论文标题
量子幅度的整体变换
An integral transform for quantum amplitudes
论文作者
论文摘要
降低过渡幅度多维积分到分析形式或至少较少数量积分维度的中心障碍是存在坐标矢量差异的幅度（多项式的平方根）$ | x} _ {2} |^{2} = \ sqrt {x_ {1}^{2}^{2} -2x_ {1} x_ {2} \cosθ+x_ {2}^{2}^{2}^{2}}} $在函数的分离产品中。傅立叶通过为每种产品引入三维动量积分来绕过这一点，随后在许多情况下进行了另一套积分变换，以将所有结果的分母移至一个平方米的一个平方器中，其正方形是我的正方形。高斯变换为每种产品引入了一个一维积分，同时将坐标矢量差异的平方根与正方形的根部进行，并将它们移至指数。添加定理也可以用于此目的，有时甚至可以直接集成。每种方法都有其优点和劣势。替代对傅立叶变换的整体变换，并在此得出并使用高斯变换。麦克唐纳函数，超几何函数和具有复杂参数的梅耶尔G功能的许多随之而来的积分。
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) $|{\bf x}_{1}-{\bf x}_{2}|^{2}=\sqrt{x_{1}^{2}-2x_{1}x_{2}\cosθ+x_{2}^{2}}$ 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.
