# 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
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
of *G* splits into the sum
of the Cartan subalgebra
of *H* and its supplement
, such that

(In physics, for instance, amount to vector generators and to axial ones.)

There exists an open neighborhood *U* of the unit of *G* such
that any element
is uniquely brought into the form

Let
be an open neighborhood of the unit of *G* such that
, and let
be an open neighborhood of the
*H*-invariant center
of the quotient *G/H* which consists of elements

Then there is a local section of over .

With this local section, one can define the induced representation, called the **nonlinear realization**, of elements
on
given by the expressions

The corresponding nonlinear realization of a Lie algebra
of *G* takes the following form.

Let , be the bases for and , respectively, together with the commutation relations

Then a desired nonlinear realization of in reads

- ,

up to the second order in .

In physical models, the coefficients are treated as Goldstone fields. Similarly, nonlinear realizations of Lie superalgebras are considered.

## See also

## References

- Coleman S., Wess J., Zumino B., Structure of phenomenological Lagrangians, I, II,
*Phys. Rev.***177**(1969) 2239. - Joseph A., Solomon A., Global and infinitesimal nonlinear chiral transformations,
*J. Math. Phys.***11**(1970) 748. - Giachetta G., Mangiarotti L., Sardanashvily G.,
*Advanced Classical Field Theory*, World Scientific, 2009, ISBN 978-981-283-895-7.