Weyl formalism and quantum stochastic processes

Emilio Santos

We show that the Weyl-Wigner formalism in the Heisenberg picture may be used for the interpretation of experiments involving entangled photon pairs produced in nonlinear crystals via spontaneous parametric down-conversion. The calculations are usually no more involved than those with the Hilbert-space formalism. The WW formalism suggests a physical picture in terms of random variables and stochastic processes, but the picture is shown untenable in some instances.

John R. Klauder

Bernard Gaveau, Lawrence S. Schulman

The time evolution of a massless particle satisfying the Weyl equation is described as a stochastic process on a space of Grassmann variables, in close formal analogy with the use of Brownian motion for Schr\"odinger evolution. The Grassmann process is then combined with a Poisson process previously used for the evolution of Dirac electrons. (In that process electrons propagate as massless left- or right-handed particles with random changes in direction occurring at an average rate given by the particle mass.) Electron motion is thus given as an expectation over the two processes and spatial position supersymmetrically acquires contributions from sums of products of Grassmann variables.

Harish Parthasarathy

Emilio Santos

We characterize the electromagnetic vacuum as a stochastic field. Some consequences, like the particle behaviour of light, are studied. The stochastic approach is connected with the standard Hilbert space formalism via the Weyl transform. Several experiments involving spontaneous parametric down conversion are studied comparing Hilbert space and Weyl–Wigner formalisms. This allows an intuitive picture of entanglement to be obtained as a correlation between field fluctuations in distant places, involving the vacuum fields. The analysis shows that the Bell definition of local realism is not general enough, whence the reported violation of Bell inequalities does not refute local realism.

C. V. Heer