Abstract
In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions.
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References
V. P. Belavkin, “A New Form and a ⋆-Algebraic Structure of Quantum Stochastic Integrals in Fock Space,” Rediconti del Sem. Mat. e Fis. di Milano LVIII, 177–193 1988.
V. P. Belavkin, “Chaotic States and Stochastic Integration in Quantum Systems,” Russ. Math. Surveys 47, 53–116 1992.
R. L. Hudson and K. R. Parathasarathy, Quantum Itô Formula and Stochastic Evolutions (Communications in Mathematical Physics, 1984).
V. P. Belavkin, “A Quantum Non-Adapted Itô Formula and Stochastic Analysis in Fock Scale,” J. Funct. Anal. 102, 414–447 1991.
V. P. Belavkin, “Quantum Stochastic Calculus and Quantum Non-Linear Filtering,” J. Multivariate Analysis 42, 171–201 1992.
V. P. Belavkin and M. F. Brown, “Q-adapted Quantum Stochastic Integrals and Differentials In Fock Scale,” Noncommutative Harmonic Analysis with Applications to Probability III, Banach Center Publications 96, 51–66 2012.
V. P. Belavkin and M. F. Brown, “Q-Adapted Integrals and Itô Formula of Noncommutative Stochastic Calculus in Fock Space,” Communications on Stochastic Analysis (KRP Volume) 6, 157–175 2012.
R. L. Hudson and K. R. Parthasarathy, “Unification of Fermion and Boson Stochastic Calculus,” Comm. Math. Phys 93, 301–323 1984.
M. F. Brown, An Investigation of The Stochastic Representation of Quantum Evolution (PhD Thesis, University of Nottingham, 2013).
M. F. Brown, The Stochastic Representation of Hamiltonian Dynamics and The Quantization of Time (arXiv:1111.7043, 2012).
V. P. Belavkin, “Quantum Trajectories, State Diffusion, and Time Asymmetric Eventum Mechanics,” Internat. J. Theoret. Phys. 42(10), 2461–2485 (2003).
V. P. Belavkin and V. N. Kolokoltsov, “Stochastic Evolutions as Boundary Value Problems,” Infinite Dimensional Analysis and Quantum Probability Theory, Research Institute for Mathematical Studies, Sūrikaisekikenkyūsho Kokyūrokū 1227, 83–95 (2001) [in Japanese].
A. Guichardet, Symmetric Hilbert Spaces and Related Topics (Springer-Verlag, 1972).
E. C. Lance, Hilbert C*-Modules (LMS Lecture Note Series, Cambridge University Press, 1995).
P. Malliavin, “Stochastic Calculus of Variation and Hypoelliptic Operators,” Proceedings of the International Symposium on Stochastic Differential Equantions (Wiley, New York-Chichester-Brisbane, 1978), pages 195–263.
V. P. Belavkin, “Quantum Stochastics, Dirac Boundary Value Problem, and the Ultrarelativistic Limit,” Rep. Math. Phys 46(3), 359–382 (2000).
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Brown, M.F. A canonical dilation of the Schrödinger equation. Russ. J. Math. Phys. 21, 316–325 (2014). https://doi.org/10.1134/S1061920814030030
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DOI: https://doi.org/10.1134/S1061920814030030