Moyal Bracket

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. In the meantime this idea was independently introduced in 1946 by Hip Groenewold.

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 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.

For instance, in a two-dimensional flat phase space, and for the Weyl-map correspondence (cf. Wigner-Weyl transform), the Moyal bracket reads,

begin{align}
{{f,g}} & stackrel{mathrm{def}}{=} frac{1}{ihbar}(fstar g-gstar 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 ( {{tfrac{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.

A popular (Fourier) integral representation for it, introduced by George Baker 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.

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

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

 =
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 second-class constraints involves an operation on equivalence classes of functions in phase space, which might be considered as a quantum deformation of the Dirac bracket.

Read more about Moyal BracketSine Backet and Cosine Bracket

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