Measures on the Hilbert space of a quantum system

Russian Journal of Mathematical Physics - Tập 24 - Trang 234-240 - 2017
A. Yu. Khrennikov1, O. G. Smolyanov2
1Linnaeus University, Kalmar, Sweden
2Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

Tóm tắt

The paper is the first in a series of papers on the use of measures and generalized measures in quantum theory. In particular, a survey of the proofs of equivalence of various definitions of the density operator is presented. The exposition is of algebraic nature, and analytic assumptions are usually omitted.

Tài liệu tham khảo

J. Kupsch, O. G. Smolyanov, and N. A. Sidorova, “States of Quantum Systems and Their Liftings,” J. Math. Phys. 42 (3), 1026–1037 (2001). L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic theory, Course of Theoretical Physics, Vol. 3 (Nauka, Moscow, 1962; Pergamon Press Ltd., London–Paris; for USA and Canada: Addison-Wesley Publishing Co., Inc., Reading, Mass, 1958). J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955; Nauka, Moscow, 1964). V. V. Kozlov and O. G. Smolyanov, Foundations of Statistical Mechanics and Works of Poincaré, Ehrenfests, and von Neumann, in: Poincaré H., Ehrenfests P. and T., von Neumann J. Works on Statistical Mechanics. With appendices and edited by V. V. Kozlov and O. G. Smolyanov (Moscow–Izhevsk, 2011), pp. 249–279. V. V. Kozlov and O. G. Smolyanov, “Infinite-Dimensional Liouville Equation with Respect to Measures,” Dokl. Akad. Nauk 432 (1), 28–32 (2010) [Dokl. Math. 81 (3), 476–480 (2010)]. O. G. Smolyanov, Analysis on Topological Linear Spaces and Its Applications (Izdat. MGU, Moscow, 1979). A. Khrennikov, “A Pre-Quantum Classical Statistical Model with Infinite-Dimensional Phase Space,” J. Phys. A: Math. Gen. 38, 9051–9073 (2005). A. Khrennikov, “Representation of Quantum Field Theory as Classical Statistical Mechanics for Field Functionals,” Dokl. Math. 74 (2), 758–761 (2006). O. G. Smolyanov, “Nonlinear Pseudodifferential Operators in Superspaces,” Abstracts of Reports of the Conference “Methods of Algebra and Analysis,” Tartu, 103–106 (1988). J. Kupsch and O.G. Smolyanov, “Bogolyubov Transformations in Wiener-Segal-Fock Space,” Mathematical Notes 68 (3), 408–414 (2000).