## ABSTRACT

In this chapter we shall again obtain the same set of the QCD Feynman rules as in the previous chapter. But now, our approach will be based on a formalism of a functional (or continual) integration. The paths integral idea introduced by R. Feynman [10] in quantum mechanics lies at the heart of this formalism (detailed presentation of this method could be found in the book [11]). Later on this method received wide recognition in statistical physics from which it was extended to quantum field theory. The functional integration method has a whole series of advantages over other quantization methods. It traces the linkage with classical dynamics. Besides, under such an approach we are operating with ordinary functions. This permits carrying out of any nonlinear transformations of fundamental dynamical variables in a more simple style and to directly see effects appearing under these transformations. The functional integration method is a general and self-contained quantization method that works for a wide class of systems including those with couplings. It easily deduces the Feynman rules and, at the same time, gives an opportunity for calculating nonperturbative Green functions. The advantage of the method is that it establishes the commonness of quantum field theory (QFT) with quantum statistical physics. This, in turn, permits the use of the quantum statistics technique in QFT. Formulation compactness of fundamental relations of QFT enables us to easily trace the structure of one or another model and its gauge invariance in particular. An analogue with Gauss integrals lies at the heart of the functional integration. In a

theory of the function of a single variable a Gauss integral has the form∫ ∞ −∞

dy√ pi exp (−ay2) = 1√

a , (2.1)

where a > 0. Let us consider a quadratic form of k variables

(y, Ay) ≡ k∑

yiAijyj ≥ 0. (2.2)

Diagonalizing this expression with the help of the orthogonal linear transformation, we can easily ascertain the validity of the following value of the k-dimensional Gauss integral:∫ ∞

−∞ dy1 . . .