Changes of Variables and the Renormalization Group

Citation

Caticha, Ariel (1985) Changes of Variables and the Renormalization Group. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/KDVQ-GB21. https://resolver.caltech.edu/CaltechETD:etd-09222005-135843

Abstract

A class of exact infinitesimal renormalization group (RG) transformations is studied. These transformations are pure changes of variables (i.e., no integration or elimination of some degrees of freedom is required) such that a saddle point approximation is more accurate, becoming, in some cases, asymptotically exact as the transformations are iterated. The formalism provides a simplified and unified approach to several known renormalization groups. The RG equations for a scalar field theory are obtained and solved both by expanding in ε = d - 4 and also by expanding in a single coupling constant. This calculation yields results in agreement with conventional methods.

Next we study the application of this kind of RG to Yang-Mills theories. A simple exact gauge covariant RG transformation is constructed; the corresponding RG equations are obtained and solved in the weak coupling regime. This calculation shows that only certain initial conditions (i.e., bare actions) are compatible with the constraint that all the RG evolution be described by a single coupling constant. It also shows that at the tree level the β function for the SU(N) gauge theory is -[(21/6)((N/(4π)²)g³]. A one-loop calculation yields the usual result [-(22/6)((N/(4π)²)g³]. Unlike the scalar theory case the iteration of the RG transformation does not lead to an asymptotic situation in which the saddle-point approximation is exact. A lattice gauge theory is proposed for which the application of this RG formalism is straightforward.

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