en>Yobot: WP:CHECKWIKI error fixes using AWB (9494)

2013-09-26T05:16:39Z

WP:CHECKWIKI error fixes using AWB (9494)

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In mathematics, '''nonlinear realization''' of a [[Lie group]] <math>G</math> possessing a [[Cartan subgroup]] <math>H</math> is a particular [[induced representation]] of <math>G</math>. In fact it is a representation of a [[Lie algebra]] <math>\mathfrak g</math> of <math>G</math> in a neighborhood of its origin.

A nonlinear realization technique is part and parcel of many [[Field theory (physics)|field theories]] with [[spontaneous symmetry breaking]], e.g., [[nonlinear sigma model]], [[chiral symmetry breaking]], [[Goldstone boson]] theory, [[Higgs field (classical)|classical Higgs field theory]], [[gauge gravitation theory]] and [[supergravity]].

Let <math>G</math> be a Lie group and <math>H</math> its Cartan subgroup which admits a linear representation in a vector space <math>V</math>. A Lie
algebra <math>\mathfrak g</math> of <math>G</math> is split into the sum <math>\mathfrak g=\mathfrak h +\mathfrak f</math> of the [[Cartan subalgebra]] <math>\mathfrak h</math> of <math>H</math> and its supplement <math>\mathfrak f</math> so that

: <math> [\mathfrak f,\mathfrak f]\subset \mathfrak h, \qquad [\mathfrak f,\mathfrak h
]\subset \mathfrak f. </math>

There exists an open neighbourhood <math> U</math> of the unit of <math> G</math> such
that any element <math> g\in U</math> is uniquely brought into the form

: <math> g=\exp(F)\exp(I), \qquad F\in\mathfrak f, \qquad I\in\mathfrak h. </math>

Let <math>U_G</math> be an open neighborhood of the unit of <math>G</math> such that
<math>U_G^2\subset U</math>, and let <math>U_0</math> be an open neighborhood of the
<math>H</math>-invariant center <math>\sigma_0</math> of the quotient <math>G/H</math> which consists of elements

: <math>\sigma=g\sigma_0=\exp(F)\sigma_0, \qquad g\in U_G. </math>

Then there is a local section <math>s(g\sigma_0)=\exp(F) </math> of <math>G\to G/H</math>
over <math>U_0</math>. With this local section, one can define the [[induced representation]], called the '''nonlinear realization''', of elements <math>g\in U_G\subset G</math> on <math>U_0\times V</math> given by the expressions

: <math> g\exp(F)=\exp(F')\exp(I'), \qquad g:(\exp(F)\sigma_0,v)\to (\exp(F')\sigma_0,\exp(I')v). </math>

The corresponding nonlinear realization of a Lie algebra
<math>\mathfrak g</math> of <math>G</math> takes the following form.
Let <math>\{F_\alpha\}</math>, <math>\{I_a\}</math> be the bases for <math>\mathfrak f</math> and <math>\mathfrak h</math>, respectively, together with the commutation relations

: <math> [I_a,I_b]= c^d_{ab}I_d, \qquad [F_\alpha,F_\beta]= c^d_{\alpha\beta}I_d,
\qquad [F_\alpha,I_b]= c^\beta_{\alpha b}F_\beta. </math>

Then a desired nonlinear realization of <math>\mathfrak g</math> in <math>\mathfrak f\times V</math> reads

: <math>F_\alpha: (\sigma^\gamma F_\gamma,v)\to (F_\alpha(\sigma^\gamma)F_\gamma,
F_\alpha(v)), \qquad I_a: (\sigma^\gamma F_\gamma,v)\to (I_a(\sigma^\gamma)F_\gamma,I_av), </math>,

: <math>F_\alpha(\sigma^\gamma)=
\delta^\gamma_\alpha +
\frac{1}{12}(c^\beta_{\alpha\mu}c^\gamma_{\beta\nu} - 3 c^b_{\alpha\mu}c^\gamma_{\nu
b})\sigma^\mu\sigma^\nu, \qquad I_a(\sigma^\gamma)=c^\gamma_{a\nu}\sigma^\nu, </math>

up to the second order in <math>\sigma^\alpha</math>. In physical models, the coefficients <math>\sigma^\alpha</math> are treated as [[Goldstone boson|Goldstone fields]]. Similarly, nonlinear realization of [[Lie superalgebra]]s is comsidered.

== See also ==
*[[Induced representation]]

== References ==
* Coleman S., Wess J., Zumino B., Structure of phenomenological Lagrangians, I, II, ''Phys. Rev.'' '''177''' (1969) 2239.
* Joseph A., Solomon A., Global and infinitesimal nonlinear chiral transformations, ''J. Math. Phys.'' '''11''' (1970) 748.
* Giachetta G., Mangiarotti L., [[Gennadi Sardanashvily|Sardanashvily G.]], ''Advanced Classical Field Theory'', World Scientific, 2009, ISBN 978-981-283-895-7.

== External links ==
* [[Gennadi Sardanashvily|Sardanashvily G.]], Reduction of principal superbundles, Higgs superfields, and supermetric. Appendix: Nonlinear realization, [http://xxx.lanl.gov/abs/hep-th/0609070 arXiv: hep-th/0609070].

[[Category:Representation theory]]
[[Category:Theoretical physics]]

General covariant transformations - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (9494)