, in: A. Khrennikov (ed.): Quantum Theory: Reconsideration of Foundations–2, pp. 267286, Växjö University Press, Sweden, 2003 (PDF)

^ M. R. Brown, B. J. Hiley: Schrodinger revisited: an algebraic approach, arXiv:quantph/0005026 (submitted 4 May 2000, version of 19 July 2004, retrieved June 3, 2011)
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