Moyal bracket

The Moyal bracket is a way of describing the commutator of observables in the phase space formulation of quantum mechanics 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. 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 of the phase-space Poisson bracket (essentially an extension of it), the deformation parameter being the reduced Planck constant {{mvar|ħ}}. Thus, its group contraction {{math|ħ→0}} 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,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"] the "method of classical analogy" for quantization.P.A.M. Dirac, "The Principles of Quantum Mechanics" (Clarendon Press Oxford, 1958) {{isbn|978-0-19-852011-5}}

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

: $\backslash begin\{align\}$

\{\{f,g\}\} & \stackrel{\mathrm{def}}{=}\ \frac{1}{i\hbar}(f\star g-g\star f) \\

& = \{f,g\} + O(\hbar^2), \\

\end{align}

where ★ 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.Conversely, the Poisson bracket is formally expressible in terms of the star product, {{math|iħ{f, g} }} = 2{{math|f (log★) g}}.

More specifically, in operational calculus language, this equals

{{Equation box 1

|indent =:

|equation =

$\backslash \{\backslash \{f,g\backslash \}\backslash \}\backslash \; =$

\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).

|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{{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 }} is

:$\backslash \{\; \backslash \{\; f,g\; \backslash \}\; \backslash \}(x,p)\; =\; \{2\; \backslash over\; \backslash hbar^3\; \backslash pi^2\; \}\; \backslash int\; dp\text{'}\; \backslash ,\; dp$ \, dx' \, dx f(x+x',p+p') g(x+x,p+p)\sin \left( \tfrac{2}{\hbar} (x'p-xp')\right)~.

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.C.Zachos, D. Fairlie, and 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 }}

The Moyal bracket specifies the eponymous infinite-dimensional

Lie algebra—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_f ≡ f}}★, so that

: $[\; T\_f\; ~,\; T\_g\; ]\; =\; T\_\{i\backslash hbar\; \backslash \{\; \backslash \{\; f,g\; \backslash \}\; \backslash \}\; \}.$

On a 2-torus phase space, {{math|T ²}}, with periodic

coordinates {{mvar|x}} and {{mvar|p}}, each in {{math|[0,2π]}}, and integer mode indices {{math|m_i}} , for basis functions {{math|exp(i (m₁x+m₂p))}}, this Lie algebra reads,{{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 }}

: $[\; T\_\{m\_1,m\_2\}\; ~\; ,\; T\_\{n\_1,n\_2\}\; ]\; =$

2i \sin \left (\tfrac{\hbar}{2}(n_1 m_2 - n_2 m_1 )\right ) ~ T_{m_1+n_1,m_2+ n_2}, ~

which reduces to SU(N) for integer {{math|N ≡ 4π/ħ}}.

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,{{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 }} which can be considered as a quantum deformation of the Dirac bracket.

Next to the sine bracket discussed, Groenewold further introduced the cosine bracket, elaborated by Baker,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]

: $\backslash begin\{align\}$

\{ \{ \{f ,g\} \} \} & \stackrel{\mathrm{def}}{=}\ \tfrac{1}{2}(f\star g+g\star f) = f g + O(\hbar^2). \\

\end{align}

Here, again, ★ 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 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 classical limit, 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.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 ([http://www.birkbeck.ac.uk/tpru/BasilHiley/ShadowPhaseVajxo03.pdf 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, [https://arxiv.org/abs/quant-ph/0005026 arXiv:quant-ph/0005026] (submitted 4 May 2000, version of 19 July 2004, retrieved June 3, 2011)

Category:Mathematical quantization

Category:Symplectic geometry