In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.
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,[4] the "method of classical analogy" for quantization.
A popular (Fourier) integral representation for it, introduced by George Baker[7] 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).
[8] 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, T 2, 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,[9] which reduces to SU(N) for integer N ≡ 4π/ħ.
Generalization of the Moyal bracket for quantum systems with second-class constraints involves an operation on equivalence classes of functions in phase space,[10] which can be considered as a quantum deformation of the Dirac bracket.
Next to the sine bracket discussed, Groenewold further introduced[3] the cosine bracket, elaborated by Baker,[7][11] Here, again, ★ is the star-product operator in phase space, f and g are differentiable phase-space functions, and f g is the ordinary product.
The sine and cosine brackets are, respectively, the results of antisymmetrizing and symmetrizing the star product.
[12] The sine and cosine bracket also stand in relation to equations of a purely algebraic description of quantum mechanics.