Quantum Stochastics, Dirac Boundary Value Problem, and the Ultra Relativistic Limit

We prove that a single-jump quantum stochastic unitary evolution is equivalent to a Dirac boundary value problem on the half line in an extra dimension. This amounts to the equivalence of the quantum measurement boundary-value problem in infinite number particles space to the stochastic calculus in Fock space. It is shown that this exactly solvable model can be obtained from a Schroedinger boundary value problem for a positive relativistic Hamiltonian in the half-line as the inductive ultra relativistic limit, correspondent to the input flow of Dirac particles with asymptotically infinite momenta. Thus the stochastic limit can be interpreted in terms of quantum stochastic scheme for time-continuous non-demolition observation. The question of microscopic time reversibility is also studied for this paper.