Quantum phase transition - Revision history

203.110.246.25: /* Classical description */

2014-07-22T15:11:17Z

Classical description

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	~~==Spin(8)==~~
	~~Like all special orthogonal groups of~~ <~~math~~>~~n > 2~~<~~/math~~>, ~~SO(8) is not [[simply connected]]~~, ~~having a [[fundamental group]] [[isomorphic]] to [[cyclic group\|'''Z'''~~<~~sub~~>2<~~/sub~~>]]. ~~The [[universal cover]] of SO(8) is the [[spin group]] '''Spin(8)'''~~.

	~~==Center==~~
	~~The~~ [~~[center of a group\|center]] of SO(8) is '''Z'''<sub>2<~~/~~sub>, the diagonal matrices {±I} (as for all SO(2''n'') for 2''n'' > 2), while the center of Spin(8) is '''Z'''<sub>2</sub>×'''Z'''<sub>2<~~/~~sub> (as for all Spin(4''n''), 4''n'' > 0)~~.

	~~==Triality==~~
	~~{{main\|Triality}}~~
	~~[[Image:Dynkin diagram D4~~.~~png\|thumb\|[[Dynkin diagram]] of SO(8), (D<sub>4<~~/~~sub>)]]~~

	~~SO(8) is unique among the [[simple Lie group]]s in that its [[Dynkin diagram]] (shown right) (D<sub>4<~~/~~sub> under the Dynkin classification) possesses a three~~-~~fold [[symmetry]~~]. ~~This gives rise to peculiar feature of Spin(8) known as [[triality]]~~. Related to this is the fact that the two [[spinor]] [[group representation\|representations]], as well as the [[fundamental representation\|fundamental]] vector representation, of Spin(8) are all eight-dimensional (for all other spin groups the spinor representation is either smaller or larger than the vector representation). ~~The triality [[automorphism]] of Spin(8) lives in the [[outer automorphism group]] of Spin(8) which is isomorphic to the [[symmetric group]] S~~<~~sub~~>3<~~/sub~~> ~~that permutes these three representations~~. ~~The automorphism group acts on the center '''Z'''~~<~~sub~~>2<~~/sub~~> ~~x '''Z'''<sub>2</sub> (which also has automorphism group isomorphic to ''S''<sub>3</sub> which may also be considered as the [[general linear group]] over the finite field with two elements~~, ~~''S''<sub>3</sub> ≅GL(2~~,~~2))~~. ~~When one quotients Spin(8) by one central '''Z'''<sub>2</sub>~~, ~~breaking this symmetry and obtaining SO(8)~~, ~~the remaining [[outer automorphism group]] is only '''Z'''<sub>2</sub>~~. ~~The triality symmetry acts again on the further quotient SO(8)/'''Z'''~~<~~sub~~>2<~~/sub~~>.

	~~Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a [[semidirect product]]~~: ~~Aut(Spin(8)) ≅ Spin (8) ⋊ ''S''~~<~~sub~~>3<~~/sub~~>.

	~~==[[Root system]]==~~
	~~<math>(\pm 1~~,~~\pm 1~~,0,0)<~~/math~~>

	<~~math~~>~~(\pm 1~~,0,~~\pm 1~~,~~0)</math>~~

	~~<math>(\pm 1~~,0,0,~~\pm 1~~)~~</math>~~

	~~<math>~~(~~0,\pm 1,\pm 1,0)~~<~~/math~~>

	<~~math~~>~~(0,\pm 1,0,\pm 1)</math>~~

	~~<math>(0,0,\pm 1,\pm 1)~~<~~/math~~>

	~~==[[Weyl group]]==~~
	~~Its [[Weyl group\|Weyl]]/[[Coxeter group]] has 4!×8=192 elements.~~

	~~==[[Cartan matrix]]==~~
	<~~math~~>
	~~\begin{pmatrix}~~
	2 ~~& -1 & -1 & -1\\~~
	~~-1 &~~ ~~2 &~~ ~~0 &~~ ~~0\\~~
	~~-1 &~~ ~~0 &~~ ~~2 & 0\\~~
	~~-1 & 0 & 0 & 2~~
	~~\end{pmatrix}~~
	<~~/math~~>

	~~==See~~ also==
	* [~~[Octonions]]~~
	* [[Clifford algebra]]
	* [[G2 (mathematics)\|''G''<sub>2</~~sub>]]~~

	~~==References==~~
	*{{citation\|last=Adams\|first=J.F.~~\|authorlink=Frank Adams\|title=Lectures~~ on ~~exceptional Lie groups\|series=Chicago Lectures in Mathematics\|publisher= [[University of Chicago Press]]\|year= 1996\|isbn= 0-226-00526-7}}~~
	*{{citation\|last=Chevalley\|first=Claude\|authorlink=Claude Chevalley\|title=The algebraic theory of spinors and Clifford algebras\|series=Collected works\|volume=2\|publisher=Springer-Verlag\|year=1997\|isbn= 3-540-57063-2}} (originally published in 1954 by [[Columbia University Press]])
	*{{citation\|last=Porteous\|first= Ian R.~~\|authorlink=Ian R. Porteous\|title=Clifford algebras and the classical groups\|series=~~
	~~Cambridge Studies in Advanced Mathematics\|volume= 50\|publisher= [[Cambridge University Press]]\|year= 1995\|isbn= 0-521-55177-3}}~~

	~~[[Category:Lie groups]]~~

209.119.70.1: The text (and a relevant link) has been updated to reflect the fact that the "order" of a phase transition refers to the derivative of free energy which is discontinuous at the transition rather than to an order of approximation.

2013-07-15T16:29:31Z

The text (and a relevant link) has been updated to reflect the fact that the "order" of a phase transition refers to the derivative of free energy which is discontinuous at the transition rather than to an order of approximation.

← Older revision		Revision as of 18:29, 15 July 2013
Line 1:		Line 1:
			{{Group theory sidebar \|Topological}}

			In [[mathematics]], '''SO(8)''' is the [[special orthogonal group]] acting on eight-dimensional [[Euclidean space]]. It could be either a real or complex [[simple Lie group]] of rank 4 and dimension 28.

	~~Greetings~~. The ~~author~~'~~s title~~ is ~~Consuelo and she completely digs that name~~. ~~Years~~ in the ~~past we moved~~ to ~~Maryland~~. ~~One~~ of the ~~extremely best issues~~ in the ~~globe for me~~ is to ~~do ceramics and now I~~'~~m attempting~~ to ~~make cash~~ with it. ~~My working~~ [~~http:~~/~~/Www~~.~~squidoo.com~~/~~search~~/~~results?q=day+occupation day occupation~~] ~~is a manufacturing and preparing officer but the promotion never arrives. He~~'~~s been operating on his website for some time now~~. ~~Check it out right here: http:~~//~~www.redesigningthepla.net~~/~~wiki~~/~~The_Appeal_Of_Nya_Internet_Casinon_P%C3%A5_N%C3%A4tet~~<br><br>~~Feel free to visit my weblog~~ - [~~http:~~/~~/www~~.~~redesigningthepla~~.~~net/wiki/The_Appeal_Of_Nya_Internet_Casinon_P%C3%A5_N%C3%A4tet nya internet svenska casino~~]		==Spin(8)==
			Like all special orthogonal groups of <math>n > 2</math>, SO(8) is not [[simply connected]], having a [[fundamental group]] [[isomorphic]] to [[cyclic group\|'''Z'''<sub>2</sub>]]. The [[universal cover]] of SO(8) is the [[spin group]] '''Spin(8)'''.

			==Center==
			The [[center of a group\|center]] of SO(8) is '''Z'''<sub>2</sub>, the diagonal matrices {±I} (as for all SO(2''n'') for 2''n'' > 2), while the center of Spin(8) is '''Z'''<sub>2</sub>×'''Z'''<sub>2</sub> (as for all Spin(4''n''), 4''n'' > 0).

			==Triality==
			{{main\|Triality}}
			[[Image:Dynkin diagram D4.png\|thumb\|[[Dynkin diagram]] of SO(8), (D<sub>4</sub>)]]

			SO(8) is unique among the [[simple Lie group]]s in that its [[Dynkin diagram]] (shown right) (D<sub>4</sub> under the Dynkin classification) possesses a three-fold [[symmetry]]. This gives rise to peculiar feature of Spin(8) known as [[triality]]. Related to this is the fact that the two [[spinor]] [[group representation\|representations]], as well as the [[fundamental representation\|fundamental]] vector representation, of Spin(8) are all eight-dimensional (for all other spin groups the spinor representation is either smaller or larger than the vector representation). The triality [[automorphism]] of Spin(8) lives in the [[outer automorphism group]] of Spin(8) which is isomorphic to the [[symmetric group]] S<sub>3</sub> that permutes these three representations. The automorphism group acts on the center '''Z'''<sub>2</sub> x '''Z'''<sub>2</sub> (which also has automorphism group isomorphic to ''S''<sub>3</sub> which may also be considered as the [[general linear group]] over the finite field with two elements, ''S''<sub>3</sub> ≅GL(2,2)). When one quotients Spin(8) by one central '''Z'''<sub>2</sub>, breaking this symmetry and obtaining SO(8), the remaining [[outer automorphism group]] is only '''Z'''<sub>2</sub>. The triality symmetry acts again on the further quotient SO(8)/'''Z'''<sub>2</sub>.

			Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a [[semidirect product]]: Aut(Spin(8)) ≅ Spin (8) ⋊ ''S''<sub>3</sub>.

			==[[Root system]]==
			<math>(\pm 1,\pm 1,0,0)</math>

			<math>(\pm 1,0,\pm 1,0)</math>

			<math>(\pm 1,0,0,\pm 1)</math>

			<math>(0,\pm 1,\pm 1,0)</math>

			<math>(0,\pm 1,0,\pm 1)</math>

			<math>(0,0,\pm 1,\pm 1)</math>

			==[[Weyl group]]==
			Its [[Weyl group\|Weyl]]/[[Coxeter group]] has 4!×8=192 elements.

			==[[Cartan matrix]]==
			<math>
			\begin{pmatrix}
			2 & -1 & -1 & -1\\
			-1 & 2 & 0 & 0\\
			-1 & 0 & 2 & 0\\
			-1 & 0 & 0 & 2
			\end{pmatrix}
			</math>

			==See also==
			* [[Octonions]]
			* [[Clifford algebra]]
			* [[G2 (mathematics)\|''G''<sub>2</sub>]]

			==References==
			*{{citation\|last=Adams\|first=J.F.\|authorlink=Frank Adams\|title=Lectures on exceptional Lie groups\|series=Chicago Lectures in Mathematics\|publisher= [[University of Chicago Press]]\|year= 1996\|isbn= 0-226-00526-7}}
			*{{citation\|last=Chevalley\|first=Claude\|authorlink=Claude Chevalley\|title=The algebraic theory of spinors and Clifford algebras\|series=Collected works\|volume=2\|publisher=Springer-Verlag\|year=1997\|isbn= 3-540-57063-2}} (originally published in 1954 by [[Columbia University Press]])
			*{{citation\|last=Porteous\|first= Ian R.\|authorlink=Ian R. Porteous\|title=Clifford algebras and the classical groups\|series=
			Cambridge Studies in Advanced Mathematics\|volume= 50\|publisher= [[Cambridge University Press]]\|year= 1995\|isbn= 0-521-55177-3}}

			[[Category:Lie groups]]

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2012-05-12T00:28:14Z

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