Local conservation laws in nonlinear sigma models based on symmetric spaces
The formalism of a class of two-dimensional field theories known as nonlinear sigma models based on a symmetric space is reviewed, and the projective representation of such a symmetric space is used to find a natural geometric interpretation for the Riccati-like equations, and the consequent infinity of local conservation laws, for these models. The inverse scattering method, which has been used to great effect in the search for exact solutions to certain nonlinear partial differential equations in two variables is reviewed. These general methods are illustrated by applying them to the Korteweg-de Vries equation. After a short mathematical digression on symmetric spaces, the inverse scattering formalism is developed for nonlinear sigma models in which the fundamental field takes values in a symmetric space G/H, where G is the global invariance group of the model, and H is a subset of G is the hidden local invariance group. The isospectral pair of the inverse scattering method is interpreted as expressing the infinitesimal linear action of the group G on itself. On the other hand, the group G can be taken to act nonlinear on one of its associated symmetric spaces G/H. This nonlinear action is taken to be infinitesimal. A pair of Riccati-like equations is found. A natural geometric interpretation for the Riccati equations which in the literature appear ex nihilo is found.
- Research Organization:
- Yale Univ., New Haven, CT (USA)
- OSTI ID:
- 7097478
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
SIGMA MODEL
CONSERVATION LAWS
NONLINEAR PROBLEMS
INVERSE SCATTERING PROBLEM
KORTEWEG-DE VRIES EQUATION
MATHEMATICAL SPACE
SYMMETRY
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL MODELS
PARTIAL DIFFERENTIAL EQUATIONS
PARTICLE MODELS
SPACE
645400* - High Energy Physics- Field Theory