Abstract
The quantum theory of Yang – Mills in four-dimensional space – time plays an important role in modern theoretical physics. Currently, this model contains many open problems, therefore, it is of great interest to mathematicians. This work consists of several parts, however, it only offers a new approach and, therefore, it is methodological. First of all, the diagram technique and the mathematical basis will be recalled in the first part. Then the process of renormalization will be explained. It is based on momentum cut-off regularization and described in [1] and [2]. However, this type of the regularization has several problems, as a result, only the first correction is calculated. After common constructions and observations, the first correction will be described in detail. Namely, the heat kernel will be considered since it plays a main role in this formalism. In particular, the method for calculating of coefficients of arbitrary order will be proposed.
Highlights
Introduction toYang – Mills theorySuppose G is a compact group of charges, G is the Lie algebra
The Killing form tr[·, ·] may be introduced on the Lie algebra with the normalization conditions tr[ta, tb] = C(N)δab, where N is the number of generators
In this case the problem of the renormalization can be formulated in the form of three conditions: Wreg(α(Λ), Λ, μ)
Summary
Suppose G is a compact group of charges, G is the Lie algebra. Assume that ta are generators of the Lie algebra and f abc are structural constants, such that the equalities [ta, tb] = f abctc are fulfilled. The Killing form tr[·, ·] may be introduced on the Lie algebra with the normalization conditions tr[ta, tb] = C(N)δab, where N is the number of generators. It is convenient to operate with space R4, where xμ , μ = 1, . The Yang – Mills field is a 1-form and has view in local coordinates. Based on the constructions (1) and (2), the classical action for the Yang – Mills theory has the form. Two differential operators can be introduced: M1 = ∇σ∇σδμν + 2[Fμν , · ],. Corrections Wn for n > 0 are related to the strongly connected vacuum corrections with n + 1 loops
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