An integral representation for quantum amplitudes

• Published in: Physica scripta Vol. 98; no. 10; pp. 105406 - 105416
• Main Author: Straton, Jack C
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.10.2023
• Subjects: integral representation; integral transform; integrals of hypergeometric functions; integrals of Macdonald functions; integrals of Meijer G-functions; quantum amplitudes
• ISSN: 0031-8949; 1402-4896
• DOI: 10.1088/1402-4896/acf531

Abstract The central impediment to reducing multidimensional integrals of transition amplitudes to analytic form, or at least to a fewer number of integral dimensions, is the presence of magnitudes of coordinate vector differences (square roots of polynomials) ∣ x 1 − x 2 ∣ = x 1 2 − 2 x 1 x 2 cos θ + x 2 2 in disjoint products of functions. Fourier transforms circumvent this by introducing a three-dimensional momentum integral for each of those products, followed in many cases by another set of integral representations to move all of the resulting denominators into a single quadratic form in one denominator whose square my be completed. Gaussian transforms introduce a one-dimensional integral for each such function while squaring the square roots of coordinate vector differences and moving them into a common exponential. Addition theorems may also be used for extracting the angular variables, and sometimes direct integration is even possible. Each method has its strengths and weaknesses. An integral representation is derived herein that stands as an alternative to these four approaches. A number of consequent integrals of Macdonald functions, hypergeometric functions, and Meijer G-functions with complicated arguments are given.