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 ${\displaystyle {\mathfrak {g}}}$ 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 Lie algebra ${\displaystyle {\mathfrak {g}}}$ of G splits into the sum ${\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus {\mathfrak {f}}}$ of the Cartan subalgebra ${\displaystyle {\mathfrak {h}}}$ of H and its supplement ${\displaystyle {\mathfrak {f}}}$, such that

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