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 ${\displaystyle {\mathfrak {g}}}$ 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

Let G be a Lie group and H its Cartan subgroup which admits a linear representation in a vector space V. A Lie algebra ${\displaystyle {\mathfrak {g}}}$ of G splits into the sum ${\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {f}}}$ of the Cartan subalgebra ${\displaystyle {\mathfrak {h}}}$ of H and its supplement ${\displaystyle {\mathfrak {f}}}$, such that

${\displaystyle [{\mathfrak {f}},{\mathfrak {f}}]\subset {\mathfrak {h}},\qquad [{\mathfrak {f}},{\mathfrak {h}}]\subset {\mathfrak {f}}.}$

(In physics, for instance, ${\displaystyle {\mathfrak {h}}}$ amount to vector generators and ${\displaystyle {\mathfrak {f}}}$ to axial ones.)

There exists an open neighborhood U of the unit of G such that any element ${\displaystyle g\in U}$ is uniquely brought into the form

${\displaystyle g=\exp(F)\exp(I),\qquad F\in {\mathfrak {f}},\qquad I\in {\mathfrak {h}}.}$

Let ${\displaystyle U_{G}}$ be an open neighborhood of the unit of G such that ${\displaystyle U_{G}^{2}\subset U}$, and let ${\displaystyle U_{0}}$ be an open neighborhood of the H-invariant center ${\displaystyle \sigma _{0}}$ of the quotient G/H which consists of elements

${\displaystyle \sigma =g\sigma _{0}=\exp(F)\sigma _{0},\qquad g\in U_{G}.}$

Then there is a local section ${\displaystyle s(g\sigma _{0})=\exp(F)}$ of ${\displaystyle G\to G/H}$ over ${\displaystyle U_{0}}$.

With this local section, one can define the induced representation, called the nonlinear realization, of elements ${\displaystyle g\in U_{G}\subset G}$ on ${\displaystyle U_{0}\times V}$ given by the expressions

${\displaystyle g\exp(F)=\exp(F')\exp(I'),\qquad g:(\exp(F)\sigma _{0},v)\to (\exp(F')\sigma _{0},\exp(I')v).}$

The corresponding nonlinear realization of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ of G takes the following form.

Let ${\displaystyle \{F_{\alpha }\}}$, ${\displaystyle \{I_{a}\}}$ be the bases for ${\displaystyle {\mathfrak {f}}}$ and ${\displaystyle {\mathfrak {h}}}$, respectively, together with the commutation relations

${\displaystyle [I_{a},I_{b}]=c_{ab}^{d}I_{d},\qquad [F_{\alpha },F_{\beta }]=c_{\alpha \beta }^{d}I_{d},\qquad [F_{\alpha },I_{b}]=c_{\alpha b}^{\beta }F_{\beta }.}$

Then a desired nonlinear realization of ${\displaystyle {\mathfrak {g}}}$ in ${\displaystyle {\mathfrak {f}}\times V}$ reads

${\displaystyle F_{\alpha }:(\sigma ^{\gamma }F_{\gamma },v)\to (F_{\alpha }(\sigma ^{\gamma })F_{\gamma },F_{\alpha }(v)),\qquad I_{a}:(\sigma ^{\gamma }F_{\gamma },v)\to (I_{a}(\sigma ^{\gamma })F_{\gamma },I_{a}v),}$,

${\displaystyle F_{\alpha }(\sigma ^{\gamma })=\delta _{\alpha }^{\gamma }+{\frac {1}{12}}(c_{\alpha \mu }^{\beta }c_{\beta \nu }^{\gamma }-3c_{\alpha \mu }^{b}c_{\nu b}^{\gamma })\sigma ^{\mu }\sigma ^{\nu },\qquad I_{a}(\sigma ^{\gamma })=c_{a\nu }^{\gamma }\sigma ^{\nu },}$

up to the second order in ${\displaystyle \sigma ^{\alpha }}$.

In physical models, the coefficients ${\displaystyle \sigma ^{\alpha }}$ 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. 177 (5). American Physical Society (APS): 2239–2247.
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• Joseph, A.; Solomon, A. I. (1970). "Global and Infinitesimal Nonlinear Chiral Transformations". Journal of Mathematical Physics. 11 (3). AIP Publishing: 748–761.
  ISSN 0022-2488
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• Giachetta G., Mangiarotti L.,
  ISBN 978-981-283-895-7
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