Quantum mechanical irreversibility

Physica A: Statistical Mechanics and its Applications

Volumn 236, Issue 3-4, 1997, Pages 485-549

  Bohm, A ^a   Maxson, S ^a   Loewe, Mark ^b   Gadella, M ^c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b The University of Texas at Austin   (United States)

^c UNIVERSITY OF VALLADOLID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; CALCULATIONS; EIGENVALUES AND EIGENFUNCTIONS; ELECTRON RESONANCE; ELECTRON SCATTERING; MATRIX ALGEBRA; POLES AND ZEROS; VECTORS;

GAMOW VECTORS; GOLDEN RULE; QUANTUM MECHANICAL IRREVERSIBILITY; RIGGED HILBERT SPACE;

QUANTUM THEORY;

EID: 0031094462     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0378-4371(96)00284-1     Document Type: Article

References (141)

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12. The question of non-quantum-mechanical versus quantum-mechanical effects for the neutral-Kaon system has been discussed in P. Huet, M.E. Peskin, Nucl. Phys. B 434 (1995) 3. See also J. Ellis, J.S. Hagelin, D.V. Nanopoulos and M. Srendicki, Nucl. Phys. B 241 (1984) 381; J. Ellis, N.E. Mavromatos and D.V. Nanopolous, Phys. Lett. B 293 (1992) 142, and CERN-TH-6755/92 (1992).
13. The question of non-quantum-mechanical versus quantum-mechanical effects for the neutral-Kaon system has been discussed in P. Huet, M.E. Peskin, Nucl. Phys. B 434 (1995) 3. See also J. Ellis, J.S. Hagelin, D.V. Nanopoulos and M. Srendicki, Nucl. Phys. B 241 (1984) 381; J. Ellis, N.E. Mavromatos and D.V. Nanopolous, Phys. Lett. B 293 (1992) 142, and CERN-TH-6755/92 (1992).
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15. J. von Neumann, Mathematical Foundations of Quantum Mechanics (English translation by R.T. Beyer) (Princeton University Press, Princeton, 1955). The mathematics of the Hilbert space are discussed many places, including M. Reed and B. Simon, Functional Analysis (Academic Press, New York, 1972).
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19. The Hamiltonian is also always assumed to be bounded from below, H ≥ 0, as required for the stability of matter. E.H. Lieb and W.E. Thirring, Phys. Rev. Lett. 35 (1975) 687 and Phys. Rev. Lett. 35 (1975) 1116.
20. I.M. Gel'fand and N. Ya. Vilenkin, Generalized Functions, Vol. 4 (Academic Press, New York, 1964); K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups (Polish Scientific Publishers, Warzawa, 1968). The word "rigged" in rigged Hilbert space has a nautical connotation, such as the phrase "fully rigged ship": it has nothing to do with any unsavory practice such as "fixing" or predetermining a result. The phrase "rigged Hilbert space" is a direct translation of the phrase osnashchyonnoe Hilbertovo prostranstvo from the original Russian of the first citation above. The rigged Hilbert space is also called a Gel'fand triplet.
21. I.M. Gel'fand and N. Ya. Vilenkin, Generalized Functions, Vol. 4 (Academic Press, New York, 1964); K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups (Polish Scientific Publishers, Warzawa, 1968). The word "rigged" in rigged Hilbert space has a nautical connotation, such as the phrase "fully rigged ship": it has nothing to do with any unsavory practice such as "fixing" or predetermining a result. The phrase "rigged Hilbert space" is a direct translation of the phrase osnashchyonnoe Hilbertovo prostranstvo from the original Russian of the first citation above. The rigged Hilbert space is also called a Gel'fand triplet.
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25. J.E. Roberts, J. Math. Phys. 7 (1966) 1097; A. Bohm, Boulder Lectures in Theoretical Physics 1966, Vol. 9A (Gordon and Breach, New York, 1967); J.-P. Antoine, J. Math. Phys. 10 (1969) 53; 10 (1969) 2276; O. Melsheimer, J. Math. Phys. 15 (1974) 902.
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28. I. Prigogine, From Being to Becoming (Freeman, New York, 1980); G. Nicholis and I. Prigogine, Exploring Complexity (Freeman, New York, 1988); I. Prigogine, Phys. Rep. 219 (1992) 93; I. Antoniou, Nature 338 (1989) 210; T. Petrosky, I. Prigogine and S. Tasaki, Physica A 173 (1991) 175; T.Y. Petrosky and I. Prigogine, Physica 147A (1988) 439; T. Petrosky and I. Prigogine, Physica A 175 (1991) 146; T. Petrosky and I. Prigogine, Chaos, Solitons and Fractals 4 (1994) 311: T. Petrosky and I. Prigogine, in: Nonlinear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers, Singapore, 1995). Whereas, in the present paper we restrict ourselves to the irreversible time evolution of simple systems (like one or two resonances) associated to first order poles of the S-matrix and hypothetical systems associated to higher order S-matrix poles, the papers quoted above are also concerned with complex systems described by density matrices and their approach to equilibrium. In the later case, one uses the von Neumann equation (2.5) and extends the eigenvalue problem of L in (2.4) to complex values.
29. I. Prigogine, From Being to Becoming (Freeman, New York, 1980); G. Nicholis and I. Prigogine, Exploring Complexity (Freeman, New York, 1988); I. Prigogine, Phys. Rep. 219 (1992) 93; I. Antoniou, Nature 338 (1989) 210; T. Petrosky, I. Prigogine and S. Tasaki, Physica A 173 (1991) 175; T.Y. Petrosky and I. Prigogine, Physica 147A (1988) 439; T. Petrosky and I. Prigogine, Physica A 175 (1991) 146; T. Petrosky and I. Prigogine, Chaos, Solitons and Fractals 4 (1994) 311: T. Petrosky and I. Prigogine, in: Nonlinear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers, Singapore, 1995). Whereas, in the present paper we restrict ourselves to the irreversible time evolution of simple systems (like one or two resonances) associated to first order poles of the S-matrix and hypothetical systems associated to higher order S-matrix poles, the papers quoted above are also concerned with complex systems described by density matrices and their approach to equilibrium. In the later case, one uses the von Neumann equation (2.5) and extends the eigenvalue problem of L in (2.4) to complex values.
30. I. Prigogine, From Being to Becoming (Freeman, New York, 1980); G. Nicholis and I. Prigogine, Exploring Complexity (Freeman, New York, 1988); I. Prigogine, Phys. Rep. 219 (1992) 93; I. Antoniou, Nature 338 (1989) 210; T. Petrosky, I. Prigogine and S. Tasaki, Physica A 173 (1991) 175; T.Y. Petrosky and I. Prigogine, Physica 147A (1988) 439; T. Petrosky and I. Prigogine, Physica A 175 (1991) 146; T. Petrosky and I. Prigogine, Chaos, Solitons and Fractals 4 (1994) 311: T. Petrosky and I. Prigogine, in: Nonlinear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers, Singapore, 1995). Whereas, in the present paper we restrict ourselves to the irreversible time evolution of simple systems (like one or two resonances) associated to first order poles of the S-matrix and hypothetical systems associated to higher order S-matrix poles, the papers quoted above are also concerned with complex systems described by density matrices and their approach to equilibrium. In the later case, one uses the von Neumann equation (2.5) and extends the eigenvalue problem of L in (2.4) to complex values.
31. I. Prigogine, From Being to Becoming (Freeman, New York, 1980); G. Nicholis and I. Prigogine, Exploring Complexity (Freeman, New York, 1988); I. Prigogine, Phys. Rep. 219 (1992) 93; I. Antoniou, Nature 338 (1989) 210; T. Petrosky, I. Prigogine and S. Tasaki, Physica A 173 (1991) 175; T.Y. Petrosky and I. Prigogine, Physica 147A (1988) 439; T. Petrosky and I. Prigogine, Physica A 175 (1991) 146; T. Petrosky and I. Prigogine, Chaos, Solitons and Fractals 4 (1994) 311: T. Petrosky and I. Prigogine, in: Nonlinear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers, Singapore, 1995). Whereas, in the present paper we restrict ourselves to the irreversible time evolution of simple systems (like one or two resonances) associated to first order poles of the S-matrix and hypothetical systems associated to higher order S-matrix poles, the papers quoted above are also concerned with complex systems described by density matrices and their approach to equilibrium. In the later case, one uses the von Neumann equation (2.5) and extends the eigenvalue problem of L in (2.4) to complex values.
32. I. Prigogine, From Being to Becoming (Freeman, New York, 1980); G. Nicholis and I. Prigogine, Exploring Complexity (Freeman, New York, 1988); I. Prigogine, Phys. Rep. 219 (1992) 93; I. Antoniou, Nature 338 (1989) 210; T. Petrosky, I. Prigogine and S. Tasaki, Physica A 173 (1991) 175; T.Y. Petrosky and I. Prigogine, Physica 147A (1988) 439; T. Petrosky and I. Prigogine, Physica A 175 (1991) 146; T. Petrosky and I. Prigogine, Chaos, Solitons and Fractals 4 (1994) 311: T. Petrosky and I. Prigogine, in: Nonlinear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers, Singapore, 1995). Whereas, in the present paper we restrict ourselves to the irreversible time evolution of simple systems (like one or two resonances) associated to first order poles of the S-matrix and hypothetical systems associated to higher order S-matrix poles, the papers quoted above are also concerned with complex systems described by density matrices and their approach to equilibrium. In the later case, one uses the von Neumann equation (2.5) and extends the eigenvalue problem of L in (2.4) to complex values.
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34. I. Prigogine, From Being to Becoming (Freeman, New York, 1980); G. Nicholis and I. Prigogine, Exploring Complexity (Freeman, New York, 1988); I. Prigogine, Phys. Rep. 219 (1992) 93; I. Antoniou, Nature 338 (1989) 210; T. Petrosky, I. Prigogine and S. Tasaki, Physica A 173 (1991) 175; T.Y. Petrosky and I. Prigogine, Physica 147A (1988) 439; T. Petrosky and I. Prigogine, Physica A 175 (1991) 146; T. Petrosky and I. Prigogine, Chaos, Solitons and Fractals 4 (1994) 311: T. Petrosky and I. Prigogine, in: Nonlinear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers, Singapore, 1995). Whereas, in the present paper we restrict ourselves to the irreversible time evolution of simple systems (like one or two resonances) associated to first order poles of the S-matrix and hypothetical systems associated to higher order S-matrix poles, the papers quoted above are also concerned with complex systems described by density matrices and their approach to equilibrium. In the later case, one uses the von Neumann equation (2.5) and extends the eigenvalue problem of L in (2.4) to complex values.
35. I. Prigogine, From Being to Becoming (Freeman, New York, 1980); G. Nicholis and I. Prigogine, Exploring Complexity (Freeman, New York, 1988); I. Prigogine, Phys. Rep. 219 (1992) 93; I. Antoniou, Nature 338 (1989) 210; T. Petrosky, I. Prigogine and S. Tasaki, Physica A 173 (1991) 175; T.Y. Petrosky and I. Prigogine, Physica 147A (1988) 439; T. Petrosky and I. Prigogine, Physica A 175 (1991) 146; T. Petrosky and I. Prigogine, Chaos, Solitons and Fractals 4 (1994) 311: T. Petrosky and I. Prigogine, in: Nonlinear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers, Singapore, 1995). Whereas, in the present paper we restrict ourselves to the irreversible time evolution of simple systems (like one or two resonances) associated to first order poles of the S-matrix and hypothetical systems associated to higher order S-matrix poles, the papers quoted above are also concerned with complex systems described by density matrices and their approach to equilibrium. In the later case, one uses the von Neumann equation (2.5) and extends the eigenvalue problem of L in (2.4) to complex values.
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51. M. de Haan, C. George and F. Mayné, Physica A 92 (1978) 584; T. Berggren, in: Resonances, eds. E. Brändas and N. Elander, Lecture Notes in Physics, Vol. 325 (Springer, Berlin, 1987), p. 105; E. Brändas, M. Rittby and N. Elander, in: Resonances, eds. E. Brändas and N. Elander, Lecture Notes in Physics, Vol. 325 (Springer, Berlin, 1987), p. 345; W. Hunziker, Ann. Inst. Henri Poincaré 45 (1986) 339; I.M. Sigal, Ann. Inst. Henri Poincaré 41 (1984) 103; B. Helffer and J. Sjöstrand, Bull, S.M.F. mémoire no 24/25, tome 114 (1986); H.L. Cycon. Helv. Phys. Acta 58 (1986) 969.
52. M. de Haan, C. George and F. Mayné, Physica A 92 (1978) 584; T. Berggren, in: Resonances, eds. E. Brändas and N. Elander, Lecture Notes in Physics, Vol. 325 (Springer, Berlin, 1987), p. 105; E. Brändas, M. Rittby and N. Elander, in: Resonances, eds. E. Brändas and N. Elander, Lecture Notes in Physics, Vol. 325 (Springer, Berlin, 1987), p. 345; W. Hunziker, Ann. Inst. Henri Poincaré 45 (1986) 339; I.M. Sigal, Ann. Inst. Henri Poincaré 41 (1984) 103; B. Helffer and J. Sjöstrand, Bull, S.M.F. mémoire no 24/25, tome 114 (1986); H.L. Cycon. Helv. Phys. Acta 58 (1986) 969.
53. M. de Haan, C. George and F. Mayné, Physica A 92 (1978) 584; T. Berggren, in: Resonances, eds. E. Brändas and N. Elander, Lecture Notes in Physics, Vol. 325 (Springer, Berlin, 1987), p. 105; E. Brändas, M. Rittby and N. Elander, in: Resonances, eds. E. Brändas and N. Elander, Lecture Notes in Physics, Vol. 325 (Springer, Berlin, 1987), p. 345; W. Hunziker, Ann. Inst. Henri Poincaré 45 (1986) 339; I.M. Sigal, Ann. Inst. Henri Poincaré 41 (1984) 103; B. Helffer and J. Sjöstrand, Bull, S.M.F. mémoire no 24/25, tome 114 (1986); H.L. Cycon. Helv. Phys. Acta 58 (1986) 969.
54. M. de Haan, C. George and F. Mayné, Physica A 92 (1978) 584; T. Berggren, in: Resonances, eds. E. Brändas and N. Elander, Lecture Notes in Physics, Vol. 325 (Springer, Berlin, 1987), p. 105; E. Brändas, M. Rittby and N. Elander, in: Resonances, eds. E. Brändas and N. Elander, Lecture Notes in Physics, Vol. 325 (Springer, Berlin, 1987), p. 345; W. Hunziker, Ann. Inst. Henri Poincaré 45 (1986) 339; I.M. Sigal, Ann. Inst. Henri Poincaré 41 (1984) 103; B. Helffer and J. Sjöstrand, Bull, S.M.F. mémoire no 24/25, tome 114 (1986); H.L. Cycon. Helv. Phys. Acta 58 (1986) 969.
55. R.E. Peierls, in: Proc. of the 1954 Glasgow Conf. on Nuclear and Meson Physics, eds. E.M. Bellamy, et al. (Pergamon Press, Oxford, 1955); G. Garcia-Calderon and R. Peierls, Nucl. Phys. A 265 (1976) 443; and E. Hernandez and A. Mondragon, Phys. Rev. C 29 (1984) 722; A. Mondragon and E. Hernandez, Ann. der Physik 48 (1991) 503; G. Garcia-Calderon, in: Symmetries in Physics (Moshinsky Symposium), eds. A. Frank and K.B. Wolf (Springer, Berlin, 1992).
56. R.E. Peierls, in: Proc. of the 1954 Glasgow Conf. on Nuclear and Meson Physics, eds. E.M. Bellamy, et al. (Pergamon Press, Oxford, 1955); G. Garcia-Calderon and R. Peierls, Nucl. Phys. A 265 (1976) 443; and E. Hernandez and A. Mondragon, Phys. Rev. C 29 (1984) 722; A. Mondragon and E. Hernandez, Ann. der Physik 48 (1991) 503; G. Garcia-Calderon, in: Symmetries in Physics (Moshinsky Symposium), eds. A. Frank and K.B. Wolf (Springer, Berlin, 1992).
57. R.E. Peierls, in: Proc. of the 1954 Glasgow Conf. on Nuclear and Meson Physics, eds. E.M. Bellamy, et al. (Pergamon Press, Oxford, 1955); G. Garcia-Calderon and R. Peierls, Nucl. Phys. A 265 (1976) 443; and E. Hernandez and A. Mondragon, Phys. Rev. C 29 (1984) 722; A. Mondragon and E. Hernandez, Ann. der Physik 48 (1991) 503; G. Garcia-Calderon, in: Symmetries in Physics (Moshinsky Symposium), eds. A. Frank and K.B. Wolf (Springer, Berlin, 1992).
58. R.E. Peierls, in: Proc. of the 1954 Glasgow Conf. on Nuclear and Meson Physics, eds. E.M. Bellamy, et al. (Pergamon Press, Oxford, 1955); G. Garcia-Calderon and R. Peierls, Nucl. Phys. A 265 (1976) 443; and E. Hernandez and A. Mondragon, Phys. Rev. C 29 (1984) 722; A. Mondragon and E. Hernandez, Ann. der Physik 48 (1991) 503; G. Garcia-Calderon, in: Symmetries in Physics (Moshinsky Symposium), eds. A. Frank and K.B. Wolf (Springer, Berlin, 1992).
59. R.E. Peierls, in: Proc. of the 1954 Glasgow Conf. on Nuclear and Meson Physics, eds. E.M. Bellamy, et al. (Pergamon Press, Oxford, 1955); G. Garcia-Calderon and R. Peierls, Nucl. Phys. A 265 (1976) 443; and E. Hernandez and A. Mondragon, Phys. Rev. C 29 (1984) 722; A. Mondragon and E. Hernandez, Ann. der Physik 48 (1991) 503; G. Garcia-Calderon, in: Symmetries in Physics (Moshinsky Symposium), eds. A. Frank and K.B. Wolf (Springer, Berlin, 1992).
60. Generalized eigenvectors of self-adjoint operators with complex eigenvalue in the RHS were systematically studied for the first time in the reduction of SO(2, 1) with respect to its noncompact subgroup, G. Lindblad and B. Nagel, Ann. Inst. Henri Poincaré A 13 (1970) 27. As poles of the resolvent, they were first introduced by H. Baumgartel, Math. Nachr. 75 (1976) 133, and in a more general setting by A. Grossmann, J. Math. Phys., 5 (1964) 1025.
61. Generalized eigenvectors of self-adjoint operators with complex eigenvalue in the RHS were systematically studied for the first time in the reduction of SO(2, 1) with respect to its noncompact subgroup, G. Lindblad and B. Nagel, Ann. Inst. Henri Poincaré A 13 (1970) 27. As poles of the resolvent, they were first introduced by H. Baumgartel, Math. Nachr. 75 (1976) 133, and in a more general setting by A. Grossmann, J. Math. Phys., 5 (1964) 1025.
62. Generalized eigenvectors of self-adjoint operators with complex eigenvalue in the RHS were systematically studied for the first time in the reduction of SO(2, 1) with respect to its noncompact subgroup, G. Lindblad and B. Nagel, Ann. Inst. Henri Poincaré A 13 (1970) 27. As poles of the resolvent, they were first introduced by H. Baumgartel, Math. Nachr. 75 (1976) 133, and in a more general setting by A. Grossmann, J. Math. Phys., 5 (1964) 1025.
63. A. Bohm, Quantum Mechanics, 1st Ed. (Springer, Berlin, 1979); 3rd Ed. (1993).
64. A. Bohm, in: Group Theoretical Methods in Physics. Lecture Notes in Physics. Vol. 94 (Springer, Berlin, 1978), p. 245.
65. A. Bohm, J. Math. Phys. 22 (1981) 2813; A. Bohm, Lett. Math. Phys. 3 (1978) 455.
66. A. Bohm, J. Math. Phys. 22 (1981) 2813; A. Bohm, Lett. Math. Phys. 3 (1978) 455.
67. M. Gadella, J. Math. Phys. 24 (1983) 2142; J. Math. Phys. 25 (1984) 2481.
68. M. Gadella, J. Math. Phys. 24 (1983) 2142; J. Math. Phys. 25 (1984) 2481.
69. M. Gadella, J. Math. Phys. 24 (1983) 1462.
70. A. Bohm and M. Gadella, Dirac Kets, Gamow Vectors, and Gel'fand Triplets, Lecture Notes in Physics, Vol. 348 (Springer, Berlin, 1989).
71. I. Antoniou, in: Proc. 2nd Internat. Wigner Symp. Goslar, 1991, eds. H.D. Doebner, et al. (World Scientific Publishers, Singapore, 1992); I.E. Antoniou, I. Prigogine, Physica A 192 (1993) 443; I. Antoniou, S. Tasaki, Int. J. Quant. Chem. 46 (1993) 425.
72. I. Antoniou, in: Proc. 2nd Internat. Wigner Symp. Goslar, 1991, eds. H.D. Doebner, et al. (World Scientific Publishers, Singapore, 1992); I.E. Antoniou, I. Prigogine, Physica A 192 (1993) 443; I. Antoniou, S. Tasaki, Int. J. Quant. Chem. 46 (1993) 425.
73. I. Antoniou, in: Proc. 2nd Internat. Wigner Symp. Goslar, 1991, eds. H.D. Doebner, et al. (World Scientific Publishers, Singapore, 1992); I.E. Antoniou, I. Prigogine, Physica A 192 (1993) 443; I. Antoniou, S. Tasaki, Int. J. Quant. Chem. 46 (1993) 425.
74. A. Bohm, in: Proc. Symp. on the Foundations of Modern Physics, Cologne, June 1, 1993. eds. P. Busch and P. Mittelstaedt (World Scientific, 1993), p. 77; A. Bohm, I. Antoniou and P. Kielanowski, Phys. Lett. A 189 (1994) 442; A. Bohm, I. Antoniou and P. Kielanowski, J. Math. Phys. 36 (1995) 2593.
75. A. Bohm, in: Proc. Symp. on the Foundations of Modern Physics, Cologne, June 1, 1993. eds. P. Busch and P. Mittelstaedt (World Scientific, 1993), p. 77; A. Bohm, I. Antoniou and P. Kielanowski, Phys. Lett. A 189 (1994) 442; A. Bohm, I. Antoniou and P. Kielanowski, J. Math. Phys. 36 (1995) 2593.
76. A. Bohm, in: Proc. Symp. on the Foundations of Modern Physics, Cologne, June 1, 1993. eds. P. Busch and P. Mittelstaedt (World Scientific, 1993), p. 77; A. Bohm, I. Antoniou and P. Kielanowski, Phys. Lett. A 189 (1994) 442; A. Bohm, I. Antoniou and P. Kielanowski, J. Math. Phys. 36 (1995) 2593.
77. G. Ludwig, Foundations of Quantum Mechanics, Vol. 1 (Springer, Berlin, 1983), Vol. II, (1985); G. Ludwig, An Axiomatic Basis for Quantum Mechanics, Vol. I (Springer, Berlin, 1983), Vol. II (1987).
78. G. Ludwig, Foundations of Quantum Mechanics, Vol. 1 (Springer, Berlin, 1983), Vol. II, (1985); G. Ludwig, An Axiomatic Basis for Quantum Mechanics, Vol. I (Springer, Berlin, 1983), Vol. II (1987).
79. G. Ludwig, in: Proc. Symp. on the Foundations of Modern Physics, Cologne, June 1, 1993, eds. P. Busch and P. Mittelstaedt (World Scientific, 1993), p. 242.
80. The importance of the initial conditions has been emphasized by R. Peierls and his school. R. Peierls, Surprises in Theoretical Physics (Princeton University Press, Princeton, 1979), Ch. 3.8, and More Surprises in Theoretical Physics, (Princeton University Press, Princeton, 1992), Ch. 5.3. Gamow's complex energy states have been defined as eigenstates of the Schrödinger equation with purely outgoing boundary conditions in Ref. [18].
81. The importance of the initial conditions has been emphasized by R. Peierls and his school. R. Peierls, Surprises in Theoretical Physics (Princeton University Press, Princeton, 1979), Ch. 3.8, and More Surprises in Theoretical Physics, (Princeton University Press, Princeton, 1992), Ch. 5.3. Gamow's complex energy states have been defined as eigenstates of the Schrödinger equation with purely outgoing boundary conditions in Ref. [18].
82. x as an extension not of the operator A† but of the operator A. One also uses linear continuous functional instead of antilinear functional, as done here. For selfadjoint. A, this distinction is not important, but for general linear operators (in particular, for unitary operators U†) the definition (3.3) is more practical.
83. K. Napiorkowski, Bull. Polish Academy Sci. 22 (1974) 1215; 23 (1975) 251. Not much is gained with this change (except that Dirac's bra-ket formalism can be made rigorous).
84. K. Napiorkowski, Bull. Polish Academy Sci. 22 (1974) 1215; 23 (1975) 251. Not much is gained with this change (except that Dirac's bra-ket formalism can be made rigorous).
85. E.P. Wigner, Symmetries and Reflections (Oxbow Press, 1979), p. 38.
86. x is essential, and script H sign is only of historical importance since properties such as the stability of matter are traditionally expressed by the Hilbert space properties of H (the Hilbert space spectrum is bounded from below).
87. This may have its origins in efforts to develop a basis for the HS quantum mechanics, e.g., J. Jauch, Foundations of Quantum Mechanics, (Addison-Wesley, Reading, MA, 1973); C. Piron, Foundations of Quantum Physics (Benjamin, New York, 1976); G. Ludwig, Die Grundlagen der Quantenmechanik (Springer, Berlin, 1954).
88. This may have its origins in efforts to develop a basis for the HS quantum mechanics, e.g., J. Jauch, Foundations of Quantum Mechanics, (Addison-Wesley, Reading, MA, 1973); C. Piron, Foundations of Quantum Physics (Benjamin, New York, 1976); G. Ludwig, Die Grundlagen der Quantenmechanik (Springer, Berlin, 1954).
89. This may have its origins in efforts to develop a basis for the HS quantum mechanics, e.g., J. Jauch, Foundations of Quantum Mechanics, (Addison-Wesley, Reading, MA, 1973); C. Piron, Foundations of Quantum Physics (Benjamin, New York, 1976); G. Ludwig, Die Grundlagen der Quantenmechanik (Springer, Berlin, 1954).
90. Note, however, thai this arrow of time - which we call the preparation → registration arrow below - is not the arrow of time originating from the irreversible act of measurement ("collapse of the wave function" axiom) [4, 6]. This later arrow is the extrinsic arrow of time [3] mentioned in Section 2.
91. A. Bohm, in: Symmetry Groups and Spectrum Generating Groups, eds. J.D. Hennig, et al., Lecture Notes in Physics, Vol. 379, (Springer, Berlin, 1991), p. 207.
92. T.D. Lee, Particle Physics and Introduction to Field Theory (Harwood Academic, Chur, 1981), Ch. 13. We shall restrict ourselves here to time reversal invariance (of the observables), though this need not always be fulfilled.
93. A. Bohm, Phys. Rev. A 51 (1995) 1758.
94. ± = {z ∈ ℂ, Im z (greater-then or less-then) 0} of the complex plane ℂ. Both areas, scattering theory in physics and the theory of Hardy class functions in mathematics, were developed independently of each other and without knowledge of the other. That they finally turned out to agree with each other [44] - except for the opposite sign of the label - is another example of what Wigner called "the unreasonable effectiveness of mathematics in natural sciences".
95. R.G. Newton, Scattering Theory of Waves and Particles (Springer, New York, 1966) Section 6.4; 2nd Ed. (1982); Ref. [20], 3rd Ed., Ch XV.
96. M.L. Goldberger and K.M. Watson, Collision Theory (Wiley, New York, 1964).
97. p Spaces (Academic Press, New York. 1970); K. Hoffman, Banach Spaces of Analytic Functions (Prentice-Hall, Englewood Cliffs, NJ, 1962; Dover Publications, Mineola, NY, 1988).
98. p Spaces (Academic Press, New York. 1970); K. Hoffman, Banach Spaces of Analytic Functions (Prentice-Hall, Englewood Cliffs, NJ, 1962; Dover Publications, Mineola, NY, 1988).
99. H. Baumgartel was the first to suggest the Hardy class functions to one of us, around 1977.
100. C. van Winter, Trans. Am. Math. Soc. 162 (1971) 103; J. Math. Anal. Appl. 47 (1974) 633.
101. C. van Winter, Trans. Am. Math. Soc. 162 (1971) 103; J. Math. Anal. Appl. 47 (1974) 633.
102. -1 = - iΓ follows from the unitarity of the S-matrix; cf. [20]. Ch. XVIII. 6.
103. G of (5.18b) does not lead to new-conditions, but leads to the same conditions (5.28) (5.29).
104. From the mathematical formulation (5.9) and (5.10) of the QAT one arrives at (5.28) and (5.29) (using a version of the Paley-Wiener theorem) for p = 2 only, since the Paley-Wiener theorem gives both necessary and sufficient conditions for p = 2 only. However, there may be other possible mathematical idealizations for the physical statement of the QAT besides (5.9) and (5.10), which would also lead to (5.28) and (5.29) with other values for p. We will assume in this paper that p = 2.
105. N.G. van Kampen, Phys. Rev. 91 (1953) 1267.
106. H.M. Nussenzweig, Causality and Dispersion Relations (Academic Press, NY, 1972); J.R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions (Wiley, New York, 1972), for potential scattering. Ref. [20], Ch. XVIII. 4, 5. For the "redundant" poles, see [42], Ch. 6.
107. H.M. Nussenzweig, Causality and Dispersion Relations (Academic Press, NY, 1972); J.R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions (Wiley, New York, 1972), for potential scattering. Ref. [20], Ch. XVIII. 4, 5. For the "redundant" poles, see [42], Ch. 6.
108. 2 splitting).
109. F.R. Gantmacher, Theory of Matrices, Vol. I, translated from the Russian by K.A. Hirsch (Chelsea, New York, 1958) Ch. VII; P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd Ed. (Academic Press, San Diego, California, 1985).
110. F.R. Gantmacher, Theory of Matrices, Vol. I, translated from the Russian by K.A. Hirsch (Chelsea, New York, 1958) Ch. VII; P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd Ed. (Academic Press, San Diego, California, 1985).
111. E.J. Brändas, C.A. Chatzidimitriou-Dreissmann. in: Resonances, E. Brändas and N. Elander eds., Lecture Notes in Physics, Vol. 325 (Springer, Berlin, 1987), p. 480-540.
112. I. Antoniou and M. Gadella, International Solvay Institute preprint (1995), to appear.
113. M.V. Berry and J.M. Robbins, Proc. R. Soc. Lond. A 442 (1993) 659.
114. F. Wilczek and A. Shapere, Geometric Phase in Physics (World Scientific, Singapore, 1989); C.A. Mead, The geometric phase in molecular systems, Rev. Mod. Phys. 64 (1992) 51; A. Bohm, The geometric phase in quantum physics, in: Integrable Systems, eds. L.A. Ibort, M.A. Rodriguez (Kluwer Academic Publishers, 1993) p. 347; Ref. [20], 3rd Ed., Ch. 22.
115. F. Wilczek and A. Shapere, Geometric Phase in Physics (World Scientific, Singapore, 1989); C.A. Mead, The geometric phase in molecular systems, Rev. Mod. Phys. 64 (1992) 51; A. Bohm, The geometric phase in quantum physics, in: Integrable Systems, eds. L.A. Ibort, M.A. Rodriguez (Kluwer Academic Publishers, 1993) p. 347; Ref. [20], 3rd Ed., Ch. 22.
116. F. Wilczek and A. Shapere, Geometric Phase in Physics (World Scientific, Singapore, 1989); C.A. Mead, The geometric phase in molecular systems, Rev. Mod. Phys. 64 (1992) 51; A. Bohm, The geometric phase in quantum physics, in: Integrable Systems, eds. L.A. Ibort, M.A. Rodriguez (Kluwer Academic Publishers, 1993) p. 347; Ref. [20], 3rd Ed., Ch. 22.
117. J.J.M. Verbaarschot, H.A. Weidenmüller and M.R. Zirnbauer, Phys. Rep. 129 (1985) 367.
118. E. Hernandez and A. Mondragon, Phys. Lett B 326 (1994) 1; A. Mondragon, et al., in: Non-linear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers. Singapore, 1995); A. Mondragon and E. Hernandez, J. Phys. A 26 (1993); 5595; E. Hernandez, A. Jauregui and A. Mondragon, Revista Mexicana de Fisica 38, Supl. 2 (1992).
119. E. Hernandez and A. Mondragon, Phys. Lett B 326 (1994) 1; A. Mondragon, et al., in: Non-linear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers. Singapore, 1995); A. Mondragon and E. Hernandez, J. Phys. A 26 (1993); 5595; E. Hernandez, A. Jauregui and A. Mondragon, Revista Mexicana de Fisica 38, Supl. 2 (1992).
120. E. Hernandez and A. Mondragon, Phys. Lett B 326 (1994) 1; A. Mondragon, et al., in: Non-linear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers. Singapore, 1995); A. Mondragon and E. Hernandez, J. Phys. A 26 (1993); 5595; E. Hernandez, A. Jauregui and A. Mondragon, Revista Mexicana de Fisica 38, Supl. 2 (1992).
121. E. Hernandez and A. Mondragon, Phys. Lett B 326 (1994) 1; A. Mondragon, et al., in: Non-linear, Deformed and Irreversible Quantum Systems, eds. H.D. Doebner et al. (World Scientific Publishers. Singapore, 1995); A. Mondragon and E. Hernandez, J. Phys. A 26 (1993); 5595; E. Hernandez, A. Jauregui and A. Mondragon, Revista Mexicana de Fisica 38, Supl. 2 (1992).
122. A. Bohm, The geometric phase of bound states and resonances, in: Workshop on Geometric Phase in Atomic and Molecular Physics, Las Vegas, Nevada, February. 1995, unpublished.
123. For definitions of Jordan operators and their properties, see H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators, Operator Theory Advances and Applications, Vol. 15 (Birkhäuser, Basel, 1985), Ch. 2. There are many other books on finite dimensional algebra and matrices, e.g. [52]. The account of the spectral theory of finite dimensional operators in this beautiful book of Baumgärtel's is the most suitable for the purpose of our generalization to the infinite dimensional case.
124. p|e〉 = 0. cf, [52].
125. Precisely, this means that there is no regeneration in a vacuum and, in particular, no regeneration without an interaction different from H.
126. s has vanished, without assuming a T-violating part in H. Whether such a mechanism is consistent with all available data requires more detailed investigations.
127. s has vanished, without assuming a T-violating part in H. Whether such a mechanism is consistent with all available data requires more detailed investigations.
128. The exact quantum theory using the Hilbert space formulation in contrast cannot avoid regeneration due to the deviation from the exponential law for any vector of the Hilbert space. L.A. Khalfin, JETP Lett. 15 (1972) 388.
129. 8Be, see another explanation in Ref. [20], Ch. XX.2b.
130. W.H. Zurek, Phys. Rev. D 26 (1982) 1862; W.H. Zurek and J.P. Paz, in: Proc. Symp. on the Foundations of Modern Physics, Cologne, June 1, 1993, Eds. P. Busch and P. Mittelstaedt (World Scientific, 1993), p. 458; H.D. Zeh [8] and references thereof: R. Omnes, Rev. Mod. Phys. 64 (1992) 339; M. Gell-Mann and J.B. Hartle, in: Complexity, Entropy and the Physics of Information, Santa Fe Institute Studies in the Science of Complexity, Vol. VIII, ed. W. Zurek (Addison-Wesley, Reading, MA, 1990); M. Gell-Mann and J.B. Hartle, Phys. Rev. D 47 (1993) 3345.
131. W.H. Zurek, Phys. Rev. D 26 (1982) 1862; W.H. Zurek and J.P. Paz, in: Proc. Symp. on the Foundations of Modern Physics, Cologne, June 1, 1993, Eds. P. Busch and P. Mittelstaedt (World Scientific, 1993), p. 458; H.D. Zeh [8] and references thereof: R. Omnes, Rev. Mod. Phys. 64 (1992) 339; M. Gell-Mann and J.B. Hartle, in: Complexity, Entropy and the Physics of Information, Santa Fe Institute Studies in the Science of Complexity, Vol. VIII, ed. W. Zurek (Addison-Wesley, Reading, MA, 1990); M. Gell-Mann and J.B. Hartle, Phys. Rev. D 47 (1993) 3345.
132. W.H. Zurek, Phys. Rev. D 26 (1982) 1862; W.H. Zurek and J.P. Paz, in: Proc. Symp. on the Foundations of Modern Physics, Cologne, June 1, 1993, Eds. P. Busch and P. Mittelstaedt (World Scientific, 1993), p. 458; H.D. Zeh [8] and references thereof: R. Omnes, Rev. Mod. Phys. 64 (1992) 339; M. Gell-Mann and J.B. Hartle, in: Complexity, Entropy and the Physics of Information, Santa Fe Institute Studies in the Science of Complexity, Vol. VIII, ed. W. Zurek (Addison-Wesley, Reading, MA, 1990); M. Gell-Mann and J.B. Hartle, Phys. Rev. D 47 (1993) 3345.
133. W.H. Zurek, Phys. Rev. D 26 (1982) 1862; W.H. Zurek and J.P. Paz, in: Proc. Symp. on the Foundations of Modern Physics, Cologne, June 1, 1993, Eds. P. Busch and P. Mittelstaedt (World Scientific, 1993), p. 458; H.D. Zeh [8] and references thereof: R. Omnes, Rev. Mod. Phys. 64 (1992) 339; M. Gell-Mann and J.B. Hartle, in: Complexity, Entropy and the Physics of Information, Santa Fe Institute Studies in the Science of Complexity, Vol. VIII, ed. W. Zurek (Addison-Wesley, Reading, MA, 1990); M. Gell-Mann and J.B. Hartle, Phys. Rev. D 47 (1993) 3345.
134. W.H. Zurek, Phys. Rev. D 26 (1982) 1862; W.H. Zurek and J.P. Paz, in: Proc. Symp. on the Foundations of Modern Physics, Cologne, June 1, 1993, Eds. P. Busch and P. Mittelstaedt (World Scientific, 1993), p. 458; H.D. Zeh [8] and references thereof: R. Omnes, Rev. Mod. Phys. 64 (1992) 339; M. Gell-Mann and J.B. Hartle, in: Complexity, Entropy and the Physics of Information, Santa Fe Institute Studies in the Science of Complexity, Vol. VIII, ed. W. Zurek (Addison-Wesley, Reading, MA, 1990); M. Gell-Mann and J.B. Hartle, Phys. Rev. D 47 (1993) 3345.
135. J. Elster and H.J.F. Geitel discovered exponential decay just before the turn of the century. In 1900, Rutherford reported the decay of thorium to be exponential in an issue of Philosophical Magazine, based on his own cloud chamber observations. We can thus fairly say that the exponential decay law has been a cornerstone of much of experimental physics since 1900.
136. D could be any vector, e.g., an element of Φ or of script H sign. In the Hilbert space theory, however, it has long been known that the theory predicts deviations from the exponential law [64, 67].
137. L. Fonda. G.C. Ghirardi and A. Rimini, Rep. Prog. Phys. 41 (1978) 587 and references thereof.
138. E.B. Norman et al., Phys. Rev. Lett. 60 (1988) 2246, and references thereof.
139. G.C. Hegerfeldt, Phys. Rev. Lett. 72 (1994) 596.
140. e dimisses this possibility of script P sign(t) = 0 as "unphysical", whereas we dismiss as "unphysical" the other alternative of Ref. [71], namely that script P sign(t) has always been non-zero.
141. The calculation uses the Lippmann-Schwinger equation (5.5). which is well accepted, but which may not be considered mathematically unimpeachable so long as one is not presented with a mathematically rigorous scattering theory based on it.