We define a class of Lorentz invariant Bohmian quantum models for N entangled but noninteracting Dirac particles. Lorentz invariance is achieved for these models through the incorporation of an additional dynamical space-time structure provided by a foliation of space-time. These models can be regarded as the extension of Bohm's model for N Dirac particles, corresponding to the foliation into the equal-time hyperplanes for a distinguished Lorentz frame, to more general foliations. As with Bohm's model, there exists for these models an equivariant measure on the leaves of the foliation. This makes possible a simple statistical analysis of position correlations analogous to the equilibrium analysis for (the nonrelativistic) Bohmian mechanics.Comment: 17 pages, 3 figures, RevTex. Completely revised versio

Abstract.The quantum probability ux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. The relation between these crossing probabilities and the usual formula for the scattering cross section is provided by the ux-acrosssurfaces theorem, which w as conjectured by Combes, Newton and Shtokhamer. 1 We p r o ve t h e u xacross-surfaces theorem for short range potentials and wave functions without energy cuto s. The proof is based on the free ux-across-surfaces theorem (Daumer et. al.), 2 and on smoothness properties of generalized eigenfunctions: It is shown that if the potential V (x) d e c a ys like jxj ; at in nity with > n 2 IN then the generalized eigenfunctions of the corresponding Hamiltonian ; 1 2 + V are n ; 2 times continuously di erentiable with respect to the momentum variable.

A theorem of Bourgain states that the harmonic measure for a domain in R d is supported on a set of Hausdorff dimension strictly less than d [2].