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

that covered everything known at that time. This survey is heavily cited; looking these up will bring you close to the state of the art on this.
They survey the construction of effective few-particle models. There are no singular interactions, hence there is no need for renormalization.

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.

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.

(Other work in this direction includes Dirac's many-time quantum theory, with a separate time coordinate for each particle; see, e.g.,

  • Marian Guenther, Phys Rev 94, 1347-1357 (1954)
  • and references there. Related multi-time work was done under the name of 'proper time quantum mechanics' or 'manifestly covariant quantum mechanics', see, e.g.,
  • L.P. Horwitz and C. Piron, Helv. Phys. Acta 48 (1973) 316
  • and later work by Horwitz, but it does not reproduce standard physics, and apparently never reached a stage useful to phenomenology.)

    Arnold Neumaier (
    A theoretical physics FAQ