A New Formulation and Regularization of Gauge Theories Using a Non-Linear Wavelet Expansion : Particles and Fields

概要

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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 A_μ(x) ≌ Σ__αc_αψ_μ^α(x), where ψ_μ^α(x) is a divergence-free wavelet and c_α 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 regular-ized by a cutoff regularization. We leave to later work relating these regularizations to the usual dimensional regularization.
理論物理学刊行会の論文
1995-12-25