Complex measures and amplitudes, generalized stochastic processes and their applications to quantum mechanics

Abstract

Complex measure theory is used to widen the scope of the study of stochastic processes and it is shown how, with such an extension, the physical concepts of superposition and diffraction follow automatically. The Dirac-Feynman path integral formalism is seen as a natural development. Several generic Markov processes are studied when extended to complex measures. The role of conditional expectations in this framework as propagating amplitudes is brought out with special reference to the Huyghens' principle. Diffusion is studied in this extended formalism and the context in which the Schrodinger or Dirac equation can be derived is stated. The Hamiltonian evolution and decay of correlations require complex measures which are boundary values of analytic functions.