Nonlinear realization

In mathematical physics, nonlinear realization of a Lie group G possessing a Cartan subgroup H is a particular induced representation of G. In fact, it is a representation of a Lie algebra 𝔤 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 with spontaneous symmetry breaking, e.g., chiral models, chiral symmetry breaking, Goldstone boson theory, classical Higgs field theory, gauge gravitation theory and 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 𝔤 of G splits into the sum 𝔤=𝔥⊕𝔣 of the Cartan subalgebra 𝔥 of H and its supplement 𝔣, such that

[𝔣,𝔣]⊂𝔥,[𝔣,𝔥]⊂𝔣.

(In physics, for instance, 𝔥 amount to vector generators and 𝔣 to axial ones.)

There exists an open neighborhood U of the unit of G such that any element g∈U is uniquely brought into the form

g=exp⁡(F)exp⁡(I),F∈𝔣,I∈𝔥.

Let UG be an open neighborhood of the unit of G such that UG2⊂U, and let U0 be an open neighborhood of the H-invariant center σ0 of the quotient G/H which consists of elements

σ=gσ0=exp⁡(F)σ0,g∈UG.

Then there is a local section s(gσ0)=exp⁡(F) of G->G/H over U0.

With this local section, one can define the induced representation, called the nonlinear realization, of elements g∈UG⊂G on U0×V given by the expressions

gexp⁡(F)=exp⁡(F′)exp⁡(I′),g:(exp⁡(F)σ0,v)->(exp⁡(F′)σ0,exp⁡(I′)v).

The corresponding nonlinear realization of a Lie algebra 𝔤 of G takes the following form.

Let {Fα}, {Ia} be the bases for 𝔣 and 𝔥, respectively, together with the commutation relations

[Ia,Ib]=cabdId,[Fα,Fβ]=cαβdId,[Fα,Ib]=cαbβFβ.

Then a desired nonlinear realization of 𝔤 in 𝔣×V reads

Fα:(σγFγ,v)->(Fα(σγ)Fγ,Fα(v)),Ia:(σγFγ,v)->(Ia(σγ)Fγ,Iav),,

Fα(σγ)=δαγ+112(cαμβcβνγ−3cαμbcνbγ)σμσν,Ia(σγ)=caνγσν,

up to the second order in σα.

In physical models, the coefficients σα are treated as Goldstone fields. Similarly, nonlinear realizations of Lie superalgebras are considered.

## See also

* Induced representation
* Chiral model

## References

* Coleman, S.; Wess, J.; Zumino, Bruno (1969-01-25). "Structure of Phenomenological Lagrangians. I". Physical Review (American Physical Society (APS)) 177 (5): 2239–2247. doi:10.1103/physrev.177.2239. ISSN 0031-899X.
* Joseph, A.; Solomon, A. I. (1970). "Global and Infinitesimal Nonlinear Chiral Transformations". Journal of Mathematical Physics (AIP Publishing) 11 (3): 748–761. doi:10.1063/1.1665205. ISSN 0022-2488.
* Giachetta G., Mangiarotti L., Sardanashvily G., Advanced Classical Field Theory, World Scientific, 2009, ISBN 978-981-283-895-7.

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