In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, 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
José Enrique Moyal ( he, יוסף הנרי מויאל; 1 October 1910 – 22 May 1998) was an Australian mathematician and mathematical physicist who contributed to aeronautical engineering, electrical engineering and statistics, among ot ...
, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
. In the meantime this idea was independently introduced in 1946 by
Hip Groenewold.
Overview
The Moyal bracket is a way of describing the
commutator of observables in the
phase space formulation
The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...
of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
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 the
Wigner–Weyl transform
In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform (after Hermann Weyl and Eugene Wigner) is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödin ...
. 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
Deformation can refer to:
* Deformation (engineering), changes in an object's shape or form due to the application of a force or forces.
** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
of the phase-space
Poisson bracket (essentially an
extension of it), the deformation parameter being the reduced
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. Thus, its
group contraction 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 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, the Moyal bracket reads,
:
where
★ is the star-product operator in phase space (cf.
Moyal product), while and are differentiable phase-space functions, and is their Poisson bracket.
More specifically, in
operational calculus Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.
History
Th ...
language, this equals
The left & right arrows over the partial derivatives denote the left & right partial derivatives. 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 and , and satisfies the
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
.
The corresponding abstract
Lie algebra is realized by
★, so that
:
On a 2-torus phase space, , with periodic
coordinates and , each in , and integer mode indices , for basis functions , this Lie algebra reads,
:
which reduces to ''SU''(''N'') for integer .
''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 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
the cosine bracket, elaborated by Baker,
[See also the citation of Baker (1958) in:]
arXiv:hep-th/9711183v3
/ref>
:
Here, again, ★ is the star-product operator in phase space, and are differentiable phase-space functions, and 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 mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
in standard quantum mechanics. Similarly, as the Moyal bracket equals the Poisson bracket up to higher orders of , the cosine bracket equals the ordinary product up to higher orders of {{mvar, ħ. In the classical limit
The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
, 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
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
.[ B. J. Hiley: Phase space descriptions of quantum phenomena, in: A. Khrennikov (ed.): ''Quantum Theory: Re-consideration of Foundations–2'', pp. 267-286, Växjö University Press, Sweden, 2003]
PDF
The sine and cosine bracket also stand in relation to equations of a purely algebraic description of quantum mechanics.[M. R. Brown, B. J. Hiley: ''Schrodinger revisited: an algebraic approach'']
arXiv:quant-ph/0005026
(submitted 4 May 2000, version of 19 July 2004, retrieved June 3, 2011)
References
Mathematical quantization
Symplectic geometry