C or a geometrical space for renormalization group flows : one step towards non-perturbative approximation methods

Generowicz, Jacek Marian (1998) C or a geometrical space for renormalization group flows : one step towards non-perturbative approximation methods. University of Southampton, Doctoral Thesis.

Abstract

The focal point of the exact renormalization group is contrasted to that of its perturbative relative, and the significance of renormalization group fixed points in the study of critical phenomena and continuum field theories is discussed.

The Polchinski formulation of the exact renormalization group is outlined and recast within the local potential approximation. Within this approximation a c-function, modeling the behaviour of the one introduced by Zamolodchikov, is constructed for general spatial dimension D > 2; this endows the space of theories with a geometry and allows for the description of renormalization group transformations as gradient flows. A promising method of finding approximations to fixed points, by looking for a stationary c-function in sub-manifolds of the full theory space, is presented.

Concentrating on O (N) symmetric scalar models, an extensive survey is performed of the local potential approximation values of the c-function and the critical exponents v and w, using both exact methods and the sub-manifold approximation method, for a wide variety of component numbers and spatial dimensions.

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