|
|
| Line 1: |
Line 1: |
| {{Quantum mechanics|cTopic=Background}}
| | Hello, my title is Felicidad but I don't like when people use my full name. Bookkeeping is what I do for a living. What she loves doing is taking part in croquet and she is trying to make it a profession. Some time in the past I chose to live in Arizona but I need to move for my family members.<br><br>my site: [http://Bulls4Bears.com/blog/2014/08/23/auto-repair-tips-youll-wish-youd-read-sooner/ extended auto warranty] |
| In [[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 [[José Enrique Moyal]], but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute with [[Paul Dirac]].<ref>{{cite doi|10.1017/S0305004100000487|noedit}}</ref><ref>
| |
| {{cite web |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)|accessdate=2010-05-02 }}</ref> In the meantime this idea was independently introduced in 1946 by [[Hilbrand J. Groenewold|Hip Groenewold]].<ref name="groenewold">{{cite doi|10.1016/S0031-8914(46)80059-4|noedit}}</ref>
| |
| | |
| 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 [[Weyl quantization]]. It underlies [[phase space formulation#Time evolution|Moyal’s dynamical equation]], an equivalent formulation of [[Heisenberg equation|Heisenberg’s quantum equation of motion]], thereby providing the quantum generalization of [[Hamiltonian mechanics|Hamilton’s equations]].
| |
| | |
| Mathematically, it is a [[Deformation theory|deformation]] of the phase-space [[Poisson bracket]], 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: 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 [[phase space]], and for the Weyl-map correspondence (cf. [[Wigner-Weyl transform]]), 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 {''f'',''g''} is their Poisson bracket.
| |
| | |
| More specifically, this equals
| |
| | |
| {{Equation box 1
| |
| |indent =:
| |
| |equation =
| |
| <math>\{\{f,g\}\}\ =
| |
| \frac{2}{\hbar} ~ f(x,p)\ \sin \left ( {{\tfrac{\hbar }{2}}(\stackrel{\leftarrow }{\partial }_x
| |
| \stackrel{\rightarrow }{\partial }_{p}-\stackrel{\leftarrow }{\partial }_{p}\stackrel{\rightarrow }{\partial }_{x})} \right )
| |
| \ g(x,p).</math>
| |
| |cellpadding= 6
| |
| |border
| |
| |border colour = #0073CF
| |
| |background colour=#F9FFF7}}
| |
| 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">G. Baker, "Formulation of Quantum Mechanics Based on the Quasi-probability Distribution Induced on Phase Space," ''Physical Review'', '''109''' (1958) pp.2198–2206. {{doi|10.1103/PhysRev.109.2198}}</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>[[Cosmas Zachos|C.Zachos]], [[David Fairlie|D. Fairlie]], and [[Thomas Curtright|T. Curtright]], "Quantum Mechanics in Phase Space" (''World Scientific'', Singapore, 2005) ISBN 978-981-238-384-6 .{{cite doi|10.1142/S2251158X12000069|noedit}}</ref>
| |
| | |
| 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<sub>f</sub> ≡ f'' ★}} , 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 doi|10.1016/0370-2693(89)91057-5|noedit}}</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'' ≡ 4''π/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|second-class constraints]] involves an operation on equivalence classes of functions in phase space,<ref>M. I. Krivoruchenko, A. A. Raduta, Amand Faessler, [http://arxiv.org/abs/hep-th/0507049 Quantum deformation of the Dirac bracket], ''Phys. Rev.'' '''D73''' (2006) 025008.</ref> which might be considered as a [[Deformation theory|quantum deformation]] of the [[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 doi|10.1103/PhysRevD.58.025002|noedit}} [http://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 [[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 [[classical limit]], the Moyal bracket helps reduction to the [[Liouville's_theorem_(Hamiltonian)#Poisson_bracket|Liouville equation (formulated in terms of the Poisson bracket)]], as the cosine bracket leads to the classical [[Hamilton–Jacobi equation]].<ref name="hiley-phase-space-description-2007">[[Basil Hiley|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])</ref>
| |
| | |
| The sine and cosine bracket also stand in relation to equations of [[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'', [http://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:Quantum mechanics]]
| |
| [[Category:Mathematical quantization]]
| |
| [[Category:Symplectic geometry]]
| |
Hello, my title is Felicidad but I don't like when people use my full name. Bookkeeping is what I do for a living. What she loves doing is taking part in croquet and she is trying to make it a profession. Some time in the past I chose to live in Arizona but I need to move for my family members.
my site: extended auto warranty