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 formula_1 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 formula_1 of "G" splits into the sum formula_3 of the Cartan subalgebra formula_4 of "H" and its supplement formula_5, such that
There exists an open neighborhood "U" of the unit of "G" such
that any element formula_9 is uniquely brought into the form
Let formula_11 be an open neighborhood of the unit of "G" such that
formula_12, and let formula_13 be an open neighborhood of the
"H"-invariant center formula_14 of the quotient "G/H" which consists of elements
Then there is a local section formula_16 of formula_17
over formula_13. 
With this local section, one can define the induced representation, called the nonlinear realization, of elements formula_19 on formula_20 given by the expressions
The corresponding nonlinear realization of a Lie algebra
formula_1 of "G" takes the following form.
Let formula_23, formula_24 be the bases for formula_5 and formula_4, respectively, together with the commutation relations
Then a desired nonlinear realization of formula_1 in formula_29 reads
up to the second order in formula_32. 
In physical models, the coefficients formula_32 are treated as Goldstone fields. Similarly, nonlinear realizations of Lie superalgebras are considered.