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Is there a multiparticle relativistic quantum mechanics?
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In his QFT book, Weinberg says no, arguing that there is no way to
implement the cluster separation property. But in fact there is:
There is a big survey by Keister and Polyzou on the subject
B.D. Keister and W.N. Polyzou,
Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics,
in: Advances in Nuclear Physics, Volume 20,
(J. W. Negele and E.W. Vogt, eds.)
Plenum Press 1991.
www.physics.uiowa.edu/~wpolyzou/papers/rev.pdf
that covered everything known at that time. This survey is heavily
cited; see
http://scholar.google.com/scholar?hl=en&as_sdt=2000&q=Keister+Polyzou
or
http://www.slac.stanford.edu/spires/find/hep?c=ANUPB,20,225
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.
The papers by Klink at
http://www.physics.uiowa.edu/~wklink/
and work by Polyzou at
http://www.physics.uiowa.edu/~wpolyzou/
contain lots of multiparticle relativistic quantum mechanics,
applied to real particles. See also the Ph.D. thesis by Krassnigg at
http://physik.uni-graz.at/~ank/dissertation-f.html
(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.)
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.)
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.