Time-asymmetric quantum physics

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 60, Issue 2, 1999, Pages 861-876

  Bohm, A ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

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EID: 0000201271     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.60.861     Document Type: Article

References (65)

1. The literature on resonances and decay is so large that it is difficult to list here even a representative selection. The standard monograph is M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964). The irreversible character of quantum-mechanical decay has rarely even been mentioned [exceptions are C. Cohen-Tannoudji et al., Quantum Mechanics (Wiley, New York, 1977), Vol. II, p. 1345; Lee 17 below] and to our knowledge has not been incorporated in a theory of decay.
2. M. Gell-Mann and J. B. Hartle, in Physical Origins of Time Asymmetry, edited by J. J. Halliwell et al (Cambridge University, Cambridge, England, 1994), p. 311.
3. We are not concerned here with irreversibility in the quantum theory of open systems for which the asymmetric time evolution is described by a Liouville equation containing terms for the effects of the external reservoir. The difference is explained in Appendix C.
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19. Some [e.g., A. Pais, in CP Violation, edited by J. Tran, Thanh Van (Editions Frontiers, Gif-sur Yvette, France, 1990)] reserve the name “particle” for an object with a unique lifetime (in addition to the unique mass), in distinction to such superpositions as (Formula presented) and (Formula presented) which are the states in which the neutral kaon system is prepared. In our theory, these exact prepared states like the (Formula presented) are the (Formula presented) of Eq. (3.25), which, in addition to the quasistable particle states like (Formula presented) representing (Formula presented), also contain a background integral.
20. V. Weisskopf and E. P. Wigner, Z. Phys. 63, 54 (1930)
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28. Nondiagonalizable complex Hamiltonian matrices (Jordan blocks) led to Jordan vectors instead of eigenvectors. Jordan vectors have a nonexponential time evolution of the order of (Formula presented) (and are therefore ruled out as decaying state vectors). H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators, Operator Theory Vol. 15 (Birkhäuser Basel, 1985), Chap. 2;
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32. A. Mondragon and E. Hernandez, J. Phys. A 26, 5595 (1993).However, the Jordan block Hamiltonians can also be shown to result from a truncation of the exact rigged Hilbert space theory to an effective theory. Its higher order Gamow states, which correspond to higher order S-matrix poles, and are not representable by vectors but by state operators have an exponential asymmetric time evolution;
33. A. Bohm, M. Loewe, S. Maxson, P. Patuleanu, C. Püntmann, and M. Gadella, J. Math. Phys. 38, 6072(1997).
34. M. Levy, Nuovo Cimento 13, 115 (1959).
35. T. D. Lee, Particle Physics and Introduction to Field Theory (Harwood Academic, New York, 1981), Chap. 13. In this reference the quantum-mechanical time reversed state is called complicated and improbable.
36. A. Kossakowski, in Irreversibility and Causality, edited by A. Bohm, H. D. Doebner, and P. Kielanowski (Springer, Berlin, 1998), p. 59, where further references to this irreversibility of quantum statistics can be found.
37. M. Gell-Mann and J. B. Hartle, in Complexity, Entropy and the Physics of Information, SFI Studies in Science and Complexity Vol. VIII, edited by W. Zurek, (Addison-Wesley, Redwood City, CA, 1990).
38. M. Gell-Mann and J. B. Hartle, University of California at Santa Barbara, Report No. UCSBTH-95-28, 1995; e-print gr-qc/9509054.
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41. A. Bohm and M. Gadella, Dirac Kets, Gamow Vectors and Gel’fand Triplets, Lecture Notes in Physics Vol. 348 (Springer-Verlag, Berlin, 1989).
42. The Gamow vectors were envisioned by G. Gamow [Z. Phys. 51, 204 (1928)] as far back as the inception of the Dirac kets. Mathematically, they were no more ill defined than the Dirac kets, though they never acquired the popularity of the latter. The reason was that, although for Dirac kets at least the probability density is finite everywhere, the probability density of Gamow’s wave functions (Formula presented) increased exponentially for large values of the distance r and large negative values of time t (the “exponential catastrophe”). The origin of this exponential catastrophe is that the emission of decay products had been assumed to go on for an arbitrarily long time, (Formula presented), as dictated by the unitary group (Formula presented) (Formula presented). In reality, the emission of decay products must have begun at some finite time (Formula presented) in the past, and the time evolution of the decaying state, as we now know, is described by the semigroup (Formula presented) (Formula presented). Therefore, in the time-asymmetric theory the probability density (Formula presented) is only defined for times (Formula presented), and for these times one can show that there exists no exponential catastrophe for the probability densities. Thus the semigroup evolution of the Gamow vectors is not only a consequence of the mathematical theory, but also a necessity for the physical interpretation of the position probabilities (Formula presented).
43. A. Bohm and N. L. Harshmann, in Irreversibility and Causality, edited by A. Bohm, H. D. Doebner, P. Kielanowski (Springer, Berlin, 1998), p. 225, Sec. 7.4.
44. The function (Formula presented) is a very well-behaved function of the upper half plane (Formula presented) if it is well behaved, i.e., (Formula presented) (Schwartz space) and if it is the boundary value of an analytic function (Formula presented) in the upper half plane (Formula presented) which vanishes faster than any power at the infinite semicircle [i.e., (Formula presented). Similarly the function (Formula presented) is a very well-behaved function of the lower half plane (Formula presented) if it is well behaved (Formula presented) and if it is the boundary value of an analytic function in the lower half plane (Formula presented) which vanishes sufficiently fast at the lower semicircle (Formula presented). For the definition of Hardy class functions and their mathematical properties needed here see Appendix A.2 of Ref. 27 and P. L. Duren, (Formula presented) Spaces, (Academic Press, New York, 1970).
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48. M. Gadella, J. Math. Phys. 24, 1462 (1983); and private communication.
49. A. Bohm, Quantum Mechanics, 3rd ed. (Springer-Verlag, New York, 1994) Sec. XVIII.6-9, XX.3.
50. For a detailed analysis of the (Formula presented)-(Formula presented) system in the framework of time-asymmetric quantum mechanics with and without (Formula presented) violation, see A. Bohm, e-print hep-th/970542.
51. This is the simplified schematic diagram of several generations of experiments measuring CP violation in the neutral kaon system; J. Christenson, J. Cronin, V. Fitch, and R. Turley, Phys. Rev. Lett. 13, 138 (1964)
52. K. Kleinknecht, in CP Violation, edited by C. Jarskog (World Scientific, 1989), p. 41, and references therein;
53. G. D. Barr et al, Experiment no. NA31, Phys. Lett. B 317, 233 (1993), Experiment no. NA31;
54. L. K. Gibbons et al, E731, Phys. Rev. Lett. 70, 1199 (1993), experiment no. E731;
55. Phys. Rev. Lett.L. K. Gibbons70, 1203 (1993).
56. This approximate agreement between a sequence of rational numbers (Formula presented) on the experimental side and a continuous function of real numbers (Formula presented) on the theoretical side is the fundamental limit to which the exponential law can be verified; experimental limitations given by the resolution, e.g., the finite size of (Formula presented), are still more important. Therefore any infinitesimal (not given in terms of the scale (Formula presented) deviations from the exponential law that are derived from a mathematical theory (Formula presented)e.g., the Hilbert space idealization, L. A. Khalfin, Pis’ma Zh. Éksp. Teor. Fiz. 15, 548 (1972) [JETP Lett. 15, 388 (1972)], or our (Formula presented)-space idealizations(Formula presented), are physically meaningless.
57. Observed deviations from the exponential law [S. R. Wilkinson et al, Nature (London) 387, 575 (1997)] are in the RHS theory explained by the basis vector expansion (3.25) and (3.28) for the experimantally prepared state (Formula presented). In addition to the exponential term, Eq. (3.30) predicts less than exponential time dependence for the background term.
58. The (Formula presented) is a relativistic decaying system and our discussion here is in terms of the non-relativistic Gamow vectors. Relativistic Gamow kets can be defined from the poles of the relativistic S matrix at the value of the invariant mass square (Formula presented). They have at rest the same semigroup time evolution as Eq. (3.20) with (Formula presented) and (Formula presented) A. Bohm, H. Kaldass, et al., e-print hetp-th/9905213;
59. A. BohmH. Kaldasse-print hetp-th/9904053.
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