Abstract
We introduce a refinement of the notion of quantum diffusion which has the advantage over the earlier definition that the algebraic conditions which are necessary for existence are always satisfied. In this framework every quantum stochastic parallel transport process induces horizontal lifts. Applications are given to group respresentations and a factorisation property is obtained for parallel transport processes driven by classical Brownian motion on the d × d matrices.
Keywords
- Rotation Number
- Parallel Transport
- Complex Hilbert Space
- Hermitian Structure
- Horizontal Lift
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References
D. Applebaum. Quantum Stochastic Parallel Transport on Non-Commutative Vector Bundles in "Quantum Probability and Applications III", Springer LNM 1303, 20–37 (1988)
D. Applebaum, Stochastic Evolution of Yang-Mills Connections on the Noncommutative Two-Torus, Lett. Math. Phys 16, 93–99 (1988)
D. Applebaum, A. Frigerio, Stationary Dilations of W*-Dynamical Systems Constructed via Quantum Stochastic Differential Equations in "From Local Times to Global Geometry, Control and Physics," Pitman Research Notes in Mathematics 150, 1–39 (1986)
A. Connes, M. Rieffel, Yang-Mills for Noncommutative Two-Tori in "Operator Algebras and Mathematical Physics," (AMS) Contemporary Mathematics 62, 237–67 (1987)
R. L. Hudson, Algebraic Theory of Quantum Diffusions in "Stochastic Mechanics and Stochastic Processes", Springer LNM 1325, 113–25 (1988)
R. L. Hudson, K. R. Parthasarathy, Quantum Ito’s Formula and Stochastic Evolution, Commun. Math. Phys 93, 301–23 (1984)
R. L. Hudson, P. Robinson, Quantum Diffusions and the Noncommutative Torus, Lett. Math. Phys 15, 47–53 (1988)
G. Lindblad, Dissipative Operators and Cohomology of Operator Algebras, Lett. Math. Phys 1, 219–24 (1976)
R. F. Streater, Current Commutation Relations, Continuous Tensor Products and Infinitely Divisible Group Representations, Rendiconti di Sc. Int. di Fisica E. Fermi Vol. XI 247–63 (1969)
P. Jorgensen, Approximately Inner Derivations, Decompositions and Vector Fields of Simple C*-Algebras, to appear in Proceedings of the 1988 U.S.-Japan Seminar on Operator Algebras.
D. Applebaum, Horizontal Lifts in Fock Space Stochastic Calculus, in preparation.
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© 1990 Springer-Verlag
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Applebaum, D. (1990). Quantum diffusions on involutive algebras. In: Accardi, L., von Waldenfels, W. (eds) Quantum Probability and Applications V. Lecture Notes in Mathematics, vol 1442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085502
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DOI: https://doi.org/10.1007/BFb0085502
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