  Dirac, Fock, and Podolsky [Ref. 1] devised a relativistic model in 1932 in
which a fixed number of $N$ Dirac electrons interact through a second-quantized
electromagnetic field. It is formulated with the help of a multi-time wave
function $\psi(t_1,\mathbf{x}_1,...,t_N,\mathbf{x}_N)$ that generalizes the
Schr\"odinger multi-particle wave function to allow for a manifestly
relativistic formulation of wave mechanics. The dynamics is given in terms of
$N$ evolution equations that have to be solved simultaneously. Integrability
imposes a rather strict constraint on the possible forms of interaction between
the $N$ particles and makes the rigorous construction of interacting dynamics a
long-standing problem, also present in the modern formulation of quantum field
theory. For a simplified version of the multi-time model, in our case
describing $N$ Dirac electrons that interact through a relativistic scalar
field, we prove well-posedness of the corresponding multi-time initial value
problem and discuss the mechanism and type of interaction between the charges.
For the sake of mathematical rigor we are forced to employ an ultraviolet
cut-off in the scalar field. Although this again breaks the desired
relativistic invariance, this violation occurs only on the arbitrary small but
finite length-scale of this cut-off. In view of recent progress in this field,
the main mathematical challenges faced in this work are, on the one hand, the
unboundedness from below of the free Dirac Hamiltonians and the unbounded,
time-dependent interaction terms, and on the other hand, the necessity of
pointwise control of the multi-time wave function.
