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.
Similar content being viewed by others
REFERENCES
S. Albeverio, V. N. Kolokoltsov, and O. G. Smolyanov, “Continuous quantum measurement: local and global approaches,” Rev. Math. Phys., 9 (1997), no. 8, 907-920.
V. P. Belavkin, “A dynamical theory of quantum measurement and spontaneous localization,” Russian J. Math. Phys., 3 (1995), no. 1, 3-23.
L. Accardi, R. Alicki, A. Frigerio, and Y. G. Lu, “An invitation to the weak coupling and low density limits,” Quantum Probability and Related Topics, 6 (1991), 3-61.
V. P. Belavkin, “A stochastic Hamiltonian approach for quantum jumps, spontaneous localizations, and continuous trajectories,” Quantum Semiclass. Opt., 8 (1996), 167-187.
V. P. Belavkin, “Nondemolition principle of quantum measurement theory,” Foundations Phys., 24 (1994), no. 5, 685-714.
A. M. Chebotarev, “The quantum stochastic equation is unitarily equivalent to a symmetric boundaryvalue problem for the Schrödinger equation,” Mat. Zametki [Math. Notes], 61 (1997), no. 4, 510-518.
R. L. Hudson and K. R. Parthasarathy, “Quantum Ito's formula and stochastic evolutions,” Comm. Math. Phys., 93 (1984), no. 3, 301-323.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Belavkin, V.P. On Quantum Stochastic Differential Equations as Dirac Boundary-Value Problems. Mathematical Notes 69, 735–748 (2001). https://doi.org/10.1023/A:1010270112937
Issue Date:
DOI: https://doi.org/10.1023/A:1010270112937