Probability conservation for multi-time integral equations

TL;DR

This paper demonstrates probability conservation in certain relativistic multi-time integral equations, ensuring consistency with quantum theory's probabilistic interpretation, through special classes of retarded interactions and asymptotic analysis.

Contribution

It establishes probability conservation for specific classes of multi-time integral equations with retarded interactions in relativistic quantum theory.

Findings

Global probability is conserved for equations with retarded light-cone interactions.

Asymptotic probability conservation holds for more general interaction kernels.

A local conservation law is derived from the global conservation results.

Abstract

In relativistic quantum theory, one sometimes considers integral equations for a wave function $\psi(x_1,x_2)$ depending on two space-time points for two particles. A serious issue with such equations is that, typically, the spatial integral over $|\psi|^2$ is not conserved in time -- which conflicts with the basic probabilistic interpretation of quantum theory. However, here it is shown that for a special class of integral equations with retarded interactions along light cones, the global probability integral is, indeed, conserved on all Cauchy surfaces. For another class of integral equations with more general interaction kernels, asymptotic probability conservation from $t=-\infty$ to $t=+\infty$ is shown to hold true. Moreover, a certain local conservation law is deduced from the first result.