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Dynamical Mappings of Density Operators in Quantum Mechanics
1.E. C. G. Sudarshan, P. M. Mathews, and J. Rau, Phys. Rev. 121, 920 (1961).
1.We refer the reader to this paper for a discussion of the physical motivation for the problem considered in the present paper and also for physical examples which illustrate the basic ideas and possible applications.
2.J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1955), Chap. IV.
2.For a general discussion of density operators see, e.g., U. Fano, Revs. Modern Phys. 29, 74 (1957).
3.Von Neumann, reference 2, p. 189.
4.This space of operators with the inner product defined by the trace has been considered by J. Schwinger, Proc. Natl. Acad. Sci. U.S. 46, 257 (1960).
5.To avoid confusion between the two types of operators we will use capital letters A for operators on L and Greek letters for operators on H (elements of L). Greek letters will denote vectors in H, while small letters a will denote scalars.
6.Another criteria for a Hamiltonian mapping, that the mapping preserve multiplication properties, has been given by J. Schwinger, Proc. Natl. Acad. Sci. U.S. 46, 570 (1960).
7.Such nonlinear mappings are used for representing antilinear discrete operations in quantum mechanics. The most familiar example is time inversion; see E. P. Wigner, Gottinger Nachr., 546, (1932);
7.J. Math. Phys. 1, 409 (1960);
7.R. G. Sachs, Nuclear Theory (Addison‐Wesley Publishing Company, Inc., Reading, Massachusetts, 1953),
7.Appendix; another example is charge conjugation in one‐particle theories, see e.g., L. L. Foldy, Phys. Rev. 102, 568 (1956).
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