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

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.

23 pages

References in corpus (3)

• Direct interaction along light cones at the quantum level
• Multi-Time Wave Functions versus Multiple Timelike Dimensions
• Interacting relativistic quantum dynamics of two particles on spacetimes with a Big Bang singularity

Cited by in corpus (2)

• Consistency Proof for Multi-Time Schrodinger Equations with Particle Creation and Ultraviolet Cut-Off
• Multi-Time Dynamics of the Dirac-Fock-Podolsky Model of QED