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Continuous‐Representation Theory. II. Generalized Relation between Quantum and Classical Dynamics
1.J. R. Klauder, J. Math. Phys. 4, 1055 (1963) (preceding paper) hereafter referred to as I.
2.A brief discussion based on this example appears in J. R. Klauder, Helv. Phys. Acta. 35, 333 (1962).
3.J. Schwinger, Proc. Natl. Acad. Sci. U.S. 46, 883, 1401 (1960).
4.E. C. G. Sudarshan, “Brandeis Summer Institute Lecture Notes,” (W. A. Benjamin, Inc., New York, 1961), Vol. 2, p. 143.
4.T. F. Jordan and E. C. G. Sudarshan, Rev. Mod. Phys. 33, 515 (1961).
4.Related ideas appear in: E. H. Wichmann, J. Math. Phys. 2, 876 (1961);
4.F. Bopp, Heisenberg‐Festschrift (Frederick Vieweg und Sohn, Braunschweig, Germany, 1961), p. 128.
5.J. E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949).
6.See, e.g., L. I. Schiff, Quantum Mechanics (McGraw‐Hill Book Company, Inc., New York, 1955), 2nd ed., p. 25.
7.J. E. Moyal and M. S. Bartlett, Proc. Cambridge Phil. Soc. 45, 545 (1949).
8.See, e.g., H. Goldstein, Classical Mechanics (Addison‐Wesley Publishing Company, Inc., Reading, Massachusetts, 1950), Chaps. 8 and 9.
9.The invariant group measure is discussed, e.g., by E. P. Wigner, Group Theory (Academic Press Inc., New York, 1959), Chap. 10.
10.It is worth remarking at this point why we do not consider a set G that includes for all p. If this set G is not complete then its utility is severely impaired; if it is considered complete then a resolution of unity in terms of these states should exist. Recourse to a Schrödinger representation shows that any kernel proposed as a matrix representation of unity fails to satisfy translational invariance unless is an eigenvector of the momentum operator P. Since such eigenvectors do not exist, one would be forced into resolutions of unity having physically undesirable characteristics.
11.V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961).
12.A related set of normalized states, which depends essentially only on one complex variable, is discussed by J. R. Klauder, Ann. Phys. (NY) 11, 123 (1960), p. 125.
13.E. P. Wigner, Phys. Rev. 40, 749 (1932);
13.see also references 3–5, and additional references therein.
14.C. Chevalley, Theory of Lie Groups I (Princeton University Press, Princeton, New Jersey, 1946), Chaps. IV and V.
15.For a recent discussion of this theorem, see: G. H. Weiss, and A. A. Maradudin, J. Math. Phys. 3, 771 (1962).
16.See, e.g., D. Bohm, R. Schiller, and J. Tiomno, Nuovo Cimento Suppl. 1, 48 (1955).
17.For further details relating to the evaluation of the action principle for the relevant restricted set of states, see J. R. Klauder, Ann. Phys. (NY) 11, 123 (1960), pp. 159 and 160.
17.Many of the formal manipulations in that reference regarding the measure on label‐space points may be eliminated by the conventional device of first working in a “box” of finite volume and later passing to the limit.
18.An alternate means to pass from a classical to a quantum theory is by means of the Feynman sum‐over‐histories. The analogue of this technique in our formalism has a somewhat different form than the usual one; it is discussed formally in general terms similar to those of the present paper in J. R. Klauder, Ann. Phys. (NY) 11, 123 (1960), pp. 142–149,
18.and in unpublished lecture notes “The Sum‐Over‐Histories: Formalism and Some Applications,” University of Bern, Switzerland, 1962.
18.A related formulation of the Schwinger Action Principle approach to quantum mechanics that is suitable only for infinite‐dimensional Hilbert spaces is discussed in reference 3. A more general statement of the Schwinger Action Principle is implicitly contained in our formalism. For example, the basic kinematical effects are contained in an expression of in terms of label differentials with the aid of the formulas in Sec. 3.
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