Probability Conservation for Multi-time Integral Equations

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.