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 $\backslash mathfrak\; g$ 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 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 $\backslash mathfrak\; g$ of G splits into the sum $\backslash mathfrak\; g=\backslash mathfrak\; h\; \backslash oplus\; \backslash mathfrak\; f$ of the Cartan subalgebra $\backslash mathfrak\; h$ of H and its supplement $\backslash mathfrak\; f$, such that

:$[\backslash mathfrak\; f,\backslash mathfrak\; f]\backslash subset\; \backslash mathfrak\; h,\; \backslash qquad\; [\backslash mathfrak\; f,\backslash mathfrak\; h$

]\subset \mathfrak f.

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

There exists an open neighborhood U of the unit of G such

that any element $g\backslash in\; U$ is uniquely brought into the form

:$g=\backslash exp(F)\backslash exp(I),\; \backslash qquad\; F\backslash in\backslash mathfrak\; f,\; \backslash qquad\; I\backslash in\backslash mathfrak\; h.$

Let $U\_G$ be an open neighborhood of the unit of G such that

$U\_G^2\backslash subset\; U$, and let $U\_0$ be an open neighborhood of the

H-invariant center $\backslash sigma\_0$ of the quotient G/H which consists of elements

:$\backslash sigma=g\backslash sigma\_0=\backslash exp(F)\backslash sigma\_0,\; \backslash qquad\; g\backslash in\; U\_G.$

Then there is a local section $s(g\backslash sigma\_0)=\backslash exp(F)$ of $G\backslash to\; G/H$

over $U\_0$.

With this local section, one can define the induced representation, called the nonlinear realization, of elements $g\backslash in\; U\_G\backslash subset\; G$ on $U\_0\backslash times\; V$ given by the expressions

:$g\backslash exp(F)=\backslash exp(F\text{'})\backslash exp(I\text{'}),\; \backslash qquad\; g:(\backslash exp(F)\backslash sigma\_0,v)\backslash to\; (\backslash exp(F\text{'})\backslash sigma\_0,\backslash exp(I\text{'})v).$

The corresponding nonlinear realization of a Lie algebra

$\backslash mathfrak\; g$ of G takes the following form.

Let $\backslash \{F\_\backslash alpha\backslash \}$, $\backslash \{I\_a\backslash \}$ be the bases for $\backslash mathfrak\; f$ and $\backslash mathfrak\; h$, respectively, together with the commutation relations

:$[I\_a,I\_b]=\; c^d\_\{ab\}I\_d,\; \backslash qquad\; [F\_\backslash alpha,F\_\backslash beta]=\; c^d\_\{\backslash alpha\backslash beta\}I\_d,$

\qquad [F_\alpha,I_b]= c^\beta_{\alpha b}F_\beta.

Then a desired nonlinear realization of $\backslash mathfrak\; g$ in $\backslash mathfrak\; f\backslash times\; V$ reads

:$F\_\backslash alpha:\; (\backslash sigma^\backslash gamma\; F\_\backslash gamma,v)\backslash to\; (F\_\backslash alpha(\backslash sigma^\backslash gamma)F\_\backslash gamma,$

F_\alpha(v)), \qquad I_a: (\sigma^\gamma F_\gamma,v)\to (I_a(\sigma^\gamma)F_\gamma,I_av), ,

:$F\_\backslash alpha(\backslash sigma^\backslash gamma)=$

\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,

up to the second order in $\backslash sigma^\backslash alpha$.

In physical models, the coefficients $\backslash sigma^\backslash alpha$ are treated as Goldstone fields. Similarly, nonlinear realizations of Lie superalgebras are considered.