  Relativistic quantum theories are usually thought of as being quantum field
theories, but this is not the only possibility. Here we consider relativistic
quantum theories with a fixed number of particles that interact neither through
potentials nor through exchange of bosons. Instead, the interaction can occur
directly along light cones, in a way similar to the Wheeler-Feynman formulation
of classical electrodynamics. For two particles, the wave function is here of
the form $\psi(x_1,x_2)$, where $x_1$ and $x_2$ are spacetime points.
Specifically, we consider a natural class of covariant equations governing the
time evolution of $\psi$ involving integration over light cones, or even more
general spacetime regions. It is not obvious, however, whether these equations
possess a unique solution for every initial datum. We prove for
Friedmann-Lemaitre-Robertson-Walker spacetimes (certain cosmological curved
spacetimes with a Big Bang singularity, i.e., with a beginning in time) that in
the case of purely retarded interactions there does, in fact, exist a unique
solution for every datum on the initial hypersurface. The proof is based on
carrying over similar results for a Minkowski half-space (i.e., the future of a
spacelike hyperplane) to curved spacetime. Furthermore, we show that also in
the case of time-symmetric interactions and for spacetimes with both a Big Bang
and a Big Crunch solutions do exist. However, initial data are then not
appropriate anymore; the solution space gets parametrized in a different way.
