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

    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 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