An integral transform for quantum amplitudes

TL;DR

This paper introduces a new integral transform to simplify multidimensional quantum amplitude integrals, reducing complexity and enabling the evaluation of complex special functions.

Contribution

A novel integral transform is derived, offering an alternative to Fourier and Gaussian transforms for simplifying quantum amplitude integrals.

Findings

Derived a new integral transform for quantum amplitudes.

Provided integrals involving Macdonald, hypergeometric, and Meijer G-functions.

Demonstrated the transform's potential to simplify complex multidimensional integrals.

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.