Moyal bracket

From Wikipedia, the free encyclopedia

In physics, the Moyal bracket is the suitably normalized antisymmetrization of the star product.

The Moyal Bracket was introduced in 1946 by Hip Groenewold and in 1949 by José Enrique Moyal. It is the phase space isomorph of the quantum commutator in Hilbert space, and plays the same crucial role in phase-space quantum mechanics, or Weyl quantization.

For example, it underlies Moyal’s dynamical equation, the isomorph of Heisenberg’s quantum equation of motion, thereby providing the quantum generalization of Hamilton’s classical equations of motion.

Mathematically, it is a deformation of the phase-space Poisson bracket, the deformation parameter being \hbar.

It specifies the eponymous infinite-dimensional Lie algebra—it is antisymmetric in its arguments f and g, and satisfies the Jacobi identity.

Up to equivalence, it 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 raised by Paul Dirac.

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

\{\{f,g\}\}\ \stackrel{\mathrm{def}}{=}\ (f*g-g*f)/ i\hbar = \{f,g\} + O(\hbar^2),

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\}\}\ \stackrel{\mathrm{def}}{=}\ \frac{2}{\hbar} ~ f(x,p)\ \sin ( {{\frac{\hbar }{2}}(\stackrel{\leftarrow }{\partial }_{x} \stackrel{\rightarrow }{\partial }_{p}-\stackrel{\leftarrow }{\partial }_{p}\stackrel{\rightarrow }{\partial }_{x})}) \ 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 in 1958, is

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

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.

[edit] References

  • H.J. Groenewold, “On the Principles of elementary quantum mechanics,” Physica,12 (1946) pp. 405-460.
  • J.E. Moyal, “Quantum mechanics as a statistical theory,” Proceedings of the Cambridge Philosophical Society, 45 (1949) pp. 99-124.
  • G. Baker, “Formulation of Quantum Mechanics Based on the Quasi-probability Distribution Induced on Phase Space,” Physical Review, 109 (1958) pp.2198-2206.
  • C. Zachos, D. Fairlie, and T. Curtright, “Quantum Mechanics in Phase Space” (World Scientific, Singapore, 2005).