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{{Short description|Suitably normalized antisymmetrization of the phase-space star product}}
{{Quantum mechanics|cTopic=Background}}
In [[Physics:Physics|physics]], the '''Moyal bracket''' is the suitably normalized antisymmetrization of the phase-space [[Moyal product|star product]].

The Moyal bracket was developed in about 1940 by [[Biography:José Enrique Moyal|José Enrique Moyal]], but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with [[Biography:Paul Dirac|Paul Dirac]].<ref>{{Cite journal | last1 = Moyal | first1 = J. E. | last2 = Bartlett | doi = 10.1017/S0305004100000487 | first2 = M. S. | title = Quantum mechanics as a statistical theory | journal = Mathematical Proceedings of the Cambridge Philosophical Society | volume = 45 | pages = 99–124 | year = 1949 | issue = 1 |bibcode = 1949PCPS...45...99M | s2cid = 124183640 }}</ref><ref>
{{cite book |url=http://epress.anu.edu.au/maverick/mobile_devices/ch03.html|title=Maverick Mathematician: The Life and Science of J.E. Moyal (Chap. 3: Battle With A Legend)|year=2006 |doi=10.22459/MM.08.2006 |access-date=2010-05-02 |last1=Moyal |first1=Ann |isbn=9781920942595 |doi-access=free }}</ref>  In the meantime this idea was independently introduced in 1946 by [[Biography:Hilbrand J. Groenewold|Hip Groenewold]].<ref name="groenewold">{{Cite journal | last1 = Groenewold | first1 = H. J. | title = On the principles of elementary quantum mechanics | doi = 10.1016/S0031-8914(46)80059-4 | journal = Physica | volume = 12 | issue = 7 | pages = 405–460 | year = 1946 |bibcode = 1946Phy....12..405G }}</ref>

==Overview==
The Moyal bracket is a way of describing the [[Commutator|commutator]] of observables in the phase space formulation of [[Physics:Quantum mechanics|quantum mechanics]] when these observables are described as functions on [[Physics:Phase space|phase space]].   It relies on schemes for identifying functions on phase space with quantum observables, the most famous of these schemes being the [[Physics:Wigner–Weyl transform|Wigner–Weyl transform]].    It underlies Moyal’s dynamical equation, an equivalent formulation of Heisenberg’s quantum equation of motion, thereby providing the quantum generalization of [[Physics:Hamiltonian mechanics|Hamilton’s equations]].

Mathematically, it is a [[Deformation theory|deformation]] of the phase-space [[Physics:Poisson bracket|Poisson bracket]] (essentially an [[Physics:Lie algebra extension|extension]] of it), the deformation parameter being the reduced [[Physics:Planck constant|Planck constant]] {{mvar|ħ}}. Thus, its [[Group contraction|group contraction]]  {{math|''ħ''→0}}  yields the [[Physics:Poisson bracket|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&ndash;van Hove theorem, which precludes such an isomorphism for the Poisson bracket, a question implicitly raised by Dirac in his 1926 doctoral thesis,<ref>P. A. M. Dirac (1926) Cambridge University Thesis [https://fsu.digital.flvc.org/islandora/object/fsu%3A641/datastream/OBJ/view/Dissertation_of_Paul_A__M__Dirac_for_Ph_D__degree.pdf "Quantum Mechanics"]</ref>  the "method of classical analogy" for quantization.<ref>P.A.M. Dirac, "The Principles of Quantum Mechanics" (''Clarendon Press Oxford'', 1958) {{isbn|978-0-19-852011-5}}</ref>

For instance, in a two-dimensional flat [[Physics:Phase space|phase space]], and for the [[Physics:Wigner–Weyl transform|Weyl-map correspondence]], the Moyal bracket reads,
: <math>\begin{align}
\{\{f,g\}\} & \stackrel{\mathrm{def}}{=}\  \frac{1}{i\hbar}(f\star  g-g\star  f) \\
            & = \{f,g\} + O(\hbar^2), \\
\end{align}</math>
where  <small>★</small> is the star-product operator in phase space (cf. [[Moyal product]]), while {{mvar|f}}  and {{mvar|g}} are differentiable phase-space functions, and {{math| {''f'', ''g''} }} is their Poisson bracket.<ref>Conversely, the Poisson bracket is formally expressible in terms of the  star product, {{math|''iħ''{''f'', ''g''} }}  = 2{{math|''f''  (log<small>★</small>) ''g''}}.</ref>

More specifically, in [[Operational calculus|operational calculus]] language, this equals
{{Equation box 1
|indent =:
|equation =  
<math>\{\{f,g\}\}\ =
\frac{2}{\hbar} ~ f(x,p)\  \sin \left ( {{\tfrac{\hbar }{2}}(\overleftarrow{\partial }_x
\overrightarrow{\partial }_{p}-\overleftarrow{\partial }_{p}\overrightarrow{\partial }_{x})} \right ) 
\  g(x,p).</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}
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<ref name="baker-1958">{{cite journal | last=Baker | first=George A. | title=Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space | journal=Physical Review | publisher=American Physical Society (APS) | volume=109 | issue=6 | date=1958-03-15 | issn=0031-899X | doi=10.1103/physrev.109.2198 | pages=2198–2206| bibcode=1958PhRv..109.2198B }}</ref> is

:<math>\{ \{ f,g \} \}(x,p) = {2 \over \hbar^3 \pi^2 } \int dp' \, dp'' \, dx' \, dx'' f(x+x',p+p') g(x+x'',p+p'')\sin \left( \tfrac{2}{\hbar} (x'p''-x''p')\right)~.</math>

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.<ref>[[Biography:Cosmas Zachos|C.Zachos]], [[Biography:David Fairlie|D. Fairlie]], and [[Biography:Thomas Curtright|T. Curtright]], "Quantum Mechanics in Phase Space" (''World Scientific'', Singapore, 2005) {{isbn|978-981-238-384-6}}.{{Cite journal | last1 = Curtright | first1 = T. L. | last2 = Zachos | first2 = C. K. | doi = 10.1142/S2251158X12000069 | title = Quantum Mechanics in Phase Space | journal = Asia Pacific Physics Newsletter | volume = 01 | pages = 37–46 | year = 2012 | arxiv = 1104.5269 | s2cid = 119230734 }}</ref>

The Moyal bracket specifies the eponymous infinite-dimensional 
[[Lie algebra]]&mdash;it is antisymmetric in its arguments {{mvar|f}}  and {{mvar|g}}, and satisfies the [[Jacobi identity]].
The corresponding abstract [[Lie algebra]] is realized by {{math|'' T<sub>f</sub> &equiv; f''}}<small>★</small>, so that 
: <math> [ T_f  ~, T_g ] = T_{i\hbar \{ \{ f,g \} \} }.  </math>
On a 2-torus phase space, {{math|''T'' <sup>2</sup>}}, with periodic 
coordinates {{mvar|x}} and {{mvar|p}}, each in {{math|[0,2''π'']}}, and integer mode indices {{math|''m<sub>i</sub>''}} , for basis functions {{math|exp(''i'' (''m''<sub>1</sub>''x''+''m''<sub>2</sub>''p''))}}, this Lie algebra reads,<ref>{{Cite journal | last1 = Fairlie | first1 = D. B. | last2 = Zachos | first2 = C. K. | doi = 10.1016/0370-2693(89)91057-5 | title = Infinite-dimensional algebras, sine brackets, and SU(∞) | journal = Physics Letters B | volume = 224 | pages = 101–107 | year = 1989 | issue = 1–2 |bibcode = 1989PhLB..224..101F | s2cid = 120159881 }}</ref>
: <math>[ T_{m_1,m_2} ~ , T_{n_1,n_2}  ] = 
2i \sin \left (\tfrac{\hbar}{2}(n_1 m_2 - n_2 m_1 )\right ) ~ T_{m_1+n_1,m_2+ n_2}, ~
</math>
which reduces to ''SU''(''N'') for integer {{math|''N''&nbsp;≡&nbsp;4''π/ħ''}}.
''SU''(''N'') then emerges as a deformation of ''SU''(∞), with deformation parameter&nbsp;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,<ref>{{cite journal | last1=Krivoruchenko | first1=M. I. | last2=Raduta | first2=A. A. | last3=Faessler | first3=Amand | title=Quantum deformation of the Dirac bracket | journal=Physical Review D | publisher=American Physical Society (APS) | volume=73 | issue=2 | date=2006-01-17 | issn=1550-7998 | doi=10.1103/physrevd.73.025008 | page=025008|arxiv=hep-th/0507049| bibcode=2006PhRvD..73b5008K | s2cid=119131374 }}</ref> which can be considered as a [[Deformation theory|quantum deformation]] of the [[Physics:Dirac bracket|Dirac bracket]].

==Sine bracket and cosine bracket==
Next to the sine bracket discussed,  Groenewold further introduced<ref name="groenewold"/> the cosine bracket,  elaborated by  Baker,<ref name="baker-1958"/><ref>See also the citation of Baker (1958) in: {{Cite journal | last1 = Curtright | first1 = T. | last2 = Fairlie | first2 = D. | last3 = Zachos | first3 = C. | title = Features of time-independent Wigner functions | doi = 10.1103/PhysRevD.58.025002 | journal = Physical Review D | volume = 58 | issue = 2 | year = 1998 | page = 025002 |arxiv = hep-th/9711183 |bibcode = 1998PhRvD..58b5002C | s2cid = 288935 }} [https://arxiv.org/abs/hep-th/9711183v3 arXiv:hep-th/9711183v3]</ref>
: <math>\begin{align}
\{ \{ \{f ,g\} \} \} & \stackrel{\mathrm{def}}{=}\  \tfrac{1}{2}(f\star  g+g\star  f)  = f g + O(\hbar^2). \\
\end{align}</math>
Here, again, <small>★</small> is the star-product operator in phase space, {{mvar|f}}  and {{mvar|g}} are differentiable phase-space functions, and {{math|''f'' ''g''}} 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 [[Physics:Wigner–Weyl transform|Wigner map]]  of the commutator, the cosine bracket is the Wigner image of the anticommutator in standard quantum mechanics. Similarly, as the Moyal bracket equals the Poisson bracket up to higher orders of {{mvar|ħ}}, the cosine bracket equals the ordinary product up to higher orders of {{mvar|ħ}}.  In the [[Physics:Classical limit|classical limit]], the Moyal bracket helps reduction to the [[Physics:Liouville's theorem (Hamiltonian)#Poisson bracket|Liouville equation (formulated in terms of the Poisson bracket)]], as the cosine bracket leads to the classical [[Physics:Hamilton–Jacobi equation|Hamilton–Jacobi equation]].<ref name="hiley-phase-space-description-2007">[[Biography:Basil Hiley|B. J. Hiley]]: Phase space descriptions of quantum phenomena, in: A. Khrennikov (ed.): ''Quantum Theory: Re-consideration of Foundations–2'', pp.&nbsp;267-286, Växjö University Press, Sweden, 2003 ([http://www.birkbeck.ac.uk/tpru/BasilHiley/ShadowPhaseVajxo03.pdf PDF])</ref>

The sine and cosine bracket also stand in relation to equations of [[Biography:Basil Hiley#Projections into shadow manifolds|a purely algebraic description]] of quantum mechanics.<ref name="hiley-phase-space-description-2007"/><ref name="brown-hiley-2004">M. R. Brown, B. J. Hiley: ''Schrodinger revisited: an algebraic approach'', [https://arxiv.org/abs/quant-ph/0005026 arXiv:quant-ph/0005026] (submitted 4 May 2000, version of 19 July 2004, retrieved June 3, 2011)</ref>

==References==
<references/>

[[Category:Mathematical quantization]]
[[Category:Symplectic geometry]]

{{Sourceattribution|Moyal bracket}}