On Quantum Stochastic Differential Equations as Dirac Boundary-Value Problems

V. P. Belavkin

Nottingham Trent University

Abstract: 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.

UDC: 517

Received: 10.02.2000

DOI: 10.4213/mzm695

 English version: Mathematical Notes, 2001, 69:6, 735–748

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