Nonlinear realization

Template:No footnotes In mathematics, nonlinear realization of a Lie group ${\displaystyle G}$ possessing a Cartan subgroup ${\displaystyle H}$ is a particular induced representation of ${\displaystyle G}$. In fact it is a representation of a Lie algebra ${\displaystyle {\mathfrak {g}}}$ of ${\displaystyle G}$ in a neighborhood of its origin.

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

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

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

There exists an open neighbourhood ${\displaystyle U}$ of the unit of ${\displaystyle G}$ 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 ${\displaystyle G}$ such that ${\displaystyle U_{G}^{2}\subset U}$, and let ${\displaystyle U_{0}}$ be an open neighborhood of the ${\displaystyle H}$-invariant center ${\displaystyle \sigma _{0}}$ of the quotient ${\displaystyle G/H}$ 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 ${\displaystyle G}$ 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 realization of Lie superalgebras is comsidered.