Application of Functional Polynomials to Approximation of Matrix-Valued Functional Integrals

The matrix-valued functional integrals, generated by solutions of the Dirac equation are considered. These integrals are defined on the one-dimensional continuous path x : |s,t|→ ℝ and take values in the space of complex d × d matrices. Matrix-valued integrals are widely used in relativistic quantum mechanics for investigation of particle in electromagnetic field. Namely integrals are applied to represent the fundamental solution of the Cauchy problem for the Dirac equation. The method of approximate evaluation of matrix-valued integrals is proposed. This method is based on the expansion of functional in a series. Terms of a series have the form of a product of linear functionals with increasing total power. Taking a finite number of terms in the series and evaluating functional integrals of a product of linear functionals we obtain approximate value of the matrix-valued functional integral. Proposed method can be used for a wide class of integrals because the series converges for a large class of functionals. Application of the suggested method in the case of small and large parameters included in the integral is considered.

functional integrals, matrix-valued integrals, functional polynomials, approximation of integrals