A new class of Volterra-type integral equations from relativistic quantum physics

TL;DR

This paper introduces a new class of Volterra-type integral equations derived from relativistic quantum physics, analyzing their mathematical properties and proving existence and uniqueness of solutions for specific cases.

Contribution

It formulates novel integral equations with singular kernels and time delay in relativistic quantum mechanics, and establishes foundational mathematical results for these equations.

Findings

Existence and uniqueness of solutions proved

Mathematical features include singular kernels and time delay

Examples formulated for scalar wave functions

Abstract

Here we study a new kind of linear integral equations for a relativistic quantum-mechanical two-particle wave function $\psi(x_1,x_2)$, where $x_1,x_2$ are spacetime points. In the case of retarded interaction, these integral equations are of Volterra-type in the in the time variables, i.e., they involve a time integration from 0 to $t_i = x_i^0,~i=1,2$. They are interesting not only in view of their applications in physics, but also because of the following mathematical features: (a) time and space variables are more interrelated than in normal time-dependent problems, (b) the integral kernels are singular, and the structure of these singularities is non-trivial, (c) they feature time delay. We formulate a number of examples of such equations for scalar wave functions and prove existence and uniqueness of solutions for them. We also point out open mathematical problems.