  For multi-time wave functions, which naturally arise as the relativistic
particle-position representation of the quantum state vector, the analog of the
Schr\"odinger equation consists of several equations, one for each time
variable. This leads to the question of how to prove the consistency of such a
system of PDEs. The question becomes more difficult for theories with particle
creation, as then different sectors of the wave function have different numbers
of time variables. Petrat and Tumulka (2014) gave an example of such a model
and a non-rigorous argument for its consistency. We give here a rigorous
version of the argument after introducing an ultraviolet cut-off into the
creation and annihilation terms of the multi-time evolution equations. These
equations form an infinite system of coupled PDEs; they are based on the Dirac
equation but are not fully relativistic (in part because of the cut-off). We
prove the existence and uniqueness of a smooth solution to this system for
every initial wave function from a certain class that corresponds to a dense
subspace in the appropriate Hilbert space.
