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
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
(In physics, for instance,
There exists an open neighborhood U of the unit of G such that any element
Let
Then there is a local section
With this local section, one can define the induced representation, called the nonlinear realization, of elements
The corresponding nonlinear realization of a Lie algebra
Let
Then a desired nonlinear realization of
up to the second order in
In physical models, the coefficients