Fourier analysis on the Lorentz group and relativistic quantum mechanics

Abstract

The non-relativistic Schrödinger and Lippman-Schwinger equations are described. The expressions of these equations are investigated in momentum and configuration spaces, using Fourier transformation. The plane wave, which is generating function for the matrix elements of three dimensional Euclidean group in spherical basis, expanded in terms of Legendre polynomials and spherical Bessel functions. Also explicit calculation of Green.s function is done.The matrix elements of the unitary irreducible representations of Lorentz group are used to introduce Fourier expansion of plane waves. And the kernel of Gelfand-Graev transformation, which is the relativistic plane wave, is expanded in to these matrix elements. Then relativistic differential difference equation in configuration space is constructed.Lippman-Schwinger equations are studied in Lobachevsky space (hyperbolic space). An analogous to the non-relativistic case, using the finite difference Schrödinger equation, one dimensional Green.s function is analyzed for the relativistic case . Also the finite difference analogue of the Heavyside step function is investigated.