On Quantum Stochastic Differential Equations as Dirac Boundary-Value Problems

We prove that a single-jump unitary quantum stochastic evolution is unitarily equivalent to the Dirac boundary-value problem on the half-line in an extended space. It is shown that this solvable model can be derived from the Schrödinger boundary-value problem for a positive relativistic Hamiltonian on the half-line as the inductive ultrarelativistic limit corresponding to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation can be reduced to a quantum mechanical boundary-value problem in the extended space. The problem of microscopic time reversibility is also discussed in the paper.