Nonlinear realization

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

[\mathfrak f,\mathfrak f]\subset \mathfrak h, \qquad [\mathfrak f,\mathfrak h ]\subset \mathfrak f.

There exists an open neighbourhood U of the unit of G such that any element g\in U is uniquely brought into the form

g=\exp(F)\exp(I), \qquad F\in\mathfrak f, \qquad I\in\mathfrak h.

Let U_G be an open neighborhood of the unit of G such that U_G^2\subset U, and let U_0 be an open neighborhood of the H-invariant center \sigma_0 of the quotient G/H which consists of elements

\sigma=g\sigma_0=\exp(F)\sigma_0, \qquad g\in U_G.

Then there is a local section s(g\sigma_0)=\exp(F) of G\to G/H over U_0. With this local section, one can define the induced representation, called the nonlinear realization, of elements g\in U_G\subset G on U_0\times V given by the expressions

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 \mathfrak g of G takes the following form. Let \{F_\alpha\}, \{I_a\} be the bases for \mathfrak f and \mathfrak h, respectively, together with the commutation relations

[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.

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

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), ,
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,

up to the second order in \sigma^\alpha. In physical models, the coefficients \sigma^\alpha are treated as Goldstone fields. Similarly, nonlinear realization of Lie superalgebras is considered.

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