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
, such that (In physics, for instance,
amount to vector generators and
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
be an open neighborhood of the H-invariant center
of the quotient G/H which consists of elements Then there is a local section
With this local section, one can define the induced representation, called the nonlinear realization, of elements
given by the expressions The corresponding nonlinear realization of a Lie algebra
, respectively, together with the commutation relations Then a desired nonlinear realization of
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.