In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product.
The Moyal Bracket was introduced in 1946 by Hip Groenewold[1] and reprised in 1949 by José Enrique Moyal.[2] 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 instance in underpinning the uncertainty principle.
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 the reduced Planck constant ħ.
Up to formal 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 implicitly raised by Paul Dirac in his 1926 doctoral thesis: the "method of classical analogy" for quantization.[3]
For instance, in a two-dimensional flat phase space, and for the Weyl-map correspondence (cf. Weyl quantization), the Moyal bracket reads,
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
Sometimes the Moyal bracket is referred to as the Sine bracket. E.g., a popular (Fourier) integral representation for it, introduced by George Baker[4] 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.[5]
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 \} \} }.](http://wpcontent.answers.com/math/c/a/8/ca8b8ac0ee89f3808e528d2bd66bc90a.png)
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(m1 x + m2 p) ) , this Lie algebra reads,[6]
![[ 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}, ~](http://wpcontent.answers.com/math/e/9/7/e977f764540de27b579d80d421140de1.png)
which reduces to SU(N) for integer N≡ 4 π / ħ. SU(N) then emerges as a deformation of SU(∞), with deformation parameter 1/N.
See also
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.
- ^ P.A.M. Dirac, "The Principles of Quantum Mechanics" (Clarendon Press Oxford, 1958) ISBN 9780198520115
- ^ 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) 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
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