# Nonlinear realization

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In mathematical physics, '''nonlinear realization''' of a [Lie group](/source/Lie_group) ''G''  possessing a [Cartan subgroup](/source/Cartan_subgroup)  ''H'' is a particular [induced representation](/source/induced_representation) of ''G''. In fact, it is a representation of a [Lie algebra](/source/Lie_algebra) <math>\mathfrak g</math> of ''G'' in a neighborhood of its origin.
A nonlinear realization, when restricted to the subgroup ''H'' reduces to a linear representation.

A nonlinear realization technique is part and parcel of many [field theories](/source/Field_theory_(physics)) with [spontaneous symmetry breaking](/source/spontaneous_symmetry_breaking), e.g., [chiral model](/source/chiral_model)s, [chiral symmetry breaking](/source/chiral_symmetry_breaking), [Goldstone boson](/source/Goldstone_boson) theory,  [classical Higgs field theory](/source/Higgs_field_(classical)), [gauge gravitation theory](/source/gauge_gravitation_theory) and [supergravity](/source/supergravity).

Let ''G'' be a Lie group and  ''H''  its Cartan subgroup which admits a linear representation in a vector space ''V''. A Lie
algebra <math>\mathfrak g</math> of ''G'' splits into the sum <math>\mathfrak g=\mathfrak h \oplus \mathfrak f</math> of the [Cartan subalgebra](/source/Cartan_subalgebra) <math>\mathfrak h</math>  of ''H'' and its supplement <math>\mathfrak f</math>, such that
:<math> [\mathfrak f,\mathfrak f]\subset \mathfrak h, \qquad [\mathfrak f,\mathfrak h
]\subset \mathfrak f. </math> 
(In physics, for instance, <math>\mathfrak h</math>  amount to vector generators and <math>\mathfrak f</math> to axial ones.)

There exists an open neighborhood  ''U'' of the unit of ''G''  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  ''G''  such that
<math>U_G^2\subset U</math>, and let <math>U_0</math> be an open neighborhood of the
''H''-invariant center <math>\sigma_0</math> of the quotient ''G/H''  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](/source/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 ''G'' 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 fields](/source/Goldstone_boson). Similarly, nonlinear realizations of [Lie superalgebra](/source/Lie_superalgebra)s  are considered.

== See also ==
*[Induced representation](/source/Induced_representation)
*[Chiral model](/source/Chiral_model)

== References ==
* {{cite journal | last1=Coleman | first1=S. | last2=Wess | first2=J. | last3=Zumino | first3=Bruno | title=Structure of Phenomenological Lagrangians. I | journal=Physical Review | publisher=American Physical Society (APS) | volume=177 | issue=5 | date=1969-01-25 | issn=0031-899X | doi=10.1103/physrev.177.2239 | pages=2239–2247| bibcode=1969PhRv..177.2239C }}
* {{cite journal | last1=Joseph | first1=A. | last2=Solomon | first2=A. I. | title=Global and Infinitesimal Nonlinear Chiral Transformations | journal=Journal of Mathematical Physics | publisher=AIP Publishing | volume=11 | issue=3 | year=1970 | issn=0022-2488 | doi=10.1063/1.1665205 | pages=748–761| bibcode=1970JMP....11..748J }}
* Giachetta G., Mangiarotti L., [Sardanashvily G.](/source/Gennadi_Sardanashvily), ''Advanced Classical Field Theory'', World Scientific, 2009, {{ISBN|978-981-283-895-7}}.

 
Category:Representation theory
Category:Theoretical physics

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