Is there a multiparticle relativistic quantum mechanics?
In his QFT book, Weinberg says no, arguing that there is no way to
implement the cluster separation property. (See the details discussed
at
Physics Overflow.)
But in fact there is:
There is a big survey by Keister and Polyzou on the subject
The models are not field theories, only Poincare-invariant few-body dynamics with cluster decomposition and phenomenological terms which can be matched to approximate form factors from experiment or some field theory. (Actually many-body dynamics also works, but the many particle case is extremely messy.) They are useful phenomenological models, but somewhat limited; for example, it is not clear how to incorporate external fields.
Papers by Klink and work by Polyzou contain lots of multiparticle relativistic quantum mechanics, applied to real particles. See also the Ph.D. thesis by Krassnigg. (Other work in this direction includes Dirac's many-time quantum theory, with a separate time coordinate for each particle; see, e.g.,
Note that in the working single-time approaches, covariance is always achieved through a representation of the Poincare group on a Hilbert space corresponding to a fixed time (or another 3D manifold in space-time), rather than through multiple times. Thus the whole theory has a single time only, whose dynamics is generated by the Hamiltonian, the generator H=P_0 of the Poincare group. (This is completely analogous to the nonrelativistic case, where multiparticle systems also have a single time only.)
See also my comments on a book by Stefanovich featuring a multiparticle QED in the FAQ entry Are there classical relativistic multiparticle theories?.
The natural manifestly covariant picture is that of a vector bundle
on Minkowski space-time, with a standard Fock space attached to each
point. An observer (i.e., formally, an orthonormal frame attached at
some space-time point) moves in space-time via the Poincare group,
and this action extends to the bundle by means of the representation
defining the Fock space.
Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ