Relativistic Invariance and Hamiltonian Theories of Interacting Particles

Abstract

Relativistic invariance may involve two different theoretical postulates: symmetry of the theory under the relativistic transformation group reflecting the invariance of physical laws under changes of reference frame, and explicit transformation properties or manifest invariance of certain quantities. A Lie group formalism, for classical or quantum mechanics of particle variables or fields, is introduced and used to show how symmetry under the Lorentz (or Galilei) group is provided by ten generators satisfying the characteristic Lie bracket equations. For a classical mechanical theory of a fixed number of particles, the Lorentz transformatlon formula is assumed for the coordinates of the space-time events that comprise the world lines of the particles as defined by their positions as a function of time. This assumptlon of manifest invariance is expressed in terms of equations involving the Poisson brackets of the canonical position coordinates with the generators of the Lorentz group. For a theory of two particles it is shown that the only generators satisfying these latter equations plus the Poisson bracket equatlons characteristic of Lorentz symmetry are those descriptive of free particle motion; the combined assumption of Lorentz symmetry and Lorentz transformation of particle positions rule out any interaction. (auth)