Global Existence of Bell's Time-Inhomogeneous Jump Process for Lattice Quantum Field Theory

We consider the time-inhomogeneous Markovian jump process introduced by John S. Bell [Phys.Rep. 137, 49] for a lattice quantum field theory, which runs on the associated configuration space. Its jump rates, tailored to give the process the quantum distribution $\mathrm{\mid }{\mathrm{\Psi }}_t{\mathrm{\mid }}^2$ at all times $t$, typically exhibit singularities. We establish the existence of a unique such process for all times, under suitable assumptions on the Hamiltonian or the initial state vector ${\mathrm{\Psi }}_0$. The proof of non-explosion takes advantage of the special role of the $\mathrm{\mid }{\mathrm{\Psi }}_t{\mathrm{\mid }}^2$ distribution.