Are there classical relativistic multiparticle
theories?
While classical relativistic particles in external fields are a
well-studied subject and occur in many textbooks - e.g. in the book
''Special Relativity in General Frames'' by Eric Gourgoulhon -,
one finds very little literature about classical relativistic
multiparticle theories. For example, Eric Gourgoulhon covers the
state of the art in 1982 [in a book from 2013!] in just 5 papges out of
784 (in Section 11.5) and then drops the topic. (The explicit action
formulations he gives - in terms of the particles eigentimes - lead to
a dynamics given by integro-differential equations that do not admit a
well-defined Cauchy problem.)
This has a good reason. Indeed, three very natural conditions one would
expect such a theory to have are:
All elements of the Poincare group can be represented by canonical
transformations.
The number of space coordinates of all particles at a given time
make up one-half of all canonical variables.
The world lines are invariant in the sense that in a transformation
into another observer frame every point of a world line transforms
like the components of a Lorentz vector.
(The analogous conditions hold in classical nonrelativistic particle
dynamics.)
But there is a no-go theorem for classical multiparticle Hamiltonian
mechanics that shows that requiring all three conditions jointly is
inconsistent unless the dynamics is free. This is the content of the
paper
D.G. Currie, T.F. Jordan and E.C.G. Sudarshan,
Relativistic invariance and Hamiltonian theories of interacting
particles,
Rev. Mod. Phys. 35 (1963), 350-375.
Erratum Rev. Mod. Phys. 35 (1963), 1032.
Thus anyone interested in constructing a good relativistic particle
picture must make at least one somewhat unnatural choice.
Nobody has so far come up with a nice theory that would have found the
acceptance of the community of physicists. One of the better models,
that has found some practical use and is based on constrained
Hamiltonian mechanics, is described in
H. Sorge, H. Stöcker and W. Greiner,
Poincare invariant Hamiltonian dynamics: Modelling multi-hadronic
interactions in a phase space approach,
Ann. Phys. 192 (1989) 266-306.
This paper also gives some references to other approaches. A more recent
approach in the context of a multiparticle quantum mechanics can be
found in the online book
(Stefanovich tries to save the situation by reinterpreting the meaning
of Lorentz transformations and allowing for a limited form of
superluminal propagation.)
Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ