Nonlinear realization explained

ak 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 Liealgebra

ak g of G splits into the sum ak g = ak h ⊕ ak f of the Cartan subalgebra ak h of H and its supplement ak f , such that[ ak f, ak f]\subset ak h, [ ak f, ak h ]\subset ak f. (In physics, for instance, ak h amount to vector generators and ak f to axial ones.)

There exists an open neighborhood U of the unit of G suchthat any element

g\in U is uniquely brought into the formg = \exp(F)\exp(I), F\in ak f, I\in ak h.

Let

U _{G} be an open neighborhood of the unit of G such thatU ^{2 \subset} _{G} U , and let U _{0} be an open neighborhood of theH-invariant center \sigma _{0} of the quotient G/H which consists of elements\sigma = g\sigma _{0=\exp(F)\sigma} _{0,} 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 x} V given by the expressionsg\exp(F) = \exp(F')\exp(I'), g:(\exp(F)\sigma _{0,v)\to} (\exp(F')\sigma _{0,\exp(I')v).}

The corresponding nonlinear realization of a Lie algebra

ak g of G takes the following form.

Let

\{F _{\alpha\}} , \{I _{a\}} be the bases for ak f and ak h , respectively, together with the commutation relations[I _{a,I} _{b]=} c ^{d} _{ab} I _{d,} [F _{\alpha,F} _{\beta]=} c ^{d} _{\alpha\beta} I _{d,} [F _{\alpha,I} _{b]=} c ^{\beta} _{\alpha b} F _{\beta.}

Then a desired nonlinear realization of

ak g in ak f x V readsF _{\alpha:} (\sigma ^{\gamma} F _{\gamma,v)\to} (F ^{\gamma)F} _{\gamma,} F _{\alpha(v)),} I _{a:} (\sigma ^{\gamma} F _{\gamma,v)\to} (I ^{\gamma)F} _{\gamma,I} _{av),} ,F ^{\gamma) = \delta} _{\alpha(\sigma} + ^{\gamma} _{\alpha} (1)/(12) (c ^{\beta} _{\alpha\mu} c ^{\gamma} _{\beta\nu} - 3 c ^{b} _{\alpha\mu} c ^{\gamma} _{\nu b} )\sigma ^{\mu\sigma} ^{\nu,} I ^{\gamma) = c} _{a(\sigma} ^{\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 realizations of Lie superalgebras are considered.

See also

• Induced representation
• Chiral model

References

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