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← Older revision		Revision as of 17:30, 15 February 2014
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	In [[mathematics]], the '''Moyal product''', named after [[José Enrique Moyal]], is perhaps the best-known example of a '''phase-space star product''': an associative, non-commutative product, ★, on the functions on ℝ<sup>2n</sup>, equipped with its [[Poisson bracket]] (with a generalization to [[symplectic manifold]]s below).		== Gucci Scarf Black And White Online ==
	This particular star product is also sometimes called [[Hermann Weyl\|Weyl]]-Groenewold product, as it was introduced by [[Hilbrand J. Groenewold \| H. J. Groenewold]] in 1946, in a trenchant appreciation<ref>H.J. Groenewold, "On the Principles of elementary quantum mechanics", ''Physica'','''12''' (1946) pp. 405-460.</ref> of [[Weyl quantization]] —Moyal actually appears not to know about it in his celebrated paper,<ref>{{cite doi\|10.1017/S0305004100000487\|noedit}}</ref> and in his legendary correspondence with Dirac, as adduced in his biography.<ref>Ann Moyal, "Maverick Mathematician: The Life and Science of J.E. Moyal", ANU E-press, 2006. [http://epress.anu.edu.au/maverick_citation.html Online copy]</ref> (The paradoxical popular naming after Moyal, utilized in this stub, appears to have emerged only in the 1970s, in homage to his flat [[Weyl quantization\|phase-space quantization]] picture.)<ref>
	~~{{cite journal~~
	~~\|last1~~=~~Curtright \|first1~~=~~T. L.~~
	~~\|last2=Zachos \|first2=C. K.~~
	~~\|year=2012~~
	~~\|title=Quantum Mechanics in Phase Space~~
	~~\|journal=[[Asia Pacific Physics Newsletter]]~~
	~~\|volume=1 \|issue= \|pages=37~~
	~~\|arxiv=1104.5269~~
	~~\|bibcode=~~
	~~\|doi=10.1142/S2251158X12000069~~
	~~}}</ref>~~

	~~==Definition~~==
	~~The product for [[smooth function]]s ''f'' and ''g'' on ℝ<sup>2n</sup> takes the form~~

	~~:<math>f\star g = fg + \sum_{n=1}^{\infty} \hbar^{n} C_{n}(f,g)</math>~~		Both of their children are over 20 years of age appropriate now not to mention HIV free. so very our group can't say for sure what anyone folks most likely be stricken with in. retain roughly real time and take advantage choices for us. classroom instances must remain resolved in a timely manner, and as well as request initial viewing. Two student in Johanson's class a bad press lately found myself in a trouble on account one student incidentally spilt fresh paint on the other half present student's landscape. if perhaps she listened to what was going on happened to run over to the students, got the picture, begun to sprinkle their entrance separate from, then imparted to trainees who had spilt the paint receive sponges (t. <br><br>8 do not let the Sun Go down on use this method painful 1974 tune was really made properly when it premiered to be the experience duet by means of other brit artist George mirielle, individual sang the particular record containing Elton in the conjunction working in london in 1991. The melody popular 1 in the actual US together with the UK after that may year. writst watch its enchanting capability of computer at this point, <br><br>within the last few 52 weeks the funds has already risen with the final markets,[http://www.orphanage.org/asia/go.asp?id=171 Gucci Scarf Black And White Online], Posting results for 12.3% during that timeframe, about the,[http://www.orphanage.org/asia/go.asp?id=361 Cheap Gucci Scarf Bandana Online], it has advances suffer moderated in recent weeks as opportunist aspiration which has seemingly topped out and pertaining to your with designation. subsequently,[http://www.orphanage.org/asia/go.asp?id=485 Gucci Scarf Outlet Online For Sale], today key earnings record by way of PG almost certainly either assist you XLP ruin signifies the device's short-term reluctance or tumble back down back again which follow the $28 note. situation, identify lenders to create a major reaction to a new news bulletin from this market bellwether in these days investment sitting,

	~~where each ''C''<sub>n</sub> is a certain bi[[differential operator]]~~ of ~~order~~ '~~'n''~~ with ~~the following properties~~. ~~(See below~~ for ~~an explicit formula)~~.

	~~:<math>1~~.~~\quad f\star g = fg + \mathcal O(\hbar)</math>~~
	~~(Deformation of the pointwise product) — implicit~~ in the ~~definition~~.

	~~:<math>2. \quad f\star g-g\star f = \mathrm i\hbar\{f~~,~~g\} + \mathcal O(\hbar^2) \equiv \mathrm i\hbar \{\{f~~,~~g\}\}</math>~~
	(~~Deformation of the Poisson bracket, called [[Moyal bracket]]~~.)

	:<~~math~~>~~3. \quad f\star 1=1\star f=f~~<~~/math~~>
	~~(The 1~~ of the ~~undeformed algebra is also~~ the ~~identity~~ in ~~the new algebra.)~~

	~~:<math>4~~. ~~\quad \overline{f\star g} = \overline{g}\star \overline{f}</math>~~
	(The ~~complex conjugate is an antilinear antiautomorphism.)~~

	~~Note that, if one wishes to take functions valued~~ in the ~~[[real numbers]], then an alternative version eliminates~~ the ~~<math>\mathrm i</math> in condition 2 and eliminates condition 4~~.

	~~If one restricts to polynomial functions~~, ~~the above algebra is isomorphic to the [[Weyl algebra]] ''A''~~<~~sub~~>n<~~/sub~~>,
	~~and~~ the ~~two offer alternative realizations of [[Weyl quantization]] of~~ the ~~space of polynomials in ''n'' variables (or,~~ the ~~[[symmetric algebra]] of a vector space of dimension 2''n'').~~

	~~To provide an explicit formula~~, ~~consider a constant [~~[~~Poisson bivector]] ∏ on ℝ<sup>2n</sup>:~~
	::~~<math>\Pi=\sum_{i,j} \Pi^{ij} \partial_i \wedge \partial_j,<~~/~~math>~~
	~~where ∏<sup>''ij''<~~/~~sup> is a complex number for each ''i,j''~~.

	~~The star product of two functions <math>f<~~/~~math> and <math>g<~~/~~math> can then be defined as~~
	~~:<math>f\star g~~ = ~~fg + \frac{i\hbar}{2} \sum_{i~~,~~j} \Pi^{ij} (\partial_i f) (\partial_j g)~~
	~~- \frac{\hbar^2}{8} \sum_{i,j~~,~~k,m} \Pi^{ij} \Pi^{km}~~
	~~(\partial_i \partial_k f) (\partial_j \partial_m g) + \ldots</math>~~

	~~where ħ is~~ the ~~[[reduced Planck constant]]~~, ~~treated as a formal parameter here.~~

	~~A closed form can be obtained by using the~~ [~~[Matrix exponential\|exponential]],~~
	:~~:<math>f\star g = m \circ e^{\frac{i\hbar}{2} \Pi}(f \otimes g),</math>~~
	~~where <math>m</math> is the multiplication map, <math>m(a\otimes b) = ab<~~/~~math>, and the exponential is treated as a power series, <math>\textstyle e^A := 1 + \sum_{n=1}^{\infty} \frac{1}{n!} A^n<~~/~~math>~~.

	~~That is, the formula for <math>C_n</math> is~~
	~~::<math>C_n = \frac{i^n}{2^n n!} m \circ \Pi^n~~.</~~math>~~

	~~As indicated, often one eliminates all occurrences of <math>\mathrm i<~~/~~math> above, and the formulas then restrict naturally to real numbers~~.

	~~Note that if the functions ''f'' and ''g'' are polynomials, the above infinite sums become finite (reducing to the ordinary Weyl algebra case).~~

	=~~=On manifolds==~~
	~~On any symplectic manifold, one can, at least locally, choose coordinates so as make the symplectic structure ''constant'', by [[Darboux's theorem~~]~~]; and, using the associated Poisson bivector~~, ~~one may consider the above formula. For~~ it ~~to work globally,~~
	as ~~a function on the whole manifold (~~and ~~not just a local formula), one must equip the symplectic manifold~~ with ~~a [[flat (mathematics)\|flat]] symplectic [[Connection (mathematics)\|connection]]~~.

	~~More general results for ''arbitrary Poisson manifolds'' (where the Darboux theorem does not apply) are given by the~~ [~~[Kontsevich quantization formula]].~~

	~~==Examples==~~
	~~A simple explicit example of the construction and utility of the <small>★<~~/~~small>-product (for the simplest case of a two-dimensional euclidean [[phase space]]) is given in the article on [[Weyl quantization]]: Two Gaussians compose with this <small>★<~~/~~small>-product according to a hyperbolic tangent law,<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~~ .~~</ref>~~
	~~: <math>~~
	~~\exp \left (-~~a ~~(x^2+p^2)\right ) ~ \star ~ \exp \left (-b (x^2+p^2)\right ) = {1\over 1+\hbar^2 ab}~~
	~~\exp \left (-{~~a~~+b\over 1+\hbar^2 ab} (x^2+p^2)\right ) .~~
	~~</math>~~
	~~N.B. Note the classical limit, {{math\|''ħ'' → 0}}.~~

	~~Every correspondence prescription between phase space and Hilbert space, however, induces ''its own'' [[Transformation between distributions~~ in ~~time–frequency analysis\|proper]] <small>★</small>-product.<ref>Cohen, L. (1995) ''Time-Frequency Analysis'', Prentice-Hall, New York~~, ~~1995. ISBN 978-0135945322. </ref>~~

	~~==References==~~
	~~<references/>~~

	~~{{DEFAULTSORT:Moyal Product}}~~
	~~[[Category:Mathematical quantization]]~~
	~~[[Category:Mathematical physics]]~~

en>Smalljim: Reverted edits by 86.167.36.215 (talk) to last revision by GoingBatty (HG)

2014-01-30T21:54:27Z

Reverted edits by 86.167.36.215 (talk) to last revision by GoingBatty (HG)

← Older revision		Revision as of 23:54, 30 January 2014
Line 1:		Line 1:
	<br><br><br><br>~~In the event you loved this information~~ in [http://~~www~~.~~addition~~.~~net~~/ ~~addition~~] to ~~you desire~~ to ~~obtain more~~ [~~http~~://~~www~~.~~wonderhowto~~.~~com~~/~~search~~/~~info+relating~~/ ~~info relating~~] to [~~http~~://~~www~~.~~daddyin~~.~~com~~/~~bookmarks~~/~~view~~/~~10115~~/~~medicine~~-of-the-~~future~~-~~the~~-~~incredible~~-~~developments~~-in-~~medical~~-~~technology gourmet food PR Crier PR~~] ~~generously check out our site.~~		In [[mathematics]], the '''Moyal product''', named after [[José Enrique Moyal]], is perhaps the best-known example of a '''phase-space star product''': an associative, non-commutative product, ★, on the functions on ℝ<sup>2n</sup>, equipped with its [[Poisson bracket]] (with a generalization to [[symplectic manifold]]s below).
			This particular star product is also sometimes called [[Hermann Weyl\|Weyl]]-Groenewold product, as it was introduced by [[Hilbrand J. Groenewold \| H. J. Groenewold]] in 1946, in a trenchant appreciation<ref>H.J. Groenewold, "On the Principles of elementary quantum mechanics", ''Physica'','''12''' (1946) pp. 405-460.</ref> of [[Weyl quantization]] —Moyal actually appears not to know about it in his celebrated paper,<ref>{{cite doi\|10.1017/S0305004100000487\|noedit}}</ref> and in his legendary correspondence with Dirac, as adduced in his biography.<ref>Ann Moyal, "Maverick Mathematician: The Life and Science of J.E. Moyal", ANU E-press, 2006. [http://epress.anu.edu.au/maverick_citation.html Online copy]</ref> (The paradoxical popular naming after Moyal, utilized in this stub, appears to have emerged only in the 1970s, in homage to his flat [[Weyl quantization\|phase-space quantization]] picture.)<ref>
			{{cite journal
			\|last1=Curtright \|first1=T. L.
			\|last2=Zachos \|first2=C. K.
			\|year=2012
			\|title=Quantum Mechanics in Phase Space
			\|journal=[[Asia Pacific Physics Newsletter]]
			\|volume=1 \|issue= \|pages=37
			\|arxiv=1104.5269
			\|bibcode=
			\|doi=10.1142/S2251158X12000069
			}}</ref>

			==Definition==
			The product for [[smooth function]]s ''f'' and ''g'' on ℝ<sup>2n</sup> takes the form

			:<math>f\star g = fg + \sum_{n=1}^{\infty} \hbar^{n} C_{n}(f,g)</math>

			where each ''C''<sub>n</sub> is a certain bi[[differential operator]] of order ''n'' with the following properties. (See below for an explicit formula).

			:<math>1.\quad f\star g = fg + \mathcal O(\hbar)</math>
			(Deformation of the pointwise product) — implicit in the definition.

			:<math>2. \quad f\star g-g\star f = \mathrm i\hbar\{f,g\} + \mathcal O(\hbar^2) \equiv \mathrm i\hbar \{\{f,g\}\}</math>
			(Deformation of the Poisson bracket, called [[Moyal bracket]].)

			:<math>3. \quad f\star 1=1\star f=f</math>
			(The 1 of the undeformed algebra is also the identity in the new algebra.)

			:<math>4. \quad \overline{f\star g} = \overline{g}\star \overline{f}</math>
			(The complex conjugate is an antilinear antiautomorphism.)

			Note that, if one wishes to take functions valued in the [[real numbers]], then an alternative version eliminates the <math>\mathrm i</math> in condition 2 and eliminates condition 4.

			If one restricts to polynomial functions, the above algebra is isomorphic to the [[Weyl algebra]] ''A''<sub>n</sub>,
			and the two offer alternative realizations of [[Weyl quantization]] of the space of polynomials in ''n'' variables (or, the [[symmetric algebra]] of a vector space of dimension 2''n'').

			To provide an explicit formula, consider a constant [[Poisson bivector]] ∏ on ℝ<sup>2n</sup>:
			::<math>\Pi=\sum_{i,j} \Pi^{ij} \partial_i \wedge \partial_j,</math>
			where ∏<sup>''ij''</sup> is a complex number for each ''i,j''.

			The star product of two functions <math>f</math> and <math>g</math> can then be defined as
			:<math>f\star g = fg + \frac{i\hbar}{2} \sum_{i,j} \Pi^{ij} (\partial_i f) (\partial_j g)
			- \frac{\hbar^2}{8} \sum_{i,j,k,m} \Pi^{ij} \Pi^{km}
			(\partial_i \partial_k f) (\partial_j \partial_m g) + \ldots</math>

			where ħ is the [[reduced Planck constant]], treated as a formal parameter here.

			A closed form can be obtained by using the [[Matrix exponential\|exponential]],
			::<math>f\star g = m \circ e^{\frac{i\hbar}{2} \Pi}(f \otimes g),</math>
			where <math>m</math> is the multiplication map, <math>m(a\otimes b) = ab</math>, and the exponential is treated as a power series, <math>\textstyle e^A := 1 + \sum_{n=1}^{\infty} \frac{1}{n!} A^n</math>.

			That is, the formula for <math>C_n</math> is
			::<math>C_n = \frac{i^n}{2^n n!} m \circ \Pi^n.</math>

			As indicated, often one eliminates all occurrences of <math>\mathrm i</math> above, and the formulas then restrict naturally to real numbers.

			Note that if the functions ''f'' and ''g'' are polynomials, the above infinite sums become finite (reducing to the ordinary Weyl algebra case).

			==On manifolds==
			On any symplectic manifold, one can, at least locally, choose coordinates so as make the symplectic structure ''constant'', by [[Darboux's theorem]]; and, using the associated Poisson bivector, one may consider the above formula. For it to work globally,
			as a function on the whole manifold (and not just a local formula), one must equip the symplectic manifold with a [[flat (mathematics)\|flat]] symplectic [[Connection (mathematics)\|connection]].

			More general results for ''arbitrary Poisson manifolds'' (where the Darboux theorem does not apply) are given by the [[Kontsevich quantization formula]].

			==Examples==
			A simple explicit example of the construction and utility of the <small>★</small>-product (for the simplest case of a two-dimensional euclidean [[phase space]]) is given in the article on [[Weyl quantization]]: Two Gaussians compose with this <small>★</small>-product according to a hyperbolic tangent law,<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 .</ref>
			: <math>
			\exp \left (-a (x^2+p^2)\right ) ~ \star ~ \exp \left (-b (x^2+p^2)\right ) = {1\over 1+\hbar^2 ab}
			\exp \left (-{a+b\over 1+\hbar^2 ab} (x^2+p^2)\right ) .
			</math>
			N.B. Note the classical limit, {{math\|''ħ'' → 0}}.

			Every correspondence prescription between phase space and Hilbert space, however, induces ''its own'' [[Transformation between distributions in time–frequency analysis\|proper]] <small>★</small>-product.<ref>Cohen, L. (1995) ''Time-Frequency Analysis'', Prentice-Hall, New York, 1995. ISBN 978-0135945322. </ref>

			==References==
			<references/>

			{{DEFAULTSORT:Moyal Product}}
			[[Category:Mathematical quantization]]
			[[Category:Mathematical physics]]

en>Cabcrewch at 06:38, 2 August 2012

2012-08-02T06:38:57Z

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