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Prog. Theor. Phys. Vol. 94 No. 6 (1995) pp. 1135-1146
A New Formulation and Regularization of Gauge Theories Using a Non-Linear Wavelet Expansion
Paul Federbush
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003
(Received June 19, 1995)
Abstract:
The Euclidean version of the Yang-Mills theory is studied in four dimensions. The field is expressed non-linearly in terms of the basic variables. The field is developed inductively, adding one excitation at a time. A given excitation is added into the “background field” of the excitations already added, the background field expressed in a radially axial gauge about the point where the excitation is centered. The linearization of the resultant expression for the field is an expansion Aµ(x) \cong\[∑α\]cαφµα(x), where φµα(x) is a divergence-free wavelet and cα is the associated basic variable, a Lie Algebra element of the gauge group. One is working in a particular gauge, regularization is simply cutoff regularization realized by omitting wavelet excitations below a certain length scale. We will prove in a later paper that only the usual gauge-invariant counterterms are required to renormalize perturbation theory.
Using related ideas, but essentially independent of the rest of the paper, we find an expression for the determinant of a gauged boson or fermion field in a fixed “small” external gauge field. This determinant is expressed in terms of explicitly gauge invariant quantities, and again may be regularized by a cutoff regularization.
We leave to later work relating these regularizations to the usual dimensional regularization.
URL :
http://ptp.ipap.jp/link?PTP/94/1135/
DOI : 10.1143/PTP.94.1135
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