Lorentz-covariant quantization of nonlinear waves

Abstract

The quantum theory of a real scalar field in one space dimension with quartic coupling is studied as an example of a field theory possessing exact classical solutions representing extended distributions of energy and momentum. It is shown how formal canonical quantization can be implemented in practice to obtain self-consistent, Lorentz-covariant descriptions of restricted portions of Hilbert space. This is done by deriving dynamical and kinematical sum rules from the equations of motion and commutation relations, respectively. These sum rules are shown to have a variational basis, which guarantees the consistency of the approach. Overall, we derive Lorentz-covariant generalizations of results found previously by Goldstone and Jackiw using the same approach, as well as some new results.