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arXiv:quant-ph/0512206

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

V. P. Belavkin

Published Dec 22, 2005• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0512206

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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.
We prove that a single-jump quantum stochastic unitary evolution is equivalent to a Dirac boundary value problem on the half line in one extra dimension. 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 ultrarelativistic limit, correspondent to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation is reduced to the to the quantum-mechanical boundary value problem in the extra dimension. The question of microscopic time reversibility is also studied for this paper.
We prove that a single-jump quantum stochastic unitary evolution is equivalent to a Dirac boundary value problem on the half line in one extra dimension. It is shown that this exactly solvable model can be obtained from a Schrödinger boundary value problem for a positive relativistic Hamiltonian in the half-line as the inductive ultrarelativistic limit, correspondent to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation is reduced to the to the quantum-mechanical boundary value problem in the extra dimension. The question of microscopic time reversibility is also studied for this paper.
We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those describing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo-Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians.
We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of Schrödinger equations in Fock space: the strong resolvent limit is unitary equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitary equivalent to the symmetric (or Stratonovich) form of QSDE. We prove that QSDE is unitary equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps of the phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.}
We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.