Geometric quantization - Revision history

en>TakuyaMurata: /* See also */

2015-01-10T13:44:23Z

See also

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	~~In [[theoretical physics]],~~ the ~~'''Bogoliubov transformation''', named after [[Nikolay Bogolyubov]]~~, is a [[unitary transformation]] {{Dubious\|date=July 2009}} from a [[unitary representation]] of some [[canonical commutation relation algebra]] or [[canonical anticommutation relation algebra]] into another unitary representation, ~~induced by an [[isomorphism]] of~~ the ~~commutation relation algebra~~. ~~The Bogoliubov transformation is often used~~ to ~~diagonalize [[Hamiltonian (quantum mechanics)\|Hamiltonian]]s, which yields~~ the ~~steady~~-~~state solutions of the corresponding [[Schrödinger equation]]. The solutions of [[BCS theory]] in a homogeneous system~~, ~~for example, are found using a Bogoliubov transformation. The Bogoliubov transformation is also important for understanding the~~ [[~~Unruh effect]], [[Hawking radiation]] and many other topics.~~		Luke is usually a celebrity during the creating as well as the career advancement initially next to his 3rd restaurant album, & , will be the resistant. He burst to the picture in 2002 together with his funny blend of down-home convenience, film celebrity luke bryan sold out [[http://lukebryantickets.omarfoundation.org http://lukebryantickets.omarfoundation.org]] very good looks and words, is defined t in a main way. The new recording Top around the nation graph and #2 in the take charts, generating it the second greatest first appearance during that time of 2006 to get a country artist. <br><br>The child of your , is aware perseverance and perseverance are important elements in terms of an effective job- . His first album, Keep Me, generated the Top hits “All My Girlfriends “Country and Say” Person,” [http://lukebryantickets.sgs-suparco.org luke bryan my kind of night tour] when his hard work, Doin’ Thing, discovered the vocalist-about three straight No. 3 single people: In addition Contacting Is a Fantastic Point.”<br><br>While in the drop of 2003, Concert tours: Luke Bryan & that had an amazing listing of , such as Urban. “It’s much like you’re getting a authorization to go to the next level, states all those artists that were a part of the Concert toursabove into a larger level of musicians.” It twisted as one of the best excursions in their twenty-year history.<br><br>my web blog ... [http://lukebryantickets.citizenswebcasting.com chip tickets]

	~~== Single bosonic mode example ==~~

	~~Consider the canonical [[Commutator\|commutation relation]] for [[bosonic]] [[creation and annihilation operators]] in the harmonic basis~~
	:~~<math>\left [ \hat{a}, \hat{a}^\dagger \right ] = 1<~~/~~math>~~

	~~Define a new pair of operators~~
	~~:<math>\hat{b} = u \hat{a} + v \hat{a}^\dagger</math>~~
	~~:<math>\hat{b}^\dagger = u^* \hat{a}^\dagger + v^* \hat{a}<~~/~~math>~~

	~~where the latter is the [[hermitian conjugate]] of the first~~. ~~The Bogoliubov transformation is a canonical transformation of these operators~~. ~~To find the conditions on the constants ''u'' and ''v'' such that the transformation remains canonical, the commutator is expanded, viz.~~

	:~~<math>\left [ \hat{b}, \hat{b}^\dagger \right ]~~
	~~= \left [ u \hat{a} + v \hat{a}^\dagger , u^* \hat{a}^\dagger + v^* \hat{a} \right ]~~
	~~= \cdots = \left ( \|u\|^2 - \|v\|^2 \right ) \left [ \hat{a}, \hat{a}^\dagger \right ]. <~~/~~math>~~

	~~It can be seen that <math>\,\|u\|^2 - \|v\|^2 = 1<~~/~~math> is the condition for which the transformation is canonical~~. ~~Since the form of this condition is reminiscent of the [[Hyperbolic function\|hyperbolic identity~~]] ~~<math>\cosh^2 x - \sinh^2 x = 1</math>, the constants ''u''~~ and ~~''v'' can be parametrized as~~

	~~:<math>u = e^{i \theta_1} \cosh r \,\!</math>~~
	~~:<math>v = e^{i \theta_2} \sinh r \~~,~~\! .</math>~~

	~~===Applications===~~
	~~The most prominent application~~ is ~~by [[Nikolai Bogoliubov]] himself~~ in ~~the context of [[superfluidity]]~~.~~<ref>[[Nikolai Bogoliubov]]: ''On~~ the ~~theory of superfluidity'', J. Phys. (USSR), 11, p. 23 (1947)</ref> Other applications comprise [[Hamiltonian (quantum mechanics)\|Hamiltonians]]~~ and ~~excitations~~ in the ~~theory~~ of ~~[[antiferromagnetism]]~~.<~~ref name="Kittel"~~>~~See e.g. the textbook by [[Charles Kittel]]: ''Quantum theory of solids'', New York, Wiley 1987.~~<~~/ref~~> ~~When calculating quantum field theory in curved space-times the definition~~ of ~~the vacuum changes and a Bogoliubov transformation between these different vacua is possible~~, ~~this~~ is ~~used~~ in ~~the derivation~~ of ~~[[Hawking radiation]]~~.

	~~== Fermionic mode ==~~

	~~For the [[Commutator\|anticommutation]] relation~~
	~~:<math>\left\{ \hat{a}~~, ~~\hat{a}^\dagger \right\} = 1</math>~~,
	the ~~same transformation with ''u''~~ and ~~''v'' becomes~~
	~~:<math>\left\{ \hat{b}~~, ~~\hat{b}^\dagger \right\} = (\|u\|^2 + \|v\|^2) \left\{ \hat{a}, \hat{a}^\dagger \right\}</math>~~

	~~To make the transformation canonical, ''u'' and ''v'' can be parameterized as~~
	:~~<math>u = e^{i \theta_1} \cos r \,\!<~~/~~math>~~
	~~:<math>v = e^{i \theta_2} \sin r \,\! .<~~/~~math>~~

	~~===Applications===~~
	~~The most prominent application is again by Nikolai Bogoliubov himself, this time for the [[BCS theory]] of [[superconductivity]] ~~.~~<ref name="Kittel" /> The point where the necessity to perform a Bogoliubov transform becomes obvious is that in mean~~-~~field approximation the Hamiltonian~~ of ~~the system can be written in both cases as a sum of bilinear terms in the original creation and destruction operators~~, ~~involving finite  <math>\~~,~~\langle a_i^+a_j^+\rangle</math>-terms, i.e. one must go beyond~~ the ~~usual [[Hartree-Fock method]] (-> [[Hartree-Fock~~-~~Bogoliubov method]])~~. ~~Also in [[nuclear physics]] this method is applicable since it may describe the "pairing energy" of nucleons in~~ a ~~heavy element~~.<~~ref~~>~~[[Vilen Mitrovanovich Strutinsky]]: ''Shell effects~~ in ~~nuclear physics and deformation energies''~~, ~~Nuclear Physics A, Vol. 95, p. 420-442 (1967), [http~~:~~//www.sciencedirect.com/science/article/pii/0375947467905106] .</ref>~~

	~~== Multimode example ==~~
	~~The [[Hilbert space]] under consideration is equipped with these operators, and henceforth describes a higher-dimensional [[quantum harmonic oscillator]] (usually~~ an ~~infinite-dimensional one).~~

	~~The [[ground state]]~~ of ~~the corresponding [[Hamiltonian (quantum mechanics)\|Hamiltonian]] is annihilated by all the annihilation operators:~~

	~~:<math>\forall i \qquad a_i \|0\rangle = 0</math>~~

	~~All excited states are obtained~~ as ~~[[linear combination]]s of the ground state excited by some creation operators:~~

	~~:<math>\prod_{k=1}^n a_{i_k}^\dagger \|0\rangle</math>~~

	~~One may redefine the creation and the annihilation operators by~~ a ~~linear redefinition:~~

	~~:<math>a'_i = \sum_j (u_{ij} a_j + v_{ij} a^\dagger_j)</math>~~

	~~where~~ the ~~coefficients <math>\~~,~~u_{ij},v_{ij}</math> must satisfy certain rules to guarantee~~ that the ~~annihilation operators and the creation operators <math>~~a~~^{\prime\dagger}_i</math>, defined by the [[Hermitian conjugate]] equation, have the same [[commutator]]s~~
	~~for bosons and anticommutators for fermions.~~

	~~The equation above defines the Bogoliubov transformation~~ of ~~the operators~~.

	~~The ground state annihilated by all <math>a'_{i}</math> is different from the original ground state <math>\|0\rangle</math> and they can be viewed~~ as ~~the Bogoliubov transformations~~ of ~~one another using~~ the ~~[[operator~~-~~state correspondence]]. They can also be defined as [[squeezed coherent state]]s. BCS wave function is an example of squeezed coherent state of fermions~~.<~~ref~~>~~Svozil, K. (1990), "Squeezed Fermion states", ''Phys. Rev. Lett.'' '''65''', 3341-3343. {{doi\|10.1103/PhysRevLett.65.3341}}~~<~~/ref~~>

	~~== References ==~~
	~~{{More footnotes\|date=February 2008}}~~

	~~{{reflist}}~~

	~~==Literature==~~
	~~The whole topic, and a lot of definite applications, are treated in the following textbooks:~~
	* J.-P. ~~Blaizot and G~~. ~~Ripka~~: ~~Quantum Theory of Finite Systems, MIT Press (1985)~~
	* A. ~~Fetter and J~~. ~~Walecka: Quantum Theory of Many-Particle Systems, Dover (2003)~~
	* Ch. Kittel: Quantum theory of solids, Wiley (1987)

	~~==External links==~~

	~~{{DEFAULTSORT:Bogoliubov Transformation}}~~
	~~[[Category:Quantum field theory]]~~
	~~[[Category:Canonical unitary transformation]~~]

en>Mathphysman at 00:15, 15 November 2013

2013-11-15T00:15:11Z

← Older revision		Revision as of 02:15, 15 November 2013
Line 1:		Line 1:
	The ~~title~~ of the ~~writer is Luther~~. ~~He presently lives~~ in ~~Idaho~~ and ~~his mothers~~ and ~~fathers reside nearby~~. ~~Interviewing~~ is ~~how he supports his family members but his promotion by no means comes~~. To ~~play croquet~~ is the ~~pastime I will by no means stop doing~~.<br><br>~~Feel free to visit my site~~ :: [~~http~~://~~infoguide~~.~~info~~/~~2779~~/~~try~~-it-~~for~~-~~yourself~~-~~auto~~-~~repair~~-~~suggestions~~ http://~~infoguide~~.~~info~~/]		In [[theoretical physics]], the '''Bogoliubov transformation''', named after [[Nikolay Bogolyubov]], is a [[unitary transformation]] {{Dubious\|date=July 2009}} from a [[unitary representation]] of some [[canonical commutation relation algebra]] or [[canonical anticommutation relation algebra]] into another unitary representation, induced by an [[isomorphism]] of the commutation relation algebra. The Bogoliubov transformation is often used to diagonalize [[Hamiltonian (quantum mechanics)\|Hamiltonian]]s, which yields the steady-state solutions of the corresponding [[Schrödinger equation]]. The solutions of [[BCS theory]] in a homogeneous system, for example, are found using a Bogoliubov transformation. The Bogoliubov transformation is also important for understanding the [[Unruh effect]], [[Hawking radiation]] and many other topics.

			== Single bosonic mode example ==

			Consider the canonical [[Commutator\|commutation relation]] for [[bosonic]] [[creation and annihilation operators]] in the harmonic basis
			:<math>\left [ \hat{a}, \hat{a}^\dagger \right ] = 1</math>

			Define a new pair of operators
			:<math>\hat{b} = u \hat{a} + v \hat{a}^\dagger</math>
			:<math>\hat{b}^\dagger = u^* \hat{a}^\dagger + v^* \hat{a}</math>

			where the latter is the [[hermitian conjugate]] of the first. The Bogoliubov transformation is a canonical transformation of these operators. To find the conditions on the constants ''u'' and ''v'' such that the transformation remains canonical, the commutator is expanded, viz.

			:<math>\left [ \hat{b}, \hat{b}^\dagger \right ]
			= \left [ u \hat{a} + v \hat{a}^\dagger , u^* \hat{a}^\dagger + v^* \hat{a} \right ]
			= \cdots = \left ( \|u\|^2 - \|v\|^2 \right ) \left [ \hat{a}, \hat{a}^\dagger \right ]. </math>

			It can be seen that <math>\,\|u\|^2 - \|v\|^2 = 1</math> is the condition for which the transformation is canonical. Since the form of this condition is reminiscent of the [[Hyperbolic function\|hyperbolic identity]] <math>\cosh^2 x - \sinh^2 x = 1</math>, the constants ''u'' and ''v'' can be parametrized as

			:<math>u = e^{i \theta_1} \cosh r \,\!</math>
			:<math>v = e^{i \theta_2} \sinh r \,\! .</math>

			===Applications===
			The most prominent application is by [[Nikolai Bogoliubov]] himself in the context of [[superfluidity]].<ref>[[Nikolai Bogoliubov]]: ''On the theory of superfluidity'', J. Phys. (USSR), 11, p. 23 (1947)</ref> Other applications comprise [[Hamiltonian (quantum mechanics)\|Hamiltonians]] and excitations in the theory of [[antiferromagnetism]].<ref name="Kittel">See e.g. the textbook by [[Charles Kittel]]: ''Quantum theory of solids'', New York, Wiley 1987.</ref> When calculating quantum field theory in curved space-times the definition of the vacuum changes and a Bogoliubov transformation between these different vacua is possible, this is used in the derivation of [[Hawking radiation]].

			== Fermionic mode ==

			For the [[Commutator\|anticommutation]] relation
			:<math>\left\{ \hat{a}, \hat{a}^\dagger \right\} = 1</math>,
			the same transformation with ''u'' and ''v'' becomes
			:<math>\left\{ \hat{b}, \hat{b}^\dagger \right\} = (\|u\|^2 + \|v\|^2) \left\{ \hat{a}, \hat{a}^\dagger \right\}</math>

			To make the transformation canonical, ''u'' and ''v'' can be parameterized as
			:<math>u = e^{i \theta_1} \cos r \,\!</math>
			:<math>v = e^{i \theta_2} \sin r \,\! .</math>

			===Applications===
			The most prominent application is again by Nikolai Bogoliubov himself, this time for the [[BCS theory]] of [[superconductivity]] .<ref name="Kittel" /> The point where the necessity to perform a Bogoliubov transform becomes obvious is that in mean-field approximation the Hamiltonian of the system can be written in both cases as a sum of bilinear terms in the original creation and destruction operators, involving finite  <math>\,\langle a_i^+a_j^+\rangle</math>-terms, i.e. one must go beyond the usual [[Hartree-Fock method]] (-> [[Hartree-Fock-Bogoliubov method]]). Also in [[nuclear physics]] this method is applicable since it may describe the "pairing energy" of nucleons in a heavy element.<ref>[[Vilen Mitrovanovich Strutinsky]]: ''Shell effects in nuclear physics and deformation energies'', Nuclear Physics A, Vol. 95, p. 420-442 (1967), [http://www.sciencedirect.com/science/article/pii/0375947467905106] .</ref>

			== Multimode example ==
			The [[Hilbert space]] under consideration is equipped with these operators, and henceforth describes a higher-dimensional [[quantum harmonic oscillator]] (usually an infinite-dimensional one).

			The [[ground state]] of the corresponding [[Hamiltonian (quantum mechanics)\|Hamiltonian]] is annihilated by all the annihilation operators:

			:<math>\forall i \qquad a_i \|0\rangle = 0</math>

			All excited states are obtained as [[linear combination]]s of the ground state excited by some creation operators:

			:<math>\prod_{k=1}^n a_{i_k}^\dagger \|0\rangle</math>

			One may redefine the creation and the annihilation operators by a linear redefinition:

			:<math>a'_i = \sum_j (u_{ij} a_j + v_{ij} a^\dagger_j)</math>

			where the coefficients <math>\,u_{ij},v_{ij}</math> must satisfy certain rules to guarantee that the annihilation operators and the creation operators <math>a^{\prime\dagger}_i</math>, defined by the [[Hermitian conjugate]] equation, have the same [[commutator]]s
			for bosons and anticommutators for fermions.

			The equation above defines the Bogoliubov transformation of the operators.

			The ground state annihilated by all <math>a'_{i}</math> is different from the original ground state <math>\|0\rangle</math> and they can be viewed as the Bogoliubov transformations of one another using the [[operator-state correspondence]]. They can also be defined as [[squeezed coherent state]]s. BCS wave function is an example of squeezed coherent state of fermions.<ref>Svozil, K. (1990), "Squeezed Fermion states", ''Phys. Rev. Lett.'' '''65''', 3341-3343. {{doi\|10.1103/PhysRevLett.65.3341}}</ref>

			== References ==
			{{More footnotes\|date=February 2008}}

			{{reflist}}

			==Literature==
			The whole topic, and a lot of definite applications, are treated in the following textbooks:
			* J.-P. Blaizot and G. Ripka: Quantum Theory of Finite Systems, MIT Press (1985)
			* A. Fetter and J. Walecka: Quantum Theory of Many-Particle Systems, Dover (2003)
			* Ch. Kittel: Quantum theory of solids, Wiley (1987)

			==External links==

			{{DEFAULTSORT:Bogoliubov Transformation}}
			[[Category:Quantum field theory]]
			[[Category:Canonical unitary transformation]]

en>Cuzkatzimhut: Clarification of where the Weyl map is unsatisfactory, and where it is quite satisfactory, indeed, a classic tool.

2012-06-07T15:46:00Z

Clarification of where the Weyl map is unsatisfactory, and where it is quite satisfactory, indeed, a classic tool.

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