Quantization of the optical phase space S2 = {ψ mod 2π, I > 0} in terms of the group SO↑ (1, 2)

Fortschritte der Physik

Volumn 51, Issue 10-11, 2003, Pages

  Kastrup, H A ^a  

^a DESY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0142169389     PISSN: 00158208     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (503)

1. P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40, 411 (1968).
2. H. Paul, Fortschr. Phys. 22, 657 (1974).
3. W. P. Schleich and S. M. Barnett (eds.), Special issue on Quantum Phase and Phase Dependent Measurements, Phys. Scr. T48 (1993).
4. A. Lukš and V. Peřinová, Quantum Opt. 6, 125 (1994).
5. U. Leonhardt and H. Paul, Prog. Quantum Electron. 19, 89 (1995).
6. R. Lynch, Phys. Rep. 256, 367 (1995).
7. M. Heni, M. Freyberger, and W. P. Schleich, in: Coherence and Quantum Optics VII, edited by J. H. Eberly, L. Mandel, and E. Wolf (Plenum Press, New York and London, 1996), p. 239.
8. D. A. Dubin, M. A. Hennings, and T. B. Smith, Int. J. Mod. Phys. B 9, 2597 (1995).
9. D. T. Pegg and S. M. Barnett, J. Mod. Opt. 44, 225 (1997).
10. V. V. Dodonov, J. Opt. B, Quantum Semiclassical Opt. 4, R1 (2002); this review contains a wealth of References.
11. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Heidelberg etc., 1994).
12. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge etc., 1995).
13. U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press, Cambridge etc., 1997).
14. P. Meystre and M. Sargent III, Elements of Quantum Optics, 3rd ed. (Springer, Berlin etc., 1998).
15. V. Peřinová, A. Lukš, and J. Peřina, Phase in Optics (World Scientific Publ. Co., Singapore, 1998).
16. D. A. Dubin, M. A. Hennings, and T. B. Smith, Mathematical Aspects of Weyl Quantization and Phase (World Scientific Publ. Co., Singapore, 2000).
17. W. Vogel, D.-G. Welsch, and S. Wallentowitz, Quantum Optics. An Introduction, 2nd ed. (Wiley-VHC Verlag, Weinheim, 2001).
18. W. P. Schleich, Quantum Optics in Phase Space (Wiley-VHC Verlag, Weinheim, 2001).
19. R. R. Puri, Mathematical Methods of Quantum Optics, Springer Series in Optical Sciences 79 (Springer, Berlin etc., 2001).
20. C. J. Isham, in: Relativity, Groups and Topology II, Les Houches Session XL, 1983, edited by B. S. Dewitt and R. Stora (North-Holland, Amsterdam etc., 1984), p. 1059.
21. V. Guillemin and S. Sternberg, Symplectic techniques in physics (Cambridge University Press, Cambridge etc., 1984); paperback edition (Cambridge University Press, Cambridge etc., 1990).
22. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). Reviews are: M. Peshkin and A. Tonomura, The Aharonov-Bohm effect, Lecture Notes in Physics 340 (Springer-Verlag, Berlin and Heidelberg, 1989). J. Hamilton, Aharonov-Bohm and other Cyclic Phenomena, Springer Tracts in Modern Physics 139 (Springer-Verlag, Berlin etc., 1997).
23. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). Reviews are: M. Peshkin and A. Tonomura, The Aharonov-Bohm effect, Lecture Notes in Physics 340 (Springer-Verlag, Berlin and Heidelberg, 1989). J. Hamilton, Aharonov-Bohm and other Cyclic Phenomena, Springer Tracts in Modern Physics 139 (Springer-Verlag, Berlin etc., 1997).
24. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959). Reviews are: M. Peshkin and A. Tonomura, The Aharonov-Bohm effect, Lecture Notes in Physics 340 (Springer-Verlag, Berlin and Heidelberg, 1989). J. Hamilton, Aharonov-Bohm and other Cyclic Phenomena, Springer Tracts in Modern Physics 139 (Springer-Verlag, Berlin etc., 1997).
25. P. A. M. Dirac, Proc. R. Soc. Lond. A 114, 243 (1927).
26. W. Heitler, The Quantum Theory of Radiation, 3rd Ed. (Oxford Univ. Press, London, 1954; reprinted by Dover Publications, New York, 1984), § 7.4.
27. F. London, Z. Phys. 37, 915 (1926).
28. F. London, Z. Phys. 40, 193 (1927).
29. P. A. M. Dirac, Proc. R. Soc. Lond. A, 109, 642 (1925); 110, 561 (1926); in the first of these papers Dirac, still a "Senior Research Student" at Cambridge, invented the formalism of matrix mechanics all by himself, after he saw Heisenberg's fundamental paper (Z. Phys. 33, 879 (1925)); he realized the close relationship of the algebraic structure of the new formalism to that of the Poisson brackets and called the commutators of the new redefined canonical quantities "quantum Poisson brackets".
30. P. A. M. Dirac, Proc. R. Soc. Lond. A, 109, 642 (1925); 110, 561 (1926); in the first of these papers Dirac, still a "Senior Research Student" at Cambridge, invented the formalism of matrix mechanics all by himself, after he saw Heisenberg's fundamental paper (Z. Phys. 33, 879 (1925)); he realized the close relationship of the algebraic structure of the new formalism to that of the Poisson brackets and called the commutators of the new redefined canonical quantities "quantum Poisson brackets".
31. P. A. M. Dirac, Proc. R. Soc. Lond. A, 109, 642 (1925); 110, 561 (1926); in the first of these papers Dirac, still a "Senior Research Student" at Cambridge, invented the formalism of matrix mechanics all by himself, after he saw Heisenberg's fundamental paper (Z. Phys. 33, 879 (1925)); he realized the close relationship of the algebraic structure of the new formalism to that of the Poisson brackets and called the commutators of the new redefined canonical quantities "quantum Poisson brackets".
32. M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic Press, New York, 1975), Chap. X. As to the impossibility to promote the classical phase to a self-adjoint operator see also G. W. Mackey, Mathematical Foundations of Quantum Mechanics (W.A. Benjamin, Inc., New York and Amsterdam, 1963), p. 103.
33. M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (Academic Press, New York, 1975), Chap. X. As to the impossibility to promote the classical phase to a self-adjoint operator see also G. W. Mackey, Mathematical Foundations of Quantum Mechanics (W.A. Benjamin, Inc., New York and Amsterdam, 1963), p. 103.
34. L. Susskind and J. Glogower, Physics 1, 49 (1964).
35. P. Carruthers and M. M. Nieto, Phys. Rev. Lett. 14, 387 (1965).
36. K. Hoffmann, Banach Spaces of Analytic Functions (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962). B. Sz-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space (North-Holland Publ. Co., Amsterdam, 1970). W. Rudin, Real and Complex Analysis, 3rd ed. (McGraw-Hill Book Co., N.Y. etc., 1987), Chap. 17. J. B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs 36 (Amer. Math. Soc., Providence, R.I., 1991).
37. K. Hoffmann, Banach Spaces of Analytic Functions (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962). B. Sz-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space (North-Holland Publ. Co., Amsterdam, 1970). W. Rudin, Real and Complex Analysis, 3rd ed. (McGraw-Hill Book Co., N.Y. etc., 1987), Chap. 17. J. B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs 36 (Amer. Math. Soc., Providence, R.I., 1991).
38. K. Hoffmann, Banach Spaces of Analytic Functions (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962). B. Sz-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space (North-Holland Publ. Co., Amsterdam, 1970). W. Rudin, Real and Complex Analysis, 3rd ed. (McGraw-Hill Book Co., N.Y. etc., 1987), Chap. 17. J. B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs 36 (Amer. Math. Soc., Providence, R.I., 1991).
39. K. Hoffmann, Banach Spaces of Analytic Functions (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962). B. Sz-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space (North-Holland Publ. Co., Amsterdam, 1970). W. Rudin, Real and Complex Analysis, 3rd ed. (McGraw-Hill Book Co., N.Y. etc., 1987), Chap. 17. J. B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs 36 (Amer. Math. Soc., Providence, R.I., 1991).
40. [3], p. 5.
41. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, New York etc., 1978), Chap. III.
42. n are to be expressed in terms of A und B, was investigated by Zassenhaus (unpublished). The "Zassenhaus formula" has been discussed by W. Magnus, Commun. Pure Appl. Math. VII, 649 (1954), and Wilcox, l.c.
43. n are to be expressed in terms of A und B, was investigated by Zassenhaus (unpublished). The "Zassenhaus formula" has been discussed by W. Magnus, Commun. Pure Appl. Math. VII, 649 (1954), and Wilcox, l.c.
44. n are to be expressed in terms of A und B, was investigated by Zassenhaus (unpublished). The "Zassenhaus formula" has been discussed by W. Magnus, Commun. Pure Appl. Math. VII, 649 (1954), and Wilcox, l.c.
45. n are to be expressed in terms of A und B, was investigated by Zassenhaus (unpublished). The "Zassenhaus formula" has been discussed by W. Magnus, Commun. Pure Appl. Math. VII, 649 (1954), and Wilcox, l.c.
46. n are to be expressed in terms of A und B, was investigated by Zassenhaus (unpublished). The "Zassenhaus formula" has been discussed by W. Magnus, Commun. Pure Appl. Math. VII, 649 (1954), and Wilcox, l.c.
47. n are to be expressed in terms of A und B, was investigated by Zassenhaus (unpublished). The "Zassenhaus formula" has been discussed by W. Magnus, Commun. Pure Appl. Math. VII, 649 (1954), and Wilcox, l.c.
48. n are to be expressed in terms of A und B, was investigated by Zassenhaus (unpublished). The "Zassenhaus formula" has been discussed by W. Magnus, Commun. Pure Appl. Math. VII, 649 (1954), and Wilcox, l.c.
49. n are to be expressed in terms of A und B, was investigated by Zassenhaus (unpublished). The "Zassenhaus formula" has been discussed by W. Magnus, Commun. Pure Appl. Math. VII, 649 (1954), and Wilcox, l.c.
50. C. M. Isham, [20].
51. P. Goddard and D. Olive, Int. J. Mod. Phys. A 1, 303 (1986).
52. H. Weyl, Z. Phys. 46, 1 (1927); Gruppentheorie und Quantenmechanik, 2. Aufl. (S. Hirzel Verlag, Leipzig, 1931); Engl. translation: The Theory of Groups and Quantum Mechanics (Dover Publications, Inc., New York, 1950), Chap. IV, § 14.
53. H. Weyl, Z. Phys. 46, 1 (1927); Gruppentheorie und Quantenmechanik, 2. Aufl. (S. Hirzel Verlag, Leipzig, 1931); Engl. translation: The Theory of Groups and Quantum Mechanics (Dover Publications, Inc., New York, 1950), Chap. IV, § 14.
54. H. Weyl, Z. Phys. 46, 1 (1927); Gruppentheorie und Quantenmechanik, 2. Aufl. (S. Hirzel Verlag, Leipzig, 1931); Engl. translation: The Theory of Groups and Quantum Mechanics (Dover Publications, Inc., New York, 1950), Chap. IV, § 14.
55. M. H. Stone, Proc. Natl. Acad. Sci. USA 15, 198 and 423 (1929); 16, 172 (1930); Linear Transformations in Hilbert Space and Their Applications to Analysis (Am. Math. Soc. Coll. Publ. XV, Providence, R.I., 1932). J. von Neumann, Math. Ann. 104, 570 (1931). C. R. Putnam, Commutation Properties of Hilbert Operators and Related Topics (Springer-Verlag, Berlin, Heidelberg, New York, 1967). See also the illuminating comments on that important theorem by G. W. Mackey, in: Functional Analysis and Related Fields, edited by F. E. Browder (Springer-Verlag, Berlin etc., 1970), p. 132. As to the harmonic analysis of the Weyl-Heisenberg group see: R. Howe, Bull. Amer. Math. Soc. (New Ser.) 3, 821 (1980). G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122 (Princeton University Press, Princeton N.J., 1989). E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 2nd printing 1995 (Princeton University Press, Princeton, N.J., 1993), Chaps. XII and XIII.
56. M. H. Stone, Proc. Natl. Acad. Sci. USA 15, 198 and 423 (1929); 16, 172 (1930); Linear Transformations in Hilbert Space and Their Applications to Analysis (Am. Math. Soc. Coll. Publ. XV, Providence, R.I., 1932). J. von Neumann, Math. Ann. 104, 570 (1931). C. R. Putnam, Commutation Properties of Hilbert Operators and Related Topics (Springer-Verlag, Berlin, Heidelberg, New York, 1967). See also the illuminating comments on that important theorem by G. W. Mackey, in: Functional Analysis and Related Fields, edited by F. E. Browder (Springer-Verlag, Berlin etc., 1970), p. 132. As to the harmonic analysis of the Weyl-Heisenberg group see: R. Howe, Bull. Amer. Math. Soc. (New Ser.) 3, 821 (1980). G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122 (Princeton University Press, Princeton N.J., 1989). E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 2nd printing 1995 (Princeton University Press, Princeton, N.J., 1993), Chaps. XII and XIII.
57. M. H. Stone, Proc. Natl. Acad. Sci. USA 15, 198 and 423 (1929); 16, 172 (1930); Linear Transformations in Hilbert Space and Their Applications to Analysis (Am. Math. Soc. Coll. Publ. XV, Providence, R.I., 1932). J. von Neumann, Math. Ann. 104, 570 (1931). C. R. Putnam, Commutation Properties of Hilbert Operators and Related Topics (Springer-Verlag, Berlin, Heidelberg, New York, 1967). See also the illuminating comments on that important theorem by G. W. Mackey, in: Functional Analysis and Related Fields, edited by F. E. Browder (Springer-Verlag, Berlin etc., 1970), p. 132. As to the harmonic analysis of the Weyl-Heisenberg group see: R. Howe, Bull. Amer. Math. Soc. (New Ser.) 3, 821 (1980). G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122 (Princeton University Press, Princeton N.J., 1989). E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 2nd printing 1995 (Princeton University Press, Princeton, N.J., 1993), Chaps. XII and XIII.
58. M. H. Stone, Proc. Natl. Acad. Sci. USA 15, 198 and 423 (1929); 16, 172 (1930); Linear Transformations in Hilbert Space and Their Applications to Analysis (Am. Math. Soc. Coll. Publ. XV, Providence, R.I., 1932). J. von Neumann, Math. Ann. 104, 570 (1931). C. R. Putnam, Commutation Properties of Hilbert Operators and Related Topics (Springer-Verlag, Berlin, Heidelberg, New York, 1967). See also the illuminating comments on that important theorem by G. W. Mackey, in: Functional Analysis and Related Fields, edited by F. E. Browder (Springer-Verlag, Berlin etc., 1970), p. 132. As to the harmonic analysis of the Weyl-Heisenberg group see: R. Howe, Bull. Amer. Math. Soc. (New Ser.) 3, 821 (1980). G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122 (Princeton University Press, Princeton N.J., 1989). E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 2nd printing 1995 (Princeton University Press, Princeton, N.J., 1993), Chaps. XII and XIII.
59. M. H. Stone, Proc. Natl. Acad. Sci. USA 15, 198 and 423 (1929); 16, 172 (1930); Linear Transformations in Hilbert Space and Their Applications to Analysis (Am. Math. Soc. Coll. Publ. XV, Providence, R.I., 1932). J. von Neumann, Math. Ann. 104, 570 (1931). C. R. Putnam, Commutation Properties of Hilbert Operators and Related Topics (Springer-Verlag, Berlin, Heidelberg, New York, 1967). See also the illuminating comments on that important theorem by G. W. Mackey, in: Functional Analysis and Related Fields, edited by F. E. Browder (Springer-Verlag, Berlin etc., 1970), p. 132. As to the harmonic analysis of the Weyl-Heisenberg group see: R. Howe, Bull. Amer. Math. Soc. (New Ser.) 3, 821 (1980). G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122 (Princeton University Press, Princeton N.J., 1989). E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 2nd printing 1995 (Princeton University Press, Princeton, N.J., 1993), Chaps. XII and XIII.
60. M. H. Stone, Proc. Natl. Acad. Sci. USA 15, 198 and 423 (1929); 16, 172 (1930); Linear Transformations in Hilbert Space and Their Applications to Analysis (Am. Math. Soc. Coll. Publ. XV, Providence, R.I., 1932). J. von Neumann, Math. Ann. 104, 570 (1931). C. R. Putnam, Commutation Properties of Hilbert Operators and Related Topics (Springer-Verlag, Berlin, Heidelberg, New York, 1967). See also the illuminating comments on that important theorem by G. W. Mackey, in: Functional Analysis and Related Fields, edited by F. E. Browder (Springer-Verlag, Berlin etc., 1970), p. 132. As to the harmonic analysis of the Weyl-Heisenberg group see: R. Howe, Bull. Amer. Math. Soc. (New Ser.) 3, 821 (1980). G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122 (Princeton University Press, Princeton N.J., 1989). E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 2nd printing 1995 (Princeton University Press, Princeton, N.J., 1993), Chaps. XII and XIII.
61. M. H. Stone, Proc. Natl. Acad. Sci. USA 15, 198 and 423 (1929); 16, 172 (1930); Linear Transformations in Hilbert Space and Their Applications to Analysis (Am. Math. Soc. Coll. Publ. XV, Providence, R.I., 1932). J. von Neumann, Math. Ann. 104, 570 (1931). C. R. Putnam, Commutation Properties of Hilbert Operators and Related Topics (Springer-Verlag, Berlin, Heidelberg, New York, 1967). See also the illuminating comments on that important theorem by G. W. Mackey, in: Functional Analysis and Related Fields, edited by F. E. Browder (Springer-Verlag, Berlin etc., 1970), p. 132. As to the harmonic analysis of the Weyl-Heisenberg group see: R. Howe, Bull. Amer. Math. Soc. (New Ser.) 3, 821 (1980). G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122 (Princeton University Press, Princeton N.J., 1989). E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 2nd printing 1995 (Princeton University Press, Princeton, N.J., 1993), Chaps. XII and XIII.
62. M. H. Stone, Proc. Natl. Acad. Sci. USA 15, 198 and 423 (1929); 16, 172 (1930); Linear Transformations in Hilbert Space and Their Applications to Analysis (Am. Math. Soc. Coll. Publ. XV, Providence, R.I., 1932). J. von Neumann, Math. Ann. 104, 570 (1931). C. R. Putnam, Commutation Properties of Hilbert Operators and Related Topics (Springer-Verlag, Berlin, Heidelberg, New York, 1967). See also the illuminating comments on that important theorem by G. W. Mackey, in: Functional Analysis and Related Fields, edited by F. E. Browder (Springer-Verlag, Berlin etc., 1970), p. 132. As to the harmonic analysis of the Weyl-Heisenberg group see: R. Howe, Bull. Amer. Math. Soc. (New Ser.) 3, 821 (1980). G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122 (Princeton University Press, Princeton N.J., 1989). E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 2nd printing 1995 (Princeton University Press, Princeton, N.J., 1993), Chaps. XII and XIII.
63. M. H. Stone, Proc. Natl. Acad. Sci. USA 15, 198 and 423 (1929); 16, 172 (1930); Linear Transformations in Hilbert Space and Their Applications to Analysis (Am. Math. Soc. Coll. Publ. XV, Providence, R.I., 1932). J. von Neumann, Math. Ann. 104, 570 (1931). C. R. Putnam, Commutation Properties of Hilbert Operators and Related Topics (Springer-Verlag, Berlin, Heidelberg, New York, 1967). See also the illuminating comments on that important theorem by G. W. Mackey, in: Functional Analysis and Related Fields, edited by F. E. Browder (Springer-Verlag, Berlin etc., 1970), p. 132. As to the harmonic analysis of the Weyl-Heisenberg group see: R. Howe, Bull. Amer. Math. Soc. (New Ser.) 3, 821 (1980). G. B. Folland, Harmonic Analysis in Phase Space, Ann. Math. Studies, no. 122 (Princeton University Press, Princeton N.J., 1989). E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, 2nd printing 1995 (Princeton University Press, Princeton, N.J., 1993), Chaps. XII and XIII.
64. [33], Chap. II.
65. R. Loll, Phys. Rev. D 41, 3785 (1990).
66. M. Bojowald, H. A. Kastrup, F. Schramm, and T. Strobl, Phys. Rev. D 62, 044026 (2000) e-print gr-qc/9906105.
67. H. A. Kastrup, Ann. Phy. (Leipzig) 9,503 (2000), e-print gr-qc/9906104.
68. H. A. Kastrup, e-print quant-ph/0005033.
69. H. A. Kastrup, report CERN-TH/2001-238, e-print quant-ph/0109013.
70. H. A. Kastrup, Quantization of Phases and Moduli in Terms of the Group SO↑ (1, 2), invited talk given at Group 24, the XXIV. Int. Colloq. on Group Theor. Methods in Physics, Paris 2002, to be published.
71. W. H. Louisell, Phys. Lett. 7, 60 (1963).
72. See Isham, [20], Chap. 4.6.
73. [48] [250], Chap. IV.
74. Orbifolds were introduced as "V-manifolds" by I. Satake, Proc. Natl. Acad. Sci. USA 42, 359 (1956); J. Math. Soc. Jpn. 9, 464 (1957). The term "orbifold" was coined in Thurston's 1978-80 Princeton lectures. The notes are available under http://www.msri.org/publications/books/gt3m/. Orbifolds are discussed in Chap. 13. In the first volume of the printed version on the subject of those lectures the termin "orbifold" does not appear, but the concept is discussed in a fascinating variety of examples: W. P. Thurston, Three-dimensional Geometry and Topology, Vol. 1 (Princeton University Press, Princeton, N.J., 1997). For a brief modern introduction see R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhäuser Verlag, Basel, Boston and Berlin, 1997), Appendix B, Sect. 2.3. Orbifolds play a larger role in string theories: J. Polchinski, String Theory I (Cambridge Univ. Press, Cambridge, 1998), Sect. 8.5. As to the state of the art see A. Adem, J. Morava, and Y. Ruan (eds.), Orbifolds in mathematics and physics, Proceedings of the Conf. Math. Aspects of Orbifold String Theory, Univ. Wisconsin May 2001, Contemp. Math. 310 (Amer. Math. Soc., Providence, R.I., 2002).
75. Orbifolds were introduced as "V-manifolds" by I. Satake, Proc. Natl. Acad. Sci. USA 42, 359 (1956); J. Math. Soc. Jpn. 9, 464 (1957). The term "orbifold" was coined in Thurston's 1978-80 Princeton lectures. The notes are available under http://www.msri.org/publications/books/gt3m/. Orbifolds are discussed in Chap. 13. In the first volume of the printed version on the subject of those lectures the termin "orbifold" does not appear, but the concept is discussed in a fascinating variety of examples: W. P. Thurston, Three-dimensional Geometry and Topology, Vol. 1 (Princeton University Press, Princeton, N.J., 1997). For a brief modern introduction see R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhäuser Verlag, Basel, Boston and Berlin, 1997), Appendix B, Sect. 2.3. Orbifolds play a larger role in string theories: J. Polchinski, String Theory I (Cambridge Univ. Press, Cambridge, 1998), Sect. 8.5. As to the state of the art see A. Adem, J. Morava, and Y. Ruan (eds.), Orbifolds in mathematics and physics, Proceedings of the Conf. Math. Aspects of Orbifold String Theory, Univ. Wisconsin May 2001, Contemp. Math. 310 (Amer. Math. Soc., Providence, R.I., 2002).
76. Orbifolds were introduced as "V-manifolds" by I. Satake, Proc. Natl. Acad. Sci. USA 42, 359 (1956); J. Math. Soc. Jpn. 9, 464 (1957). The term "orbifold" was coined in Thurston's 1978-80 Princeton lectures. The notes are available under http://www.msri.org/publications/books/gt3m/. Orbifolds are discussed in Chap. 13. In the first volume of the printed version on the subject of those lectures the termin "orbifold" does not appear, but the concept is discussed in a fascinating variety of examples: W. P. Thurston, Three-dimensional Geometry and Topology, Vol. 1 (Princeton University Press, Princeton, N.J., 1997). For a brief modern introduction see R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhäuser Verlag, Basel, Boston and Berlin, 1997), Appendix B, Sect. 2.3. Orbifolds play a larger role in string theories: J. Polchinski, String Theory I (Cambridge Univ. Press, Cambridge, 1998), Sect. 8.5. As to the state of the art see A. Adem, J. Morava, and Y. Ruan (eds.), Orbifolds in mathematics and physics, Proceedings of the Conf. Math. Aspects of Orbifold String Theory, Univ. Wisconsin May 2001, Contemp. Math. 310 (Amer. Math. Soc., Providence, R.I., 2002).
77. Orbifolds were introduced as "V-manifolds" by I. Satake, Proc. Natl. Acad. Sci. USA 42, 359 (1956); J. Math. Soc. Jpn. 9, 464 (1957). The term "orbifold" was coined in Thurston's 1978-80 Princeton lectures. The notes are available under http://www.msri.org/publications/books/gt3m/. Orbifolds are discussed in Chap. 13. In the first volume of the printed version on the subject of those lectures the termin "orbifold" does not appear, but the concept is discussed in a fascinating variety of examples: W. P. Thurston, Three-dimensional Geometry and Topology, Vol. 1 (Princeton University Press, Princeton, N.J., 1997). For a brief modern introduction see R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhäuser Verlag, Basel, Boston and Berlin, 1997), Appendix B, Sect. 2.3. Orbifolds play a larger role in string theories: J. Polchinski, String Theory I (Cambridge Univ. Press, Cambridge, 1998), Sect. 8.5. As to the state of the art see A. Adem, J. Morava, and Y. Ruan (eds.), Orbifolds in mathematics and physics, Proceedings of the Conf. Math. Aspects of Orbifold String Theory, Univ. Wisconsin May 2001, Contemp. Math. 310 (Amer. Math. Soc., Providence, R.I., 2002).
78. Orbifolds were introduced as "V-manifolds" by I. Satake, Proc. Natl. Acad. Sci. USA 42, 359 (1956); J. Math. Soc. Jpn. 9, 464 (1957). The term "orbifold" was coined in Thurston's 1978-80 Princeton lectures. The notes are available under http://www.msri.org/publications/books/gt3m/. Orbifolds are discussed in Chap. 13. In the first volume of the printed version on the subject of those lectures the termin "orbifold" does not appear, but the concept is discussed in a fascinating variety of examples: W. P. Thurston, Three-dimensional Geometry and Topology, Vol. 1 (Princeton University Press, Princeton, N.J., 1997). For a brief modern introduction see R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhäuser Verlag, Basel, Boston and Berlin, 1997), Appendix B, Sect. 2.3. Orbifolds play a larger role in string theories: J. Polchinski, String Theory I (Cambridge Univ. Press, Cambridge, 1998), Sect. 8.5. As to the state of the art see A. Adem, J. Morava, and Y. Ruan (eds.), Orbifolds in mathematics and physics, Proceedings of the Conf. Math. Aspects of Orbifold String Theory, Univ. Wisconsin May 2001, Contemp. Math. 310 (Amer. Math. Soc., Providence, R.I., 2002).
79. Orbifolds were introduced as "V-manifolds" by I. Satake, Proc. Natl. Acad. Sci. USA 42, 359 (1956); J. Math. Soc. Jpn. 9, 464 (1957). The term "orbifold" was coined in Thurston's 1978-80 Princeton lectures. The notes are available under http://www.msri.org/publications/books/gt3m/. Orbifolds are discussed in Chap. 13. In the first volume of the printed version on the subject of those lectures the termin "orbifold" does not appear, but the concept is discussed in a fascinating variety of examples: W. P. Thurston, Three-dimensional Geometry and Topology, Vol. 1 (Princeton University Press, Princeton, N.J., 1997). For a brief modern introduction see R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems (Birkhäuser Verlag, Basel, Boston and Berlin, 1997), Appendix B, Sect. 2.3. Orbifolds play a larger role in string theories: J. Polchinski, String Theory I (Cambridge Univ. Press, Cambridge, 1998), Sect. 8.5. As to the state of the art see A. Adem, J. Morava, and Y. Ruan (eds.), Orbifolds in mathematics and physics, Proceedings of the Conf. Math. Aspects of Orbifold String Theory, Univ. Wisconsin May 2001, Contemp. Math. 310 (Amer. Math. Soc., Providence, R.I., 2002).
80. L. V. Prokhorov, Sov. J. Nucl. Phys. 35, 129 (1982). S. V. Shabanov, Theor. Math. Phys. 78, 292 (1989). L. V. Prokhorov and S. V. Shabanov, Phys. Lett. B 216, 341 (1989). S. V. Shabanov, Int. J. Mod. Phys. A 6, 845 (1991). L. V. Prokhorov and S. V. Shabanov, Sov. Phys.-Usp. 34, 108 (1991).
81. L. V. Prokhorov, Sov. J. Nucl. Phys. 35, 129 (1982). S. V. Shabanov, Theor. Math. Phys. 78, 292 (1989). L. V. Prokhorov and S. V. Shabanov, Phys. Lett. B 216, 341 (1989). S. V. Shabanov, Int. J. Mod. Phys. A 6, 845 (1991). L. V. Prokhorov and S. V. Shabanov, Sov. Phys.-Usp. 34, 108 (1991).
82. L. V. Prokhorov, Sov. J. Nucl. Phys. 35, 129 (1982). S. V. Shabanov, Theor. Math. Phys. 78, 292 (1989). L. V. Prokhorov and S. V. Shabanov, Phys. Lett. B 216, 341 (1989). S. V. Shabanov, Int. J. Mod. Phys. A 6, 845 (1991). L. V. Prokhorov and S. V. Shabanov, Sov. Phys.-Usp. 34, 108 (1991).
83. L. V. Prokhorov, Sov. J. Nucl. Phys. 35, 129 (1982). S. V. Shabanov, Theor. Math. Phys. 78, 292 (1989). L. V. Prokhorov and S. V. Shabanov, Phys. Lett. B 216, 341 (1989). S. V. Shabanov, Int. J. Mod. Phys. A 6, 845 (1991). L. V. Prokhorov and S. V. Shabanov, Sov. Phys.-Usp. 34, 108 (1991).
84. L. V. Prokhorov, Sov. J. Nucl. Phys. 35, 129 (1982). S. V. Shabanov, Theor. Math. Phys. 78, 292 (1989). L. V. Prokhorov and S. V. Shabanov, Phys. Lett. B 216, 341 (1989). S. V. Shabanov, Int. J. Mod. Phys. A 6, 845 (1991). L. V. Prokhorov and S. V. Shabanov, Sov. Phys.-Usp. 34, 108 (1991).
85. S. V. Shabanov, Phys. Rep. 326, 1 (2000).
86. The realization (1.104) was apparently first discussed by L. D. Mlodinow and N. Papanicoulaou, Ann. Phys. (USA) 128, 314 (1980). See also C. C. Gerry, J. Phys. A, Math. Gen. 16, L1 (1983). J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). H. Bacry, J. Math. Phys. 31, 2061 (1990). C. C. Gerry and R. Grobe, Quantum Semiclass. Opt. 9, 59 (1997). A. Wünsche, Acta Phys. Slovaca 49, 771 (1999); J. Opt. B: Quantum Semiclass. Opt. 5, S429 (2003).
87. The realization (1.104) was apparently first discussed by L. D. Mlodinow and N. Papanicoulaou, Ann. Phys. (USA) 128, 314 (1980). See also C. C. Gerry, J. Phys. A, Math. Gen. 16, L1 (1983). J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). H. Bacry, J. Math. Phys. 31, 2061 (1990). C. C. Gerry and R. Grobe, Quantum Semiclass. Opt. 9, 59 (1997). A. Wünsche, Acta Phys. Slovaca 49, 771 (1999); J. Opt. B: Quantum Semiclass. Opt. 5, S429 (2003).
88. The realization (1.104) was apparently first discussed by L. D. Mlodinow and N. Papanicoulaou, Ann. Phys. (USA) 128, 314 (1980). See also C. C. Gerry, J. Phys. A, Math. Gen. 16, L1 (1983). J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). H. Bacry, J. Math. Phys. 31, 2061 (1990). C. C. Gerry and R. Grobe, Quantum Semiclass. Opt. 9, 59 (1997). A. Wünsche, Acta Phys. Slovaca 49, 771 (1999); J. Opt. B: Quantum Semiclass. Opt. 5, S429 (2003).
89. The realization (1.104) was apparently first discussed by L. D. Mlodinow and N. Papanicoulaou, Ann. Phys. (USA) 128, 314 (1980). See also C. C. Gerry, J. Phys. A, Math. Gen. 16, L1 (1983). J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). H. Bacry, J. Math. Phys. 31, 2061 (1990). C. C. Gerry and R. Grobe, Quantum Semiclass. Opt. 9, 59 (1997). A. Wünsche, Acta Phys. Slovaca 49, 771 (1999); J. Opt. B: Quantum Semiclass. Opt. 5, S429 (2003).
90. The realization (1.104) was apparently first discussed by L. D. Mlodinow and N. Papanicoulaou, Ann. Phys. (USA) 128, 314 (1980). See also C. C. Gerry, J. Phys. A, Math. Gen. 16, L1 (1983). J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). H. Bacry, J. Math. Phys. 31, 2061 (1990). C. C. Gerry and R. Grobe, Quantum Semiclass. Opt. 9, 59 (1997). A. Wünsche, Acta Phys. Slovaca 49, 771 (1999); J. Opt. B: Quantum Semiclass. Opt. 5, S429 (2003).
91. The realization (1.104) was apparently first discussed by L. D. Mlodinow and N. Papanicoulaou, Ann. Phys. (USA) 128, 314 (1980). See also C. C. Gerry, J. Phys. A, Math. Gen. 16, L1 (1983). J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). H. Bacry, J. Math. Phys. 31, 2061 (1990). C. C. Gerry and R. Grobe, Quantum Semiclass. Opt. 9, 59 (1997). A. Wünsche, Acta Phys. Slovaca 49, 771 (1999); J. Opt. B: Quantum Semiclass. Opt. 5, S429 (2003).
92. The realization (1.104) was apparently first discussed by L. D. Mlodinow and N. Papanicoulaou, Ann. Phys. (USA) 128, 314 (1980). See also C. C. Gerry, J. Phys. A, Math. Gen. 16, L1 (1983). J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). H. Bacry, J. Math. Phys. 31, 2061 (1990). C. C. Gerry and R. Grobe, Quantum Semiclass. Opt. 9, 59 (1997). A. Wünsche, Acta Phys. Slovaca 49, 771 (1999); J. Opt. B: Quantum Semiclass. Opt. 5, S429 (2003).
93. T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).
94. A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971).
95. A. M. Perelomov, Commun. Math. Phys. 26, 222 (1972); see also [111,249].
96. E. Schrödinger, Die Naturwissenschaften 14, 664 (1926).
97. R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963); Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963). The coherent states (3.63) were introduced by J. R. Klauder in connection with problems in quantum field theory (Ann. Phys. (USA) 11, 123 (1960)). Their importance for quantum optical problems was first and thoroughly discussed by Glauber.
98. R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963); Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963). The coherent states (3.63) were introduced by J. R. Klauder in connection with problems in quantum field theory (Ann. Phys. (USA) 11, 123 (1960)). Their importance for quantum optical problems was first and thoroughly discussed by Glauber.
99. R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963); Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963). The coherent states (3.63) were introduced by J. R. Klauder in connection with problems in quantum field theory (Ann. Phys. (USA) 11, 123 (1960)). Their importance for quantum optical problems was first and thoroughly discussed by Glauber.
100. R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963); Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963). The coherent states (3.63) were introduced by J. R. Klauder in connection with problems in quantum field theory (Ann. Phys. (USA) 11, 123 (1960)). Their importance for quantum optical problems was first and thoroughly discussed by Glauber.
101. See, e.g. the following textbooks and reviews with many References: [13], Chap. 6.3. [15], Chap. 4.7. [16], Chap. 10. [17], Chap. 3.5. [18], Chap. 8.5.
102. G. J. Milburn, J. Phys. A, Math. Gen. 17, 737 (1984).
103. R. A. Fisher, M. M. Nieto, and V. D. Sandberg, Phys. Rev. D 29, 1107 (1984).
104. K. Wódkiewicz and J. H. Eberly, J. Opt. Soc. Am. B 2,458 (1985). K. Wödkiewicz, J. Mod. Opt. 34, 941 (1987).
105. K. Wódkiewicz and J. H. Eberly, J. Opt. Soc. Am. B 2,458 (1985). K. Wödkiewicz, J. Mod. Opt. 34, 941 (1987).
106. B. L. Schumaker and C. M. Caves, Phys. Rev. A 31, 3093 (1985). B. L. Schumaker, Phys. Rep. 135, 317 (1986).
107. B. L. Schumaker and C. M. Caves, Phys. Rev. A 31, 3093 (1985). B. L. Schumaker, Phys. Rep. 135, 317 (1986).
108. C. C. Gerry, Phys. Rev. A 31, 2721 (1985); 35, 2146 (1987); 38, 1734 (1988).
109. C. C. Gerry, Phys. Rev. A 31, 2721 (1985); 35, 2146 (1987); 38, 1734 (1988).
110. C. C. Gerry, Phys. Rev. A 31, 2721 (1985); 35, 2146 (1987); 38, 1734 (1988).
111. R. F. Bishop and A. Vourdas, J. Phys. A, Math. Gen. 19, 2525 (1986); 20, 3727 (1987).
112. R. F. Bishop and A. Vourdas, J. Phys. A, Math. Gen. 19, 2525 (1986); 20, 3727 (1987).
113. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
114. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
115. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
116. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
117. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
118. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
119. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
120. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
121. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
122. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
123. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
124. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early
125. E. H. Kennard, Z. Phys. 44, 326 (1927). H. Takahashi, in: Advances in Communication Systems, Theory and Applications, Vol. 1, edited by A. V. Balakrishnan (Academic Press, New York and London, 1965), p. 227. D. Stoler, Phys. Rev. D 1, 3217 (1970); 4, 1925 (1971). E. Y. C. Lu, Lett. Nuovo Cimento 2, 1241 (1971); 3, 585 (1972). H. P. Yuen, Phys. Rev. A 13, 2226 (1976). J. N. Hollenhorst, Phys. Rev. D 19, 1669 (1979). For more see the textbooks [11-19] and the vast literature on squeezed states quoted by Dodonov in [10]. See also Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics, Lecture Notes in Physics Series, Vol. 40 (World Scientific, Singapore, 1991). Early reviews on squeezed states are D. F. Walls, Nature 306, 141 (1983). R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987). R. W. Henry and S. C. Glotzer, Am. J. Phys. 56, 318 (1988).
126. R. J. Glauber, in: Quantum Optics and Electronics, XIV. Les Houches Summer School 1964, edited by C. De-Witt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965), p. 63; in: Fundamental Problems in Statistical Mechanics II, Proceedings of the 2nd NUFFIC Int. Summer Course at Noordwijk, The Netherlands, 1967, edited by E. G. D. Cohen (North-Holland Publ. Co., Amsterdam, 1968), p. 140.
127. R. J. Glauber, in: Quantum Optics and Electronics, XIV. Les Houches Summer School 1964, edited by C. De-Witt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965), p. 63; in: Fundamental Problems in Statistical Mechanics II, Proceedings of the 2nd NUFFIC Int. Summer Course at Noordwijk, The Netherlands, 1967, edited by E. G. D. Cohen (North-Holland Publ. Co., Amsterdam, 1968), p. 140.
128. J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics (W.A. Benjamin, Inc., New York, Amsterdam, 1968).
129. J. R. Klauder and B.-S. Skagerstam, Coherent States - Applications in Physics and Mathematical Physics (World Scientific Publ. Co., Singapore, 1985).
130. W.-M. Zhang, D. H. Feng, and R. Gilmore, Rev. Mod. Phys. 62, 867 (1990).
131. S. T. Ali, J.-P. Antoine, J.-P. Gazeau, and U. A. Mueller, Rev. Math. Phys. 7, 1013 (1995).
132. D. A. Trifonov, J. Opt. Soc. Am. A 17, 2486 (2000), quant-ph/0012072.
133. B. C. Hall, Contemp. Math. 260, 1 (2000), quant-ph/9912054.
134. E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). H. P. Yuen, Phys. Lett. A 91, 101 (1982). Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986). S. L. Braunstein, C. M. Caves, and G. J. Milburn, Phys. Rev. A 43, 1153 (1991). S. Stenholm, Ann. Phys. (USA) 218, 233 (1992). U. Leonhardt and H. Paul, J. Mod. Opt. 40, 1745 (1993). M. G. Raymer, Am. J. Phys. 68, 986 (1994). M. Freiberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). U. Leonhardt, B. Bohmer, and H. Paul, Opt. Commun. 119, 296 (1995).
135. E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). H. P. Yuen, Phys. Lett. A 91, 101 (1982). Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986). S. L. Braunstein, C. M. Caves, and G. J. Milburn, Phys. Rev. A 43, 1153 (1991). S. Stenholm, Ann. Phys. (USA) 218, 233 (1992). U. Leonhardt and H. Paul, J. Mod. Opt. 40, 1745 (1993). M. G. Raymer, Am. J. Phys. 68, 986 (1994). M. Freiberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). U. Leonhardt, B. Bohmer, and H. Paul, Opt. Commun. 119, 296 (1995).
136. E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). H. P. Yuen, Phys. Lett. A 91, 101 (1982). Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986). S. L. Braunstein, C. M. Caves, and G. J. Milburn, Phys. Rev. A 43, 1153 (1991). S. Stenholm, Ann. Phys. (USA) 218, 233 (1992). U. Leonhardt and H. Paul, J. Mod. Opt. 40, 1745 (1993). M. G. Raymer, Am. J. Phys. 68, 986 (1994). M. Freiberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). U. Leonhardt, B. Bohmer, and H. Paul, Opt. Commun. 119, 296 (1995).
137. E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). H. P. Yuen, Phys. Lett. A 91, 101 (1982). Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986). S. L. Braunstein, C. M. Caves, and G. J. Milburn, Phys. Rev. A 43, 1153 (1991). S. Stenholm, Ann. Phys. (USA) 218, 233 (1992). U. Leonhardt and H. Paul, J. Mod. Opt. 40, 1745 (1993). M. G. Raymer, Am. J. Phys. 68, 986 (1994). M. Freiberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). U. Leonhardt, B. Bohmer, and H. Paul, Opt. Commun. 119, 296 (1995).
138. E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). H. P. Yuen, Phys. Lett. A 91, 101 (1982). Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986). S. L. Braunstein, C. M. Caves, and G. J. Milburn, Phys. Rev. A 43, 1153 (1991). S. Stenholm, Ann. Phys. (USA) 218, 233 (1992). U. Leonhardt and H. Paul, J. Mod. Opt. 40, 1745 (1993). M. G. Raymer, Am. J. Phys. 68, 986 (1994). M. Freiberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). U. Leonhardt, B. Bohmer, and H. Paul, Opt. Commun. 119, 296 (1995).
139. E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). H. P. Yuen, Phys. Lett. A 91, 101 (1982). Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986). S. L. Braunstein, C. M. Caves, and G. J. Milburn, Phys. Rev. A 43, 1153 (1991). S. Stenholm, Ann. Phys. (USA) 218, 233 (1992). U. Leonhardt and H. Paul, J. Mod. Opt. 40, 1745 (1993). M. G. Raymer, Am. J. Phys. 68, 986 (1994). M. Freiberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). U. Leonhardt, B. Bohmer, and H. Paul, Opt. Commun. 119, 296 (1995).
140. E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). H. P. Yuen, Phys. Lett. A 91, 101 (1982). Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986). S. L. Braunstein, C. M. Caves, and G. J. Milburn, Phys. Rev. A 43, 1153 (1991). S. Stenholm, Ann. Phys. (USA) 218, 233 (1992). U. Leonhardt and H. Paul, J. Mod. Opt. 40, 1745 (1993). M. G. Raymer, Am. J. Phys. 68, 986 (1994). M. Freiberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). U. Leonhardt, B. Bohmer, and H. Paul, Opt. Commun. 119, 296 (1995).
141. E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). H. P. Yuen, Phys. Lett. A 91, 101 (1982). Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986). S. L. Braunstein, C. M. Caves, and G. J. Milburn, Phys. Rev. A 43, 1153 (1991). S. Stenholm, Ann. Phys. (USA) 218, 233 (1992). U. Leonhardt and H. Paul, J. Mod. Opt. 40, 1745 (1993). M. G. Raymer, Am. J. Phys. 68, 986 (1994). M. Freiberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). U. Leonhardt, B. Bohmer, and H. Paul, Opt. Commun. 119, 296 (1995).
142. E. Arthurs and J. L. Kelly, Jr., Bell Syst. Tech. J. 44, 725 (1965). H. P. Yuen, Phys. Lett. A 91, 101 (1982). Y. Yamamoto and H. A. Haus, Rev. Mod. Phys. 58, 1001 (1986). S. L. Braunstein, C. M. Caves, and G. J. Milburn, Phys. Rev. A 43, 1153 (1991). S. Stenholm, Ann. Phys. (USA) 218, 233 (1992). U. Leonhardt and H. Paul, J. Mod. Opt. 40, 1745 (1993). M. G. Raymer, Am. J. Phys. 68, 986 (1994). M. Freiberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). U. Leonhardt, B. Bohmer, and H. Paul, Opt. Commun. 119, 296 (1995).
143. A. Erdélyi et al. (Eds.), Higher Transcendental Functions I (McGraw-Hill Book Co., Inc., New York etc., 1953), p. 52.
144. J. Spanier and K. B. Oldham, An Atlas of Functions (Hemisphere Publ. Corpor. - Taylor & Francis Group -, New York etc., 1987), Chap. 18.
145. A. Erdélyi et al. (eds.), Higher Transcendental Functions II (McGraw-Hill Book Co., Inc., New York etc., 1953), Chap. VII.
146. [74], Chap. IV.
147. E. Hille, Analytic Function Theory II, 2nd ed. (Chelsea Publ. Co., New York, 1987), Chap. 14.
148. [76], p. 51, Eq. (27).
149. N. Aronszajn, Trans. Amer. Math. Soc. 68, 337 (1950).
150. V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961); 20, 1 (1967).
151. V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961); 20, 1 (1967).
152. K.-H. Neeb, Holomorphy and Convexity in Lie Theory (Walter de Gruyter, Berlin, 2000).
153. [74], p. 251, Eq. (13).
154. [74], Chap. VI.
155. I. Segal, Mathematical problems of relativistic physics, in: Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, Vol. II, Colorado, 1960, edited by M. Kac (Amer. Math. Soc., Providence, R.I., 1963); Illinois J. Math. 6, 500 (1962).
156. I. Segal, Mathematical problems of relativistic physics, in: Lectures in Applied Mathematics, Proceedings of the Summer Seminar, Boulder, Vol. II, Colorado, 1960, edited by M. Kac (Amer. Math. Soc., Providence, R.I., 1963); Illinois J. Math. 6, 500 (1962).
157. H. Paul, Photonen, 2., durchges. Aufl. (B.G. Teubner, Stuttgart und Leipzig, 1999), p. 179.
158. See [12], p. 627.
159. G. S. Agarwal, J. Opt. Soc. Am. B 5, 1940 (1988).
160. V. Bužek, J. Mod. Opt. 37, 303 (1990).
161. A. Joshi and R. R. Puri, Phys. Rev. A 42, 4336 (1990).
162. C. C. Gerry and R. F. Welch, J. Opt. Soc. Am. B 8, 868 (1991).
163. M. Ban, J. Math. Phys. 33, 3213 (1992); J. Opt. Soc. Am. B 10, 1347 (1993); Phys. Rev. A 47, 5093 (1993), the exponent in Eq. (2.6) has to be -2 instead of 2.
164. M. Ban, J. Math. Phys. 33, 3213 (1992); J. Opt. Soc. Am. B 10, 1347 (1993); Phys. Rev. A 47, 5093 (1993), the exponent in Eq. (2.6) has to be -2 instead of 2.
165. M. Ban, J. Math. Phys. 33, 3213 (1992); J. Opt. Soc. Am. B 10, 1347 (1993); Phys. Rev. A 47, 5093 (1993), the exponent in Eq. (2.6) has to be -2 instead of 2.
166. D. A. Trifonov, J. Math. Phys. 35, 2297 (1994).
167. G. S. Prakash and G. S. Agarwal, Phys. Rev. A 50, 4258 (1994).
168. M. Ban, Phys. Lett. A 193, 121 (1994).
169. C. Brif and Y. Ben-Aryeh, Quantum Opt. 6, 391 (1994); J. Phys. A, Math. Gen. 27, 8185 (1994).
170. C. Brif and Y. Ben-Aryeh, Quantum Opt. 6, 391 (1994); J. Phys. A, Math. Gen. 27, 8185 (1994).
171. B. A. Bambah and G. S. Agarwal, Phys. Rev. A 51, 4918 (1995).
172. C. C. Gerry and R. Grobe, Phys. Rev. A 51, 1698 (1995).
173. C. C. Gerry and R. Grobe, Phys. Rev. A 51, 4123 (1995).
174. C. Brif, Quantum Semiclass. Opt. 7, 803 (1995); Ann. Phys. (USA) 251, 180 (1996); Int. J. Theor. Phys. 36, 1651 (1997).
175. C. Brif, Quantum Semiclass. Opt. 7, 803 (1995); Ann. Phys. (USA) 251, 180 (1996); Int. J. Theor. Phys. 36, 1651 (1997).
176. C. Brif, Quantum Semiclass. Opt. 7, 803 (1995); Ann. Phys. (USA) 251, 180 (1996); Int. J. Theor. Phys. 36, 1651 (1997).
177. C. Brif and A. Mann, Phys. Lett. A 219, 257 (1996); Quantum Semiclass. Opt. 9, 899 (1997).
178. C. Brif and A. Mann, Phys. Lett. A 219, 257 (1996); Quantum Semiclass. Opt. 9, 899 (1997).
179. C. Brif, A. Vourdas, and A. Mann, J. Phys. A, Math. Gen. 29, 5873 and 5587 (1996).
180. D. A. Trifonov, J. Phys. A, Math. Gen. 30, 5941 (1997); 31, 5673 (1998).
181. D. A. Trifonov, J. Phys. A, Math. Gen. 30, 5941 (1997); 31, 5673 (1998).
182. V. Peřinová, A. Lukš, and J. Křepelka, J. Opt. B: Semiclass. Opt. 2, 81 (2000).
183. X. Wang, B. Sanders, and S.-h. Pan, J. Phys. A, Math. Gen. 33, 7451 (2000).
184. M. S. Abdalla, F. A. A. El-Orany, and J. Peřina, Acta Phys. Slovaca 50, 613 (2000).
185. [74], p. 52, formulae (5) and (6).
186. [75], p. 45, formula 6:5:1 and p. 150, formula 18:3:3.
187. S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York and London, 1962), p. 316, Lemma 7.7; [33], p. 387, Lemma 7.7.
188. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972). R. Gilmore, Rev. Mex. Fis. 23, 143 (1974); J. Math. Phys. 15, 2090 (1974); Lie Groups, Lie Algebras, and Some of Their Applications (John Wiley & Sons, New York etc., 1974), Chap. 5, Sec. VI.
189. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972). R. Gilmore, Rev. Mex. Fis. 23, 143 (1974); J. Math. Phys. 15, 2090 (1974); Lie Groups, Lie Algebras, and Some of Their Applications (John Wiley & Sons, New York etc., 1974), Chap. 5, Sec. VI.
190. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972). R. Gilmore, Rev. Mex. Fis. 23, 143 (1974); J. Math. Phys. 15, 2090 (1974); Lie Groups, Lie Algebras, and Some of Their Applications (John Wiley & Sons, New York etc., 1974), Chap. 5, Sec. VI.
191. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972). R. Gilmore, Rev. Mex. Fis. 23, 143 (1974); J. Math. Phys. 15, 2090 (1974); Lie Groups, Lie Algebras, and Some of Their Applications (John Wiley & Sons, New York etc., 1974), Chap. 5, Sec. VI.
192. A. M. Perelomov, Sov. Phys.-Usp. 20, 703 (1978).
193. A. Wünsche, [52].
194. X.-G. Wang, Int. J. Mod. Phys. B 14, 1093 (2000).
195. V. I. Man'ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, in: Proceedings of the IV. Wigner Symposium, Guadalajara 1995, edited by N. Atakishiyev, T. Seligman, and K. B. Wolf (World Scientific, Singapore, 1996), p. 421; Phys. Scr. 55, 528 (1997).
196. V. I. Man'ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, in: Proceedings of the IV. Wigner Symposium, Guadalajara 1995, edited by N. Atakishiyev, T. Seligman, and K. B. Wolf (World Scientific, Singapore, 1996), p. 421; Phys. Scr. 55, 528 (1997).
197. R. L. de Matos Filho and W. Vogel, Phys. Rev. A 54, 4560 (1996).
198. E. C. Lerner, H. W. Huang, and G. E. Walters, J. Math. Phys. 11, 1679 (1970). Y. Aharonov, H. W. Huang, J. M. Knight, and E. C. Lerner, Lett. Nuovo Cimento 2, 1317 (1971). Y. Aharonov, E. C. Lerner, H. W. Huang, and J. M. Knight, J. Math. Phys. 14, 746 (1973).
199. E. C. Lerner, H. W. Huang, and G. E. Walters, J. Math. Phys. 11, 1679 (1970). Y. Aharonov, H. W. Huang, J. M. Knight, and E. C. Lerner, Lett. Nuovo Cimento 2, 1317 (1971). Y. Aharonov, E. C. Lerner, H. W. Huang, and J. M. Knight, J. Math. Phys. 14, 746 (1973).
200. E. C. Lerner, H. W. Huang, and G. E. Walters, J. Math. Phys. 11, 1679 (1970). Y. Aharonov, H. W. Huang, J. M. Knight, and E. C. Lerner, Lett. Nuovo Cimento 2, 1317 (1971). Y. Aharonov, E. C. Lerner, H. W. Huang, and J. M. Knight, J. Math. Phys. 14, 746 (1973).
201. J.-M. Lévy-Leblond, Ann. Phys. (USA) 101, 319 (1976).
202. J. H. Shapiro and S. R. Shepard, Phys. Rev. A 43, 3795 (1991). G. S. Agarwal, Phys. Rev. A 44, 8398 (1991).
203. J. H. Shapiro and S. R. Shepard, Phys. Rev. A 43, 3795 (1991). G. S. Agarwal, Phys. Rev. A 44, 8398 (1991).
204. A. Vourdas, Phys. Rev. A 45, 1943 (1992). G. S. Agarwal, Opt. Commun. 100, 479 (1993).
205. A. Vourdas, Phys. Rev. A 45, 1943 (1992). G. S. Agarwal, Opt. Commun. 100, 479 (1993).
206. G. S. Agarwal, Phys. Rev. A 45, 1787 (1992).
207. See [74], p. 9.
208. D. Basu, Proc. R. Soc. Lond. A 455, 975 (1999); see also the Refs. therein.
209. [75], p. 151, formula 18:5:6.
210. See [74], p. 27-31.
211. L. Lewin, Polylogarithms and Associated Functions (North Holland, New York, Oxford, 1981). See also http://functions.wolfram.com/ZetafunctionsandPolylogarithms.
212. See [74], p. 115.
213. E. W. Barnes, Proc. London Math. Soc. (Ser. 2) 4, 284 (1906).
214. See, e.g. Messiah, [34], Chap. XII and Appendix B III.
215. J. C. Garrison and J. Wong, J. Math. Phys. 11, 2242 (1970).
216. R. Jost, The General Theory of Quantized Fields (Amer. Math. Soc. Providence, Rhode Island, 1965), Chap. II.3.
217. D. T. Pegg and S. M. Barnett, Europhys. Lett. 6, 483 (1988). S. M. Barnett and D. T. Pegg, J. Mod. Opt. 36, 7 (1989). D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1615 (1989).
218. D. T. Pegg and S. M. Barnett, Europhys. Lett. 6, 483 (1988). S. M. Barnett and D. T. Pegg, J. Mod. Opt. 36, 7 (1989). D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1615 (1989).
219. D. T. Pegg and S. M. Barnett, Europhys. Lett. 6, 483 (1988). S. M. Barnett and D. T. Pegg, J. Mod. Opt. 36, 7 (1989). D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1615 (1989).
220. [8], Chap. 15. [16], Chaps. 10 and 16. See also M. G. A. Paris, Fizika B 6, 63 (1997).
221. [8], Chap. 15. [16], Chaps. 10 and 16. See also M. G. A. Paris, Fizika B 6, 63 (1997).
222. See, e.g. I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., edited by A. Jeffrey and D. Zwillinger (Academic Press, San Diego etc. 2000), p. 62.
223. ± and χk,n(ψ) have been chosen differently.
224. C. P. Boyer and K. B. Wolf, J. Math. Phys. 16, 1493 (1975).
225. E. C. Lerner, Nuovo Cimento B 56, 183 (1968); Errat. Nuovo Cimento B 57, 251 (1968).
226. E. C. Lerner, Nuovo Cimento B 56, 183 (1968); Errat. Nuovo Cimento B 57, 251 (1968).
227. R. Lynch, J. Opt. Soc. Am. B 3, 1006 (1986).
228. R. Simon and N. Mukunda, Opt. Commun. 95, 39 (1993).
229. E. Merzbacher, Quantum Mechanics, 2nd Ed. (J. Wiley & Sons, Inc. New York etc., 1970), Chap. 8, § 8.
230. D. Stoler, [65].
231. E. Y. C. Lu, [65].
232. H. P. Yuen, [65].
233. J. N. Hollenhorst, [65].
234. H. P. Robertson, Phys. Rev. 34, 163 (1929); Phys. Rev. 35, 667 (1930).
235. H. P. Robertson, Phys. Rev. 34, 163 (1929); Phys. Rev. 35, 667 (1930).
236. E. Schrödinger, Sitz.ber. Preuss. Akad. Wiss. Berl., Phys.-Math. Kl. 1930, p. 296.
237. R. Jackiw, J. Math. Phys. 9, 339 (1968).
238. T. F. Jordan, Linear Operators for Quantum Mechanics (John Wiley & Sons, Inc., New York etc., 1969), Chap. V.
239. [139], Chap. 8, § 6.
240. V. V. Dodonov, E. V. Kurmyshev, and V. I. Man'ko, Phys. Lett. A 79, 150 (1980).
241. C. Brif and A. Mann, Quantum Semiclass. Opt. 9, 899 (1997).
242. see, e.g. E. Hewitt and K. Stromberg, Real and Abstract Analysis (Springer-Verlag, Berlin etc., 1969), p. 234.
243. S. Goshen (Goldstein) and H. J. Lipkin, Ann. Phys. (USA) 6, 301 (1959). H. J. Lipkin, Lie Groups for Pedestrians (North-Holland Publ. Co., Amsterdam, 1965), Chap. 5.
244. S. Goshen (Goldstein) and H. J. Lipkin, Ann. Phys. (USA) 6, 301 (1959). H. J. Lipkin, Lie Groups for Pedestrians (North-Holland Publ. Co., Amsterdam, 1965), Chap. 5.
245. This case is a special one in Bargmann's analysis [263] of constructing Lie algebras s-fraktur sign p-fraktur sign(2n, ℝ) in terms of creation and anihilation operators.
246. U. Niederer, Helv. Phys. Acta 46, 191 (1973).
247. M. Moshinsky, SIAM J. Appl. Math. 25, 193 (1973).
248. R. H. Cushman and L. M. Bates, [49], p. 315, Example 4.
249. C. Brif, A. Mann, and A. Vourdas, J. Phys. A, Math. Gen. 29, 2053 (1996).
250. M. Hillery, Opt. Commun. 62, 135 (1987); Phys. Rev. A 36, 3796 (1987); Phys. Rev. A 40, 3147 (1989).
251. M. Hillery, Opt. Commun. 62, 135 (1987); Phys. Rev. A 36, 3796 (1987); Phys. Rev. A 40, 3147 (1989).
252. M. Hillery, Opt. Commun. 62, 135 (1987); Phys. Rev. A 36, 3796 (1987); Phys. Rev. A 40, 3147 (1989).
253. C. C. Gerry and E. R. Vrscay, Phys. Rev. 37, 1779 (1988).
254. D. Yu and M. Hillery, Quantum Opt. 6, 37 (1994).
255. W. J. Holman III and L. C. Biedenharn Jr., Ann. Phys. (USA) 39, 1 (1966).
256. A. O. Barut, in: Mathematical Methods in Theoretical Physics, Boulder Lectures in Theoretical Physics, Vol. IX A, edited by W. E. Brittin, A. O. Barut, and M. Guenin (Gordon and Breach, Science Publishers, Inc., New York 1967), p. 125.
257. A. I. Solomon, J. Math. Phys. 12, 390 (1971).
258. R. F. Bishop and A. Vourdas, Z. Physik B - Condensed Matter 71, 527 (1988).
259. J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). J. Katriel, M. Rasetti, and A. I. Solomon, Phys. Rev. D 35, 1248 (1987). G. D'Ariano, S. Morosi, M. Rasetti, J. Katriel, and A. I. Solomon, Phys. Rev. D 36, 2399 (1987).
260. J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). J. Katriel, M. Rasetti, and A. I. Solomon, Phys. Rev. D 35, 1248 (1987). G. D'Ariano, S. Morosi, M. Rasetti, J. Katriel, and A. I. Solomon, Phys. Rev. D 36, 2399 (1987).
261. J. Katriel, A. I. Solomon, G. D'Ariano, and M. Rasetti, Phys. Rev. D 34, 2332 (1986). J. Katriel, M. Rasetti, and A. I. Solomon, Phys. Rev. D 35, 1248 (1987). G. D'Ariano, S. Morosi, M. Rasetti, J. Katriel, and A. I. Solomon, Phys. Rev. D 36, 2399 (1987).
262. B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986). U. Leonhardt, Phys. Rev. A 49, 1231 (1994). C. Brif and Y. Ben-Aryeh, Quantum Semiclass. Opt. 8, 1 (1996). C. Brif and A. Mann, Phys. Lett. A 219, 257 (1996) ; Phys. Rev. A 54, 4505 (1996). V. Peřinová, A. Lukš, and J. Křepelka, J. Opt. B, Quantum Semiclassical Opt. 2, 81 (2000).
263. B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986). U. Leonhardt, Phys. Rev. A 49, 1231 (1994). C. Brif and Y. Ben-Aryeh, Quantum Semiclass. Opt. 8, 1 (1996). C. Brif and A. Mann, Phys. Lett. A 219, 257 (1996) ; Phys. Rev. A 54, 4505 (1996). V. Peřinová, A. Lukš, and J. Křepelka, J. Opt. B, Quantum Semiclassical Opt. 2, 81 (2000).
264. B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986). U. Leonhardt, Phys. Rev. A 49, 1231 (1994). C. Brif and Y. Ben-Aryeh, Quantum Semiclass. Opt. 8, 1 (1996). C. Brif and A. Mann, Phys. Lett. A 219, 257 (1996) ; Phys. Rev. A 54, 4505 (1996). V. Peřinová, A. Lukš, and J. Křepelka, J. Opt. B, Quantum Semiclassical Opt. 2, 81 (2000).
265. B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986). U. Leonhardt, Phys. Rev. A 49, 1231 (1994). C. Brif and Y. Ben-Aryeh, Quantum Semiclass. Opt. 8, 1 (1996). C. Brif and A. Mann, Phys. Lett. A 219, 257 (1996) ; Phys. Rev. A 54, 4505 (1996). V. Peřinová, A. Lukš, and J. Křepelka, J. Opt. B, Quantum Semiclassical Opt. 2, 81 (2000).
266. B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986). U. Leonhardt, Phys. Rev. A 49, 1231 (1994). C. Brif and Y. Ben-Aryeh, Quantum Semiclass. Opt. 8, 1 (1996). C. Brif and A. Mann, Phys. Lett. A 219, 257 (1996) ; Phys. Rev. A 54, 4505 (1996). V. Peřinová, A. Lukš, and J. Křepelka, J. Opt. B, Quantum Semiclassical Opt. 2, 81 (2000).
267. B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A 33, 4033 (1986). U. Leonhardt, Phys. Rev. A 49, 1231 (1994). C. Brif and Y. Ben-Aryeh, Quantum Semiclass. Opt. 8, 1 (1996). C. Brif and A. Mann, Phys. Lett. A 219, 257 (1996) ; Phys. Rev. A 54, 4505 (1996). V. Peřinová, A. Lukš, and J. Křepelka, J. Opt. B, Quantum Semiclassical Opt. 2, 81 (2000).
268. See, e.g. [11], Chap. 5. [12], Chap. 22. [15], Chap. 5.4. [17], Chap. 8.
269. E. P. Wigner, Phys. Rev. 40, 749 (1932); in a footnote Wigner says that "the expression (7.4) was found by L. Szilard and the present author some years ago for another purpose". Important follow-up papers are: J. E. Moyal, Proc. Camb. Philos. Soc. 45, 99 (1949). R. L. Stratonovich, Soviet Phys.-JETP 4, 891 (1957); 5, 1206 (1957).
270. E. P. Wigner, Phys. Rev. 40, 749 (1932); in a footnote Wigner says that "the expression (7.4) was found by L. Szilard and the present author some years ago for another purpose". Important follow-up papers are: J. E. Moyal, Proc. Camb. Philos. Soc. 45, 99 (1949). R. L. Stratonovich, Soviet Phys.-JETP 4, 891 (1957); 5, 1206 (1957).
271. E. P. Wigner, Phys. Rev. 40, 749 (1932); in a footnote Wigner says that "the expression (7.4) was found by L. Szilard and the present author some years ago for another purpose". Important follow-up papers are: J. E. Moyal, Proc. Camb. Philos. Soc. 45, 99 (1949). R. L. Stratonovich, Soviet Phys.-JETP 4, 891 (1957); 5, 1206 (1957).
272. E. P. Wigner, Phys. Rev. 40, 749 (1932); in a footnote Wigner says that "the expression (7.4) was found by L. Szilard and the present author some years ago for another purpose". Important follow-up papers are: J. E. Moyal, Proc. Camb. Philos. Soc. 45, 99 (1949). R. L. Stratonovich, Soviet Phys.-JETP 4, 891 (1957); 5, 1206 (1957).
273. V. I. Tatarskiǐ, Sov. Phys.-Usp. 26, 311 (1983).
274. M. Hillery, R. F. O'Connel, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121 (1984).
275. G. B. Folland, [38], Chap. 1.
276. Y. S. Kim and M. E. Noz, [65].
277. H.-W. Lee, Phys. Rep. 259, 147 (1995).
278. M. Gadella, Fortschr. Physik 43, 229 (1995).
279. D. Dragoman, Prog. Opt. 43, 433 (2002).
280. G. E. Uhlenbeck, J. Math. Phys. 14, 10 (1934).
281. K. Husimi, Proc. Physico-Mathematical Soc. Jpn. (3rd Ser.) 22, 264 (1940); related papers by Husimi are: Prog. Theor. Phys. 9, 238 and 381 (1953).
282. K. Husimi, Proc. Physico-Mathematical Soc. Jpn. (3rd Ser.) 22, 264 (1940); related papers by Husimi are: Prog. Theor. Phys. 9, 238 and 381 (1953).
283. Y. Kano, Proc. Phys. Soc. Jpn. 19, 1555 (1964); J. Math. Phys. 6, 1913 (1965).
284. Y. Kano, Proc. Phys. Soc. Jpn. 19, 1555 (1964); J. Math. Phys. 6, 1913 (1965).
285. E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
286. E. C. G. Sudarshan, Int. J. Theor. Phys. 32, 1069 (1993).
287. R. J. Glauber, Phys. Rev. 131, 2766 (1963).
288. J. R. Klauder, J. McKenna, and D. G. Currie, J. Math. Phys. 5, 734 (1965). J. R. Klauder, Phys. Rev. Lett. 16, 534 (1966).
289. J. R. Klauder, J. McKenna, and D. G. Currie, J. Math. Phys. 5, 734 (1965). J. R. Klauder, Phys. Rev. Lett. 16, 534 (1966).
290. C. L. Mehta and E. C. G. Sudarshan, Phys. Rev. B 138, 274 (1965).
291. J. R. Klauder and E. C. G. Sudarshan, [67], Chap. 8-4.
292. M. M. Miller and E. A. Mishkin, Phys. Rev. 164, 1610 (1967). M. M. Miller, J. Math. Phys. 9, 1270 (1968).
293. M. M. Miller and E. A. Mishkin, Phys. Rev. 164, 1610 (1967). M. M. Miller, J. Math. Phys. 9, 1270 (1968).
294. I. M. Gel'fand and G. E. Shilov, Generalized Functions Vol. I (Academic Press, New York and London, 1964), Chap. II; Vol. II (Academic Press, New York and London, 1968), Chap. III.
295. A. H. Zemanian, Distribution Theory and Transform Analysis (McGraw-Hill Book Co., New York etc., 1965), Sects. 7.6-7.10.
296. E. T. Jaynes, Phys. Rev. 106, 620 (1957); 108, 171 (1957); in: Statistical Physics, Vol. 3 of the 1962 Brandeis Summer Institute Lectures in Theor. Physics, edited by K. W. Ford (W.A. Benjamin, Inc., New York and Amsterdam, 1963), p. 181.
297. E. T. Jaynes, Phys. Rev. 106, 620 (1957); 108, 171 (1957); in: Statistical Physics, Vol. 3 of the 1962 Brandeis Summer Institute Lectures in Theor. Physics, edited by K. W. Ford (W.A. Benjamin, Inc., New York and Amsterdam, 1963), p. 181.
298. E. T. Jaynes, Phys. Rev. 106, 620 (1957); 108, 171 (1957); in: Statistical Physics, Vol. 3 of the 1962 Brandeis Summer Institute Lectures in Theor. Physics, edited by K. W. Ford (W.A. Benjamin, Inc., New York and Amsterdam, 1963), p. 181.
299. G. S. Agarwal and E. Wolf, Phys. Rev. D 2, 2161, 2187, 2206 (1970).
300. H.-Yi Fan, Y. Fan, and J.-h. Chen, J. Opt. B, Quantum Semiclassical Opt. 4, 336 (2002).
301. F. Rocca, J. Phys. (France) 28, 113 (1967).
302. J. Peřina and L. Mišta, Phys. Lett. A 27, 217 (1968); Ann. Phys. (Germany) 22(7), 372 (1969). See also J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, 2nd completely rev. ed. (Kluwer Academic Publ., Dordrecht etc., 1991), Sect. 4.3.
303. J. Peřina and L. Mišta, Phys. Lett. A 27, 217 (1968); Ann. Phys. (Germany) 22(7), 372 (1969). See also J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, 2nd completely rev. ed. (Kluwer Academic Publ., Dordrecht etc., 1991), Sect. 4.3.
304. J. Peřina and L. Mišta, Phys. Lett. A 27, 217 (1968); Ann. Phys. (Germany) 22(7), 372 (1969). See also J. Peřina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, 2nd completely rev. ed. (Kluwer Academic Publ., Dordrecht etc., 1991), Sect. 4.3.
305. A. Lukš, Czech. J. Phys. B 26, 1095 (1976).
306. A. Peřina, Czech. J. Phys. B 21, 731 (1971).
307. A. Wünsche, Acta physica slovaca 48, 385 (1998); J. Phys. A, Math. Gen. 32, 3179 (1999).
308. A. Wünsche, Acta physica slovaca 48, 385 (1998); J. Phys. A, Math. Gen. 32, 3179 (1999).
309. Th. Richter, J. Mod. Optics 48, 1881 (2001).
310. C. L. Mehta and E. C. G. Sudarshan, Phys. Rev. B 138, 274 (1965).
311. K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 and 1882 (1969).
312. M. A. M. Santiago and A. N. Vaidya, J. Phys. A, Math. Gen. 9, 897 (1976) ; in the numerator of Eq. (2.20) a factor 2 is missing.
313. D. R. Truax, Phys. Rev. D 31, 1988 (1985).
314. W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Die Grundlehren der mathem. Wissenschaften, Vol. 52 (Springer-Verlag, Berlin etc., 1966), p. 70.
315. See, e.g. [187], p. 55, problem 10.
316. See [186], Vol. I, Appendix 1 of Chap. I. [187], Sect. 1.2.
317. [74], p. 187, Eq. (12).
318. [76], p. 176, Eq. (20); see also [74], Sect. 3.15.
319. C. Brif and A. Mann, J. Phys. A, Math. Gen. 31, L9 (1998); Phys. Rev. A 59, 971 (1999).
320. C. Brif and A. Mann, J. Phys. A, Math. Gen. 31, L9 (1998); Phys. Rev. A 59, 971 (1999).
321. S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces (Progress in Mathematics Vol. 13, Birkhäuser, Boston etc., 1981), § 4; [247], Introduction.
322. See [74], p. 175, Eq. (1).
323. C. L. Mehta, Phys. Rev. Lett. 18, 752 (1967); see also the first paper of [189].
324. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge Univ. Press, Cambridge, 1999).
325. Y. Choquet-Bruhat and C. DeWitt-Morette, Analysis, Manifolds and Physics, rev. ed. (North-Holland Publ. Co., Amsterdam, New York and Oxford, 1982), Chaps. III and IV.
326. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed. (Springer-Verlag, New York, Berlin and Heidelberg, 1999), Chap. 4.
327. P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, No. 2 (Yeshiva Univ., New York, 1964). Later reviews and monographs on the subject are: E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (John Wiley & Sons, New York etc., 1974), Chaps. 8 and 9. A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale del Lincei, Roma, 1976). K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, (Springer-Verlag, Berlin, Heidelberg and New York, 1982). M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, N.J., 1992). [212], Chap. 8. J. R. Klauder, in: Methods of Quantization, Lectures held at the 39. Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, Lecture notes in physics 572, edited by H. Latal and W. Schweiger (Springer-Verlag, Berlin etc., 2001), p. 143.
328. P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, No. 2 (Yeshiva Univ., New York, 1964). Later reviews and monographs on the subject are: E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (John Wiley & Sons, New York etc., 1974), Chaps. 8 and 9. A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale del Lincei, Roma, 1976). K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, (Springer-Verlag, Berlin, Heidelberg and New York, 1982). M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, N.J., 1992). [212], Chap. 8. J. R. Klauder, in: Methods of Quantization, Lectures held at the 39. Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, Lecture notes in physics 572, edited by H. Latal and W. Schweiger (Springer-Verlag, Berlin etc., 2001), p. 143.
329. P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, No. 2 (Yeshiva Univ., New York, 1964). Later reviews and monographs on the subject are: E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (John Wiley & Sons, New York etc., 1974), Chaps. 8 and 9. A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale del Lincei, Roma, 1976). K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, (Springer-Verlag, Berlin, Heidelberg and New York, 1982). M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, N.J., 1992). [212], Chap. 8. J. R. Klauder, in: Methods of Quantization, Lectures held at the 39. Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, Lecture notes in physics 572, edited by H. Latal and W. Schweiger (Springer-Verlag, Berlin etc., 2001), p. 143.
330. P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, No. 2 (Yeshiva Univ., New York, 1964). Later reviews and monographs on the subject are: E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (John Wiley & Sons, New York etc., 1974), Chaps. 8 and 9. A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale del Lincei, Roma, 1976). K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, (Springer-Verlag, Berlin, Heidelberg and New York, 1982). M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, N.J., 1992). [212], Chap. 8. J. R. Klauder, in: Methods of Quantization, Lectures held at the 39. Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, Lecture notes in physics 572, edited by H. Latal and W. Schweiger (Springer-Verlag, Berlin etc., 2001), p. 143.
331. P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, No. 2 (Yeshiva Univ., New York, 1964). Later reviews and monographs on the subject are: E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (John Wiley & Sons, New York etc., 1974), Chaps. 8 and 9. A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale del Lincei, Roma, 1976). K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, (Springer-Verlag, Berlin, Heidelberg and New York, 1982). M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, N.J., 1992). [212], Chap. 8. J. R. Klauder, in: Methods of Quantization, Lectures held at the 39. Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, Lecture notes in physics 572, edited by H. Latal and W. Schweiger (Springer-Verlag, Berlin etc., 2001), p. 143.
332. P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, No. 2 (Yeshiva Univ., New York, 1964). Later reviews and monographs on the subject are: E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (John Wiley & Sons, New York etc., 1974), Chaps. 8 and 9. A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale del Lincei, Roma, 1976). K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, (Springer-Verlag, Berlin, Heidelberg and New York, 1982). M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, N.J., 1992). [212], Chap. 8. J. R. Klauder, in: Methods of Quantization, Lectures held at the 39. Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, Lecture notes in physics 572, edited by H. Latal and W. Schweiger (Springer-Verlag, Berlin etc., 2001), p. 143.
333. P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, No. 2 (Yeshiva Univ., New York, 1964). Later reviews and monographs on the subject are: E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (John Wiley & Sons, New York etc., 1974), Chaps. 8 and 9. A. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale del Lincei, Roma, 1976). K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, (Springer-Verlag, Berlin, Heidelberg and New York, 1982). M. Henneaux and C. Teitelboim, Quantization of Gauge Systems (Princeton University Press, Princeton, N.J., 1992). [212], Chap. 8. J. R. Klauder, in: Methods of Quantization, Lectures held at the 39. Universitätswochen für Kern- und Teilchenphysik, Schladming, Austria, Lecture notes in physics 572, edited by H. Latal and W. Schweiger (Springer-Verlag, Berlin etc., 2001), p. 143.
334. R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed. (The Benjamin/Cummings Publ. Co., Inc., Reading, Mass, etc., 1978), pp. 301-302.
335. [214], pp. 722-723.
336. N. Steenrod, The Topology of Fiber Bundles (Princeton University Press, Princeton, N.J., 1951), p. 105.
337. R. H. Cushman and L. M. Bates, [49], Sect. 1.4.
338. W. P. Thurston, [49], Sect. 2.7.
339. R. Cushman and R. Sjamaar, in: Symplectic Geometry and Mathematical Physics, Actes du colloque en l'honneur de Jean-Marie Souriau, edited by P. Donato, C. Duval, J. Elhadad, and G. M. Tuynman (Birkhäuser, Boston, Basel and Berlin, 1991), p. 114. E. Lerman, R. Montgomery, R. Sjamaar, in: Symplectic Geometry, London Math. Soc. Lecture Note Series, No. 192, edited by D. Salamon (Cambridge Univ. Press, Cambridge, 1993), p. 127.
340. R. Cushman and R. Sjamaar, in: Symplectic Geometry and Mathematical Physics, Actes du colloque en l'honneur de Jean-Marie Souriau, edited by P. Donato, C. Duval, J. Elhadad, and G. M. Tuynman (Birkhäuser, Boston, Basel and Berlin, 1991), p. 114. E. Lerman, R. Montgomery, R. Sjamaar, in: Symplectic Geometry, London Math. Soc. Lecture Note Series, No. 192, edited by D. Salamon (Cambridge Univ. Press, Cambridge, 1993), p. 127.
341. [210], Sects. 6.2 and 6.3. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, paperback ed. (North-Holland Personal Library, North-Holland, Amsterdam etc., 1987), Chap. 1.
342. [210], Sects. 6.2 and 6.3. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, paperback ed. (North-Holland Personal Library, North-Holland, Amsterdam etc., 1987), Chap. 1.
343. N. G. Walker and J. E. Carroll, Electron. Lett. 20, 981 (1984); Opt. Quantum Electron. (UK) 18, 355 (1986). N. G. Walker, J. Mod. Opt. 34, 15 (1987).
344. N. G. Walker and J. E. Carroll, Electron. Lett. 20, 981 (1984); Opt. Quantum Electron. (UK) 18, 355 (1986). N. G. Walker, J. Mod. Opt. 34, 15 (1987).
345. N. G. Walker and J. E. Carroll, Electron. Lett. 20, 981 (1984); Opt. Quantum Electron. (UK) 18, 355 (1986). N. G. Walker, J. Mod. Opt. 34, 15 (1987).
346. L. Mandel, in: Foundations of Quantum Mechanics, Santa Fe Workshop 1991, edited by T. D. Black, M. M. Nieto, H. S. Pilloff, M. O. Scully, and R. M. Sinclair (World Scientific, Singapore etc., 1992), p. 55.
347. M. Freyberger and W. Schleich, Phys. Rev. A 47, R30 (1993). M. Freyberger, K. Vogel, and W. Schleich, Quantum Opt. 5, 65 (1993); Phys. Lett. A 176, 41 (1993). M. Freyberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). M. T. Fontanelle, S. L. Braunstein, W. P. Schleich, and M. Hillery, e-print quant-ph/9712032.
348. M. Freyberger and W. Schleich, Phys. Rev. A 47, R30 (1993). M. Freyberger, K. Vogel, and W. Schleich, Quantum Opt. 5, 65 (1993); Phys. Lett. A 176, 41 (1993). M. Freyberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). M. T. Fontanelle, S. L. Braunstein, W. P. Schleich, and M. Hillery, e-print quant-ph/9712032.
349. M. Freyberger and W. Schleich, Phys. Rev. A 47, R30 (1993). M. Freyberger, K. Vogel, and W. Schleich, Quantum Opt. 5, 65 (1993); Phys. Lett. A 176, 41 (1993). M. Freyberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). M. T. Fontanelle, S. L. Braunstein, W. P. Schleich, and M. Hillery, e-print quant-ph/9712032.
350. M. Freyberger and W. Schleich, Phys. Rev. A 47, R30 (1993). M. Freyberger, K. Vogel, and W. Schleich, Quantum Opt. 5, 65 (1993); Phys. Lett. A 176, 41 (1993). M. Freyberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). M. T. Fontanelle, S. L. Braunstein, W. P. Schleich, and M. Hillery, e-print quant-ph/9712032.
351. M. Freyberger and W. Schleich, Phys. Rev. A 47, R30 (1993). M. Freyberger, K. Vogel, and W. Schleich, Quantum Opt. 5, 65 (1993); Phys. Lett. A 176, 41 (1993). M. Freyberger, M. Heni, and W. P. Schleich, Quantum Semiclass. Opt. 7, 187 (1995). M. T. Fontanelle, S. L. Braunstein, W. P. Schleich, and M. Hillery, e-print quant-ph/9712032.
352. U. Leonhardt and H. Paul, Phys. Rev. A 48, 4598 (1993); Phys. Rev. Lett. 72, 4086 (1994). U. Leonhardt, B. Böhmer, and H. Paul, Opt. Commun. 119, 296 (1995). Th. Richter, Phys. Rev. A 56, 3134 (1997).
353. U. Leonhardt and H. Paul, Phys. Rev. A 48, 4598 (1993); Phys. Rev. Lett. 72, 4086 (1994). U. Leonhardt, B. Böhmer, and H. Paul, Opt. Commun. 119, 296 (1995). Th. Richter, Phys. Rev. A 56, 3134 (1997).
354. U. Leonhardt and H. Paul, Phys. Rev. A 48, 4598 (1993); Phys. Rev. Lett. 72, 4086 (1994). U. Leonhardt, B. Böhmer, and H. Paul, Opt. Commun. 119, 296 (1995). Th. Richter, Phys. Rev. A 56, 3134 (1997).
355. U. Leonhardt and H. Paul, Phys. Rev. A 48, 4598 (1993); Phys. Rev. Lett. 72, 4086 (1994). U. Leonhardt, B. Böhmer, and H. Paul, Opt. Commun. 119, 296 (1995). Th. Richter, Phys. Rev. A 56, 3134 (1997).
356. R. Loudon, in: Coherence, cooperation and fluctuations, Proceedings of the Sympos. 60th Birthday Prof. R. J. Glauber 1985, Cambridge Studies in Modern Optics: 5, edited by F. Haake, L. M. Narducci, and D. Walls (Cambridge Univ. Press, Cambridge etc., 1986), p. 240.
357. D.-G. Welsch, W. Vogel, and T. Opatrný, Prog. Opt. 39, 63 (1999).
358. T. Opatrný, in: Coherence and Statistics of Photons and Atoms, Wiley Series in Lasers and Applications 3, edited by J. Peřina (John Wiley & Sons, Inc., New York etc., 2001), p. 159.
359. See, e.g. the textbooks [15], Chap. 4.12. [17], Chap. 6.5.
360. [18], Chap. 13.
361. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
362. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
363. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
364. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
365. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
366. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
367. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
368. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
369. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
370. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
371. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
372. J. W. Noh, A. Fougères, and L. Mandel, Phys. Rev. Lett. 67, 1426 (1991); Phys. Rev. A 45, 424 (1992); Phys. Rev. A 46, 2840 (1992); Phys. Rev. A 47, 4535; 4541 (1993); Phys. Rev. Lett. 71, 2579 (1993); Phys. Rev. A 48, 1719 (1993); Phys. Scr. T48, 29 (1993). A. Fougères, J. W. Noh, T. P. Grayson, and L. Mandel, Phys. Rev. A 49, 530 (1994). A. Fougères, J. R. Torgerson, and L. Mandel, Opt. Commun. 105, 199 (1994). J. R. Torgerson and L. Mandel, Phys. Rev. Lett. 76, 3939 (1995); Opt. Commun. 133, 153 (1997); Phys. Scr. T76, 110 (1998).
373. D. T. Smithey, M. Beck, A. Faridani, and M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993). M. Beck, D. T. Smithey, and M. G. Raymer, Phys. Rev. A 48, R890 and 3159 (1993). M. G. Raymer, J. Cooper, H. J. Carmichael, M. Beck, and D. T. Smithey, J. Opt. Soc. Am. B 12, 1801 (1995).
374. D. T. Smithey, M. Beck, A. Faridani, and M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993). M. Beck, D. T. Smithey, and M. G. Raymer, Phys. Rev. A 48, R890 and 3159 (1993). M. G. Raymer, J. Cooper, H. J. Carmichael, M. Beck, and D. T. Smithey, J. Opt. Soc. Am. B 12, 1801 (1995).
375. D. T. Smithey, M. Beck, A. Faridani, and M. G. Raymer, Phys. Rev. Lett. 70, 1244 (1993). M. Beck, D. T. Smithey, and M. G. Raymer, Phys. Rev. A 48, R890 and 3159 (1993). M. G. Raymer, J. Cooper, H. J. Carmichael, M. Beck, and D. T. Smithey, J. Opt. Soc. Am. B 12, 1801 (1995).
376. J. Řeháček, Z. Hradil, M. Zawisky, S. Pascazio, H. Rauch, and J. Peřina, Phys. Rev. A 60, 473 (1999). J. Řeháček, Z. Hradil, M. Dušek, O. Haderka, and M. Hendrych, J. Opt. B, Quantum Semiclassical Opt. 2, 237 (2000).
377. J. Řeháček, Z. Hradil, M. Zawisky, S. Pascazio, H. Rauch, and J. Peřina, Phys. Rev. A 60, 473 (1999). J. Řeháček, Z. Hradil, M. Dušek, O. Haderka, and M. Hendrych, J. Opt. B, Quantum Semiclassical Opt. 2, 237 (2000).
378. T. Hakioǧlu, A. S. Shumovsky, and O. Aytür, Phys. Lett. A 194, 304 (1994). A. Luis and J. Per̈ina, Quantum Semiclass. Opt. 8, 873 and 887 (1996). T. Hakioǧlu, Phys. Rev. A 59, 1586 (1999). A. Cives-Esclop, A. Luis, and L. L. Sánchez-Soto, Opt. Commun. 175, 153 (2000); J. Opt. B, Quantum Semiclassical Opt. 2, 526 (2000).
379. T. Hakioǧlu, A. S. Shumovsky, and O. Aytür, Phys. Lett. A 194, 304 (1994). A. Luis and J. Per̈ina, Quantum Semiclass. Opt. 8, 873 and 887 (1996). T. Hakioǧlu, Phys. Rev. A 59, 1586 (1999). A. Cives-Esclop, A. Luis, and L. L. Sánchez-Soto, Opt. Commun. 175, 153 (2000); J. Opt. B, Quantum Semiclassical Opt. 2, 526 (2000).
380. T. Hakioǧlu, A. S. Shumovsky, and O. Aytür, Phys. Lett. A 194, 304 (1994). A. Luis and J. Per̈ina, Quantum Semiclass. Opt. 8, 873 and 887 (1996). T. Hakioǧlu, Phys. Rev. A 59, 1586 (1999). A. Cives-Esclop, A. Luis, and L. L. Sánchez-Soto, Opt. Commun. 175, 153 (2000); J. Opt. B, Quantum Semiclassical Opt. 2, 526 (2000).
381. T. Hakioǧlu, A. S. Shumovsky, and O. Aytür, Phys. Lett. A 194, 304 (1994). A. Luis and J. Per̈ina, Quantum Semiclass. Opt. 8, 873 and 887 (1996). T. Hakioǧlu, Phys. Rev. A 59, 1586 (1999). A. Cives-Esclop, A. Luis, and L. L. Sánchez-Soto, Opt. Commun. 175, 153 (2000); J. Opt. B, Quantum Semiclassical Opt. 2, 526 (2000).
382. T. Hakioǧlu, A. S. Shumovsky, and O. Aytür, Phys. Lett. A 194, 304 (1994). A. Luis and J. Per̈ina, Quantum Semiclass. Opt. 8, 873 and 887 (1996). T. Hakioǧlu, Phys. Rev. A 59, 1586 (1999). A. Cives-Esclop, A. Luis, and L. L. Sánchez-Soto, Opt. Commun. 175, 153 (2000); J. Opt. B, Quantum Semiclassical Opt. 2, 526 (2000).
383. J. R. Klauder, Ann. Phys. (USA) 254, 419 (1997). J. Govaerts and J. R. Klauder, Ann. Phys. (USA) 274, 251 (1999). J. R. Klauder, see [213].
384. J. R. Klauder, Ann. Phys. (USA) 254, 419 (1997). J. Govaerts and J. R. Klauder, Ann. Phys. (USA) 274, 251 (1999). J. R. Klauder, see [213].
385. J. R. Klauder, Ann. Phys. (USA) 254, 419 (1997). J. Govaerts and J. R. Klauder, Ann. Phys. (USA) 274, 251 (1999). J. R. Klauder, see [213].
386. J.-M. Souriau, Structure of Dynamical Systems, A Symplectic View of Physics, Progress in Mathematics 149 (Birkhäuser, Boston, Basel and Berlin, 1997); translation of Structure des systèmes dynamique (Dunod, Paris, 1970), § 11.
387. J.-M. Souriau, Structure of Dynamical Systems, A Symplectic View of Physics, Progress in Mathematics 149 (Birkhäuser, Boston, Basel and Berlin, 1997); translation of Structure des systèmes dynamique (Dunod, Paris, 1970), § 11.
388. [21], Chap. II.
389. [212], Chap. 11.
390. R. H. Cushman and L. M. Bates, [49], Appendix B.3.
391. J. E. Marsden and A. Weinstein, in: Quantization of Singular Symplectic Quotients, Progress in Mathematics 198, edited by N. P. Landsman, M. Pflaum, and M. Schlichenmaier (Birkhäuser Verlag, Basel, Boston and Berlin, 2001), p. 1.
392. M. Bojowald and T. Strobl, J. Math. Phys. 41, 2537 (2000); e-print quant-ph/9912048.
393. R. H. Cushman and L. M. Bates, [49], p. 315, example 5.
394. V. Bargmann, Ann. Math. 48, 568 (1947).
395. I. M. Gel'fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. 5; Integral Geometry and Representation Theory (Academic Press, Inc., New York and London, 1966), Chap. 7.
396. P. J. Sally, Jr., Analytic continuation of the irreducible unitary representations of the universal covering group of SL(2, ℝ), Memoirs of the Amer. Math. Soc. 69 (Amer. Math. Soc., Providence, R.I., 1967).
397. G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I (Springer-Verlag, Berlin etc., 1972).
398. 2(ℝ), Graduate Texts in Mathematics 105 (Springer-Verlag, New York etc., 1975).
399. S. Helgason, Groups and Geometric Analysis (Academic Press, Inc., Orlando etc., 1984); reprinted in the series Mathematical Surveys and Monographs, No. 83 (Amer. Math. Soc., Providence, R.I., 2000).
400. S. Helgason, Groups and Geometric Analysis (Academic Press, Inc., Orlando etc., 1984); reprinted in the series Mathematical Surveys and Monographs, No. 83 (Amer. Math. Soc., Providence, R.I., 2000).
401. A. W. Knapp, Representation Theory of Semisimple Groups, an Overview based on Examples (Princeton University Press, Princeton, N.J., 1986).
402. A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin etc., 1986), Chap. 5.
403. M. Sugiura, Unitary Representations and Harmonic Analysis, North-Holland Math. Library 44, 2nd ed. (North-Holland, Amsterdam etc., 1990), Chap. 5.
404. N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie Groups and Special Functions, Vol. 1: Simplest Lie Groups, Special Functions and Integral Transforms, Mathematics and Its Applications (Soviet Series) 72 (Kluwer Academic Publishers, Dordrecht etc., 1991).
405. R. Howe and E. C. Tan, Non-Abelian Harmonic Analysis, Applications of SL(2, ℝ) (Springer-Verlag, New York etc., 1992).
406. R. Simon and N. Mukunda, in: Symmetries in Science VI, From the Rotation Group to Quantum Algebras, edited by B. Gruber (Plenum Press, New York and London, 1993), p. 659.
407. [33], Chaps. VI and IX.
408. [245], Sect 1.2.
409. [246], Chaps. XI and XIII.
410. [245], p. 58.
411. As to the other series of irreducible unitary representations of the group SU(1, 1) see, e.g. [242, 244, 246, 250]
412. L. Pukánsky, Math. Ann. 156, 96 (1964).
413. P. J. Sally, Jr., J. Funct. Anal. 6, 441 (1970).
414. C. L. Siegel, Am. J. Math. 65, 1 (1943); the article was reprinted as a book: Symplectic Geometry (Academic Press, N.Y. and London, 1964).
415. C. L. Siegel, Am. J. Math. 65, 1 (1943); the article was reprinted as a book: Symplectic Geometry (Academic Press, N.Y. and London, 1964).
416. H. Cartan, in: Fonctions Automorphes (Séminaire Henri CARTAN, 10e année: 1957/1958), éd. Ecole Normale Supérieure (Secrétariat mathématique, Paris, 1958), Vol. 1, exposé 3.
417. V. Bargmann, in: Analytical methods in mathematical physics, based on the conference held at Indiana University, Bloomington, Indiana, 1968, edited by R. P. Gilbert and R. G. Newton (Gordon and Breach Science Publ., N.Y., London and Paris, 1970), p. 27.
418. G. B. Folland, [38].
419. W. H. Greub, Multilinear Algebra (Springer-Verlag, Berlin, Heidelberg and N.Y., 1967), Chap. V.
420. R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications (Addison-Wesley Publ. Comp., London etc., 1983), Chap. 6.
421. See [248], p. 130.
422. S. Helgason, [33].
423. [245], here especially p. 84.
424. G. Rosensteel and D. J. Rowe, Int. J. Theor. Phys. 15, 453 (1976).
425. As to properties of this group, its unitary representations and the related literature till 1968 see T. O. Philips and E. P. Wigner, in: Group Theory and Its Applications, Vol. I, edited by E. M. Loebl (Academic Press, New York and London, 1968), p. 631.
426. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973), p. 131.
427. J. Polchinski, String Theory, Vol. II (Cambridge University Press, Cambridge, 1998), p. 223.
428. E. Cartan, Ann. Sci. École Norm. Sup. 31, 263 (1914).
429. [261], Nos. 56-58; due to the use of ornamental Gothic capital letters there are several confusing mix-ups of the letters "V" and "B".
430. P. A. M. Dirac, J. Math. Phys. 4, 901 (1963).
431. E. Cartan, Abhandl. aus dem Math. Seminar der Hansischen Universität (Hamburg) 11, 116 (1936).
432. A. Weil, Acta Math. 111, 143 (1964).
433. C. Itzykson, Commun. Math. Phys. 4, 92 (1967). See also D. Shale, Trans. Amer. Math. Soc. 103, 149 (1962).
434. C. Itzykson, Commun. Math. Phys. 4, 92 (1967). See also D. Shale, Trans. Amer. Math. Soc. 103, 149 (1962).
435. J. B. Ehrman, Proc. Camb. Philos. Soc. 53, 290 (1957).
436. C. Fronsdal, Rev. Mod. Phys. 37, 221 (1965).
437. N. T. Evans, J. Math. Phys. 8, 170 (1967).
438. R. Godement, in: Fonctions Automorphes, Vol. 1, see [262], exposés 5, 6 et 10.
439. E. Majorana, Nuovo Cimento 9, 335 (1932). As to the early history of this type of equations see D. M. Fradkin, Am. J. Phys. 34, 314 (1966).
440. E. Majorana, Nuovo Cimento 9, 335 (1932). As to the early history of this type of equations see D. M. Fradkin, Am. J. Phys. 34, 314 (1966).
441. A. O. Barut and H. Kleinert, Phys. Rev. 156, 1546 (1967). A. Böhm, in: High Energy Physics and Fundamental Particles, Lectures in Theor. Physics X-B, edited by A. O. Barut and W. E. Brittin (Gordon and Breach Science Publ., N.Y., London and Paris, 1968), p. 483. A. Böhm, Phys. Rev. 175, 1767 (1968); Phys. Rev. D 3, 367 (1971). A. O. Barut and A. Böhm, J. Math. Phys. 11, 2938 (1970). L. Jaffe, J. Math. Phys. 12, 882 (1971); in: De Sitter and Conformal Groups and Their Applications, Lectures in Theor. Physics XIII, edited by A. O. Barut and W. E. Brittin (Colorado Assoc. Univ. Press, Boulder, Colorado, 1971), p. 125.
442. A. O. Barut and H. Kleinert, Phys. Rev. 156, 1546 (1967). A. Böhm, in: High Energy Physics and Fundamental Particles, Lectures in Theor. Physics X-B, edited by A. O. Barut and W. E. Brittin (Gordon and Breach Science Publ., N.Y., London and Paris, 1968), p. 483. A. Böhm, Phys. Rev. 175, 1767 (1968); Phys. Rev. D 3, 367 (1971). A. O. Barut and A. Böhm, J. Math. Phys. 11, 2938 (1970). L. Jaffe, J. Math. Phys. 12, 882 (1971); in: De Sitter and Conformal Groups and Their Applications, Lectures in Theor. Physics XIII, edited by A. O. Barut and W. E. Brittin (Colorado Assoc. Univ. Press, Boulder, Colorado, 1971), p. 125.
443. A. O. Barut and H. Kleinert, Phys. Rev. 156, 1546 (1967). A. Böhm, in: High Energy Physics and Fundamental Particles, Lectures in Theor. Physics X-B, edited by A. O. Barut and W. E. Brittin (Gordon and Breach Science Publ., N.Y., London and Paris, 1968), p. 483. A. Böhm, Phys. Rev. 175, 1767 (1968); Phys. Rev. D 3, 367 (1971). A. O. Barut and A. Böhm, J. Math. Phys. 11, 2938 (1970). L. Jaffe, J. Math. Phys. 12, 882 (1971); in: De Sitter and Conformal Groups and Their Applications, Lectures in Theor. Physics XIII, edited by A. O. Barut and W. E. Brittin (Colorado Assoc. Univ. Press, Boulder, Colorado, 1971), p. 125.
444. A. O. Barut and H. Kleinert, Phys. Rev. 156, 1546 (1967). A. Böhm, in: High Energy Physics and Fundamental Particles, Lectures in Theor. Physics X-B, edited by A. O. Barut and W. E. Brittin (Gordon and Breach Science Publ., N.Y., London and Paris, 1968), p. 483. A. Böhm, Phys. Rev. 175, 1767 (1968); Phys. Rev. D 3, 367 (1971). A. O. Barut and A. Böhm, J. Math. Phys. 11, 2938 (1970). L. Jaffe, J. Math. Phys. 12, 882 (1971); in: De Sitter and Conformal Groups and Their Applications, Lectures in Theor. Physics XIII, edited by A. O. Barut and W. E. Brittin (Colorado Assoc. Univ. Press, Boulder, Colorado, 1971), p. 125.
445. A. O. Barut and H. Kleinert, Phys. Rev. 156, 1546 (1967). A. Böhm, in: High Energy Physics and Fundamental Particles, Lectures in Theor. Physics X-B, edited by A. O. Barut and W. E. Brittin (Gordon and Breach Science Publ., N.Y., London and Paris, 1968), p. 483. A. Böhm, Phys. Rev. 175, 1767 (1968); Phys. Rev. D 3, 367 (1971). A. O. Barut and A. Böhm, J. Math. Phys. 11, 2938 (1970). L. Jaffe, J. Math. Phys. 12, 882 (1971); in: De Sitter and Conformal Groups and Their Applications, Lectures in Theor. Physics XIII, edited by A. O. Barut and W. E. Brittin (Colorado Assoc. Univ. Press, Boulder, Colorado, 1971), p. 125.
446. A. O. Barut and H. Kleinert, Phys. Rev. 156, 1546 (1967). A. Böhm, in: High Energy Physics and Fundamental Particles, Lectures in Theor. Physics X-B, edited by A. O. Barut and W. E. Brittin (Gordon and Breach Science Publ., N.Y., London and Paris, 1968), p. 483. A. Böhm, Phys. Rev. 175, 1767 (1968); Phys. Rev. D 3, 367 (1971). A. O. Barut and A. Böhm, J. Math. Phys. 11, 2938 (1970). L. Jaffe, J. Math. Phys. 12, 882 (1971); in: De Sitter and Conformal Groups and Their Applications, Lectures in Theor. Physics XIII, edited by A. O. Barut and W. E. Brittin (Colorado Assoc. Univ. Press, Boulder, Colorado, 1971), p. 125.
447. A. O. Barut and H. Kleinert, Phys. Rev. 156, 1546 (1967). A. Böhm, in: High Energy Physics and Fundamental Particles, Lectures in Theor. Physics X-B, edited by A. O. Barut and W. E. Brittin (Gordon and Breach Science Publ., N.Y., London and Paris, 1968), p. 483. A. Böhm, Phys. Rev. 175, 1767 (1968); Phys. Rev. D 3, 367 (1971). A. O. Barut and A. Böhm, J. Math. Phys. 11, 2938 (1970). L. Jaffe, J. Math. Phys. 12, 882 (1971); in: De Sitter and Conformal Groups and Their Applications, Lectures in Theor. Physics XIII, edited by A. O. Barut and W. E. Brittin (Colorado Assoc. Univ. Press, Boulder, Colorado, 1971), p. 125.
448. P. A. M. Dirac, Proc. R. Soc. Lond. A 322, 435 (1971); 328, 1 (1972).
449. P. A. M. Dirac, Proc. R. Soc. Lond. A 322, 435 (1971); 328, 1 (1972).
450. E. C. G. Sudarshan and N. Mukunda, Phys. Rev. D 1, 571 (1970). N. Mukunda, H. van Dam and L. C. Biedenharn, Relativistic Models of Extended Hadrons Obeying a Mass-Spin Trajectory Constraint, Lectures, Lecture Notes in Physics 165, edited by A. Böhm and J. D. Dollard (Springer-Verlag, Berlin, Heidelberg and N.Y., 1982).
451. E. C. G. Sudarshan and N. Mukunda, Phys. Rev. D 1, 571 (1970). N. Mukunda, H. van Dam and L. C. Biedenharn, Relativistic Models of Extended Hadrons Obeying a Mass-Spin Trajectory Constraint, Lectures, Lecture Notes in Physics 165, edited by A. Böhm and J. D. Dollard (Springer-Verlag, Berlin, Heidelberg and N.Y., 1982).
452. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
453. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
454. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
455. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
456. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
457. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
458. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
459. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
460. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
461. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
462. C. Fronsdal, Phys. Rev. D 10, 589 (1974). C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975). C. Fronsdal, Phys. Rev. D 12, 3819 (1975). J. Fang and C. Fronsdal, Lett. Math. Phys. 2, 391 (1978); Phys. Rev. D 18, 3630 (1978); Phys. Rev. D 22, 1361 (1980). M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978). W. Heidenreich, J. Math. Phys. 22, 1566 (1981). E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer, Phys. Rev. D 23, 1278 (1981). M. Flato and C. Fronsdal, J. Math. Phys. 22, 1100 (1981); Phys. Lett. B 97, 236 (1980).
463. R. Howe, in: Application of Group Theory in Physics and Mathematical Physics, Lectures in Appl. Math. 21, edited by M. Flato, P. Sally, and G. Zuckerman (Amer. Math. Soc., Providence, R.I., 1985), p. 179.
464. S. Goshen and H. J. Lipkin, in: Spectroscopic and Group Theoretical Methods in Physics, Racah Memorial Volume, edited by F. Bloch, S. G. Cohen, A. De-Shalit, S. Sambursky, and I. Talmi (North-Holland Publ. Co., Amsterdam, 1968), p. 245.
465. M. Moshinsky and C. Quesne, J. Math. Phys. 12, 1772 and 1780 (1971).
466. A. A. Kirillov, Elements of the Theory of Representations, Grundlehren der mathematischen Wissenschaften 220 (Springer-Verlag, Berlin, Heidelberg and N.Y., 1976), pp. 287-290; translation of the Russian edition from 1972.
467. S. Gelbart, Inventiones Math. 19, 49 (1973); closely related papers of the same author are: Bull. Am. Math. Soc. 78, 451 (1972); Trans. Am. Math. Soc. 192, 29 (1974).
468. S. Gelbart, Inventiones Math. 19, 49 (1973); closely related papers of the same author are: Bull. Am. Math. Soc. 78, 451 (1972); Trans. Am. Math. Soc. 192, 29 (1974).
469. S. Gelbart, Inventiones Math. 19, 49 (1973); closely related papers of the same author are: Bull. Am. Math. Soc. 78, 451 (1972); Trans. Am. Math. Soc. 192, 29 (1974).
470. G. Rosensteel and D. J. Howe, Int. J. Theor. Phys. 16, 63 (1977).
471. E. Angelopoulos, in: Quantum Theory, Groups, Fields and Particles, edited by A. O. Barut (D. Reidel Publ. Co., Dordrecht, Boston and London, 1983), p. 101.
472. R. F. Bishop and A. Vourdas, Z. Phys. B, Condens. Matter 71, 527 (1988). A. Luis and L. L. Sánchez-Soto, Quantum Semiclass. Opt. 7, 153 (1995). D. A. Trifonov, J. Phys. A, Math. Gen. 31, 5673 (1998). J. Fiurášek and J. Peřina, Phys. Rev. A 62, 033808-1 (2000).
473. R. F. Bishop and A. Vourdas, Z. Phys. B, Condens. Matter 71, 527 (1988). A. Luis and L. L. Sánchez-Soto, Quantum Semiclass. Opt. 7, 153 (1995). D. A. Trifonov, J. Phys. A, Math. Gen. 31, 5673 (1998). J. Fiurášek and J. Peřina, Phys. Rev. A 62, 033808-1 (2000).
474. R. F. Bishop and A. Vourdas, Z. Phys. B, Condens. Matter 71, 527 (1988). A. Luis and L. L. Sánchez-Soto, Quantum Semiclass. Opt. 7, 153 (1995). D. A. Trifonov, J. Phys. A, Math. Gen. 31, 5673 (1998). J. Fiurášek and J. Peřina, Phys. Rev. A 62, 033808-1 (2000).
475. R. F. Bishop and A. Vourdas, Z. Phys. B, Condens. Matter 71, 527 (1988). A. Luis and L. L. Sánchez-Soto, Quantum Semiclass. Opt. 7, 153 (1995). D. A. Trifonov, J. Phys. A, Math. Gen. 31, 5673 (1998). J. Fiurášek and J. Peřina, Phys. Rev. A 62, 033808-1 (2000).
476. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985); Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988). R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994). Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995). B. Arvind and N. Mukunda, J. Phys. A, Math. Gen. 29, 5855 (1996). Arvind, N. Mukunda, and R. Simon, J. Phys. A, Math. Gen. 31, 565 (1998). E. M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A 60, 3397 (1999).
477. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985); Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988). R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994). Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995). B. Arvind and N. Mukunda, J. Phys. A, Math. Gen. 29, 5855 (1996). Arvind, N. Mukunda, and R. Simon, J. Phys. A, Math. Gen. 31, 565 (1998). E. M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A 60, 3397 (1999).
478. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985); Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988). R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994). Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995). B. Arvind and N. Mukunda, J. Phys. A, Math. Gen. 29, 5855 (1996). Arvind, N. Mukunda, and R. Simon, J. Phys. A, Math. Gen. 31, 565 (1998). E. M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A 60, 3397 (1999).
479. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985); Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988). R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994). Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995). B. Arvind and N. Mukunda, J. Phys. A, Math. Gen. 29, 5855 (1996). Arvind, N. Mukunda, and R. Simon, J. Phys. A, Math. Gen. 31, 565 (1998). E. M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A 60, 3397 (1999).
480. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985); Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988). R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994). Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995). B. Arvind and N. Mukunda, J. Phys. A, Math. Gen. 29, 5855 (1996). Arvind, N. Mukunda, and R. Simon, J. Phys. A, Math. Gen. 31, 565 (1998). E. M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A 60, 3397 (1999).
481. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985); Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988). R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994). Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995). B. Arvind and N. Mukunda, J. Phys. A, Math. Gen. 29, 5855 (1996). Arvind, N. Mukunda, and R. Simon, J. Phys. A, Math. Gen. 31, 565 (1998). E. M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A 60, 3397 (1999).
482. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985); Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988). R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994). Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995). B. Arvind and N. Mukunda, J. Phys. A, Math. Gen. 29, 5855 (1996). Arvind, N. Mukunda, and R. Simon, J. Phys. A, Math. Gen. 31, 565 (1998). E. M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A 60, 3397 (1999).
483. R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Rev. A 31, 2419 (1985); Phys. Rev. A 36, 3868 (1987); Phys. Rev. A 37, 3028 (1988). R. Simon, N. Mukunda, and B. Dutta, Phys. Rev. A 49, 1567 (1994). Arvind, B. Dutta, N. Mukunda, and R. Simon, Phys. Rev. A 52, 1609 (1995). B. Arvind and N. Mukunda, J. Phys. A, Math. Gen. 29, 5855 (1996). Arvind, N. Mukunda, and R. Simon, J. Phys. A, Math. Gen. 31, 565 (1998). E. M. Rabei, Arvind, N. Mukunda, and R. Simon, Phys. Rev. A 60, 3397 (1999).
484. D. Han, Y. S. Kim, and M. E. Noz, Phys. Rev. A 41, 6233 (1990). D. Han, Y. S. Kim, M. E. Noz, and L. Yeh, J. Math. Phys. 34, 5493 (1993). A. Wünsche, J. Opt. B, Quantum Semiclassical Opt. 2, 73 (2000); 4, 1 (2002).
485. D. Han, Y. S. Kim, and M. E. Noz, Phys. Rev. A 41, 6233 (1990). D. Han, Y. S. Kim, M. E. Noz, and L. Yeh, J. Math. Phys. 34, 5493 (1993). A. Wünsche, J. Opt. B, Quantum Semiclassical Opt. 2, 73 (2000); 4, 1 (2002).
486. D. Han, Y. S. Kim, and M. E. Noz, Phys. Rev. A 41, 6233 (1990). D. Han, Y. S. Kim, M. E. Noz, and L. Yeh, J. Math. Phys. 34, 5493 (1993). A. Wünsche, J. Opt. B, Quantum Semiclassical Opt. 2, 73 (2000); 4, 1 (2002).
487. D. Han, Y. S. Kim, and M. E. Noz, Phys. Rev. A 41, 6233 (1990). D. Han, Y. S. Kim, M. E. Noz, and L. Yeh, J. Math. Phys. 34, 5493 (1993). A. Wünsche, J. Opt. B, Quantum Semiclassical Opt. 2, 73 (2000); 4, 1 (2002).
488. [248], pp. 130, 147, 152f., 404-407, 498, 502-511, 606f., 717f..
489. T. Przebinda, The oscillator duality correspondence for the pair O(2, 2), Sp(2, ℝ), Memoirs Amer. Math. Soc. Vol. 79, No. 403 (Amer. Math. Soc., Providence, R.I., 1989).
490. [300], p. 36.
491. H. Weyl, The Theory of Groups and Quantum Mechanics, [37], p. 137.
492. B. L. van der Waerden, Die gruppentheoretische Methode in der Quantenmechanik (Verlag Julius Springer, Berlin, 1932), § 16.
493. T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Mathematics 98 (Springer-Verlag, New York etc., 1985), pp. 84-87.
494. [283], exposé 10.
495. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University Press, Cambridge, 1966), p. 79, formula (4).
496. [306], p. 79, formula (9).
497. See, e.g. [74], p. 1, formula (1) and p. 4, formula (10).
498. G. H. Hardy, Proc. Lond. Math. Soc. (Ser. 2) 2, 401 (1905).
499. E. W. Barnes, Proc. R. Soc. Lond. A 206, 249 (1906); Part III.
500. N. Nielsen, Handbuch der Theorie der Gammafunktion, Part I, in: Die Gammafunktion (Chelsea Publ. Co. New York, 1965), p. 74, formula (14); originally published (B.G. Teubner, Leipzig, 1906).
501. [311], p. 76.
502. E. R. Hansen, A Table of Series and Products (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975), p. 149, formula (6.12.1).
503. [310], Part V.