Quantum stochastic Dirac boundary value problem, and the ultrarelativistic limit

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