• DOI:10.1103/PhysRevD.48.748
• Corpus ID: 16381314

Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology.

@article{Halliwell1993SumoverhistoriesOO,
  title={Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology.},
  author={Halliwell and Ort{\'i}z},
  journal={Physical review. D, Particles and fields},
  year={1993},
  volume={48 2},
  pages={
          748-768
        }
}

• Halliwell, Ortíz
• Published 4 November 1992
• Physics
• Physical review. D, Particles and fields

This paper is concerned with the question of the existence of composition laws in the sum-over-histories approach to relativistic quantum mechanics and quantum cosmology, and its connection with the existence of a canonical formulation. In nonrelativistic quantum mechanics, the propagator is represented by a sum over histories in which the paths move forward in time. The composition law of the propagator then follows from the fact that the paths intersect an intermediate surface of constant time once and only once, and a partition of the paths according to their crossing position may be affected. In relativistic quantum mechanics, by contrast, the propagators (or Green functions) may be represented by sums over histories in which the paths move backward and forward in time. They therefore intersect surfaces of constant time more than once, and the relativistic composition law, involving a normal derivative term, is not readily recovered. The principal technical aim of this paper is to show that the relativistic composition law may, in fact, be derived directly from a sum over histories by partitioning the paths according to their first crossing position of an intermediate surface. We review the various Green functions of the Klein-Gordon equation, and derive their composition laws. We obtain path-integral representations for all Green functions except the causal one. We use the proper time representation, in which the path integral has the form of a nonrelativistic sum over histories but is integrated over time.