In mathematics, nonlinear realization of a Lie group possessing a Cartan subgroup is a particular induced representation of . In fact it is a representation of a Lie algebra of in a neighborhood of its origin.
A nonlinear realization technique is part and parcel of many field theories with spontaneous symmetry breaking, e.g., nonlinear sigma model, chiral symmetry breaking, Goldstone boson theory, classical Higgs field theory, gauge gravitation theory and supergravity.
Let be a Lie group and its Cartan subgroup which admits a linear representation in a vector space . A Lie
algebra of is split into the sum of the Cartan subalgebra of and its supplement so that
There exists an open neighbourhood of the unit of such
that any element is uniquely brought into the form
Let be an open neighborhood of the unit of such that
, and let be an open neighborhood of the
-invariant center of the quotient 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 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 realization of Lie superalgebras is comsidered.
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- Giachetta G., Mangiarotti L., Sardanashvily G., Advanced Classical Field Theory, World Scientific, 2009, ISBN 978-981-283-895-7.