An integral transform for quantum amplitudes

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) $|{\bf x}_{1}-{\bf x}_{2}|^{2}=\sqrt{x_{1}^{2}-2x_{1}x_{2}\cos\theta+x_{2}^{2}}$ 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 transforms 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 product while squaring the square roots of coordinate vector differences and moving them into an exponential. Addition theorems may also be used for this purpose, and sometimes direct integration is even possible. Each method has its strengths and weaknesses. An alternative integral transform to Fourier transforms and Gaussian transforms is derived herein and utilized. A number of consequent integrals of Macdonald functions, hypergeometric functions, and Meijer G-functions with complicated arguments is given.