Abstract
After a review of the arrows of time, we describe the possibilities of a time-asymmetry in quantum theory. Whereas Hilbert space quantum mechanics is time-symmetric, the rigged Hilbert space formulation, which arose from Dirac’s braket formalism, allows the choice of asymmetric boundary conditions analogous to the retarded solutions of the Maxwell equations for the radiation arrow of time. This led to irreversibility on the microphysical level as exemplified by decaying states or resonances. Resonances are mathematically represented by Gamow kets, functionals over a space of very well-behaved (Hardy class) vectors, which have been chosen by a boundary condition (outgoing for decaying states). Gamow states have all the properties that one heuristically needs for quasistable states. For them a Golden Rule can be derived from the fundamental probabilities P(t)=Tr(Λ(t)WGamow (t0)) that fulfills the time-asymmetry condition t≥t 0 which could not be realized in the Hilbert space.
Preview
Unable to display preview. Download preview PDF.
References
Physical Origins of Time Asymmetry, J.J. Halliwell, et al., eds., (Cambridge University Press, 1994).
P.C.W. Davis, The Physics of Time Asymmetry, (University of California Press, 1977); R. Penrose, Singularities and Time-asymmetry, in General Relativity: Einstein Centenary Survey, S.W. Hawking, et al., eds., (Cambridge University Press, 1979); H.D. Zeh, The Physical Basis of the Direction of Time, (Springer-Verlag, 1989).
R. Peierls, Surprises in Theoretical Physics, Sect. 3.8, (Princeton University Press, Princeton, 1979).
R. Ritz, Physikalische Zeitschrift 9 (1908) 903
A. Einstein, Physikalische Zeitschrift 10 (1909) 185
R. Ritz, Physikalische Zeitschrift 10 (1909) 224
R. Ritz and A. Einstein, Physikalische Zeitschrift 10 (1909) 323.
J.A. Wheeler and R.P. Feynman, Rev. Mod. Phys. 21 (1949) 425.
G. Süssmann, in Nonlinear, Deformed and Irreversible Quantum Systems, H.D. Doebner, et al., eds., (World Scientific Publ., 1995), p. 98. See also F. Hoyle and J.V. Narlikar, Proc. Roy. Soc. A 277 (1964) 1, and J.E. Hogarth, Proc. Roy. Soc. A 267 (1962) 365, who discuss these questions within the framework of ref. [5].
G. Ludwig, Foundations of Quantum Mechanics, Volume I, (Springer-Verlag, Berlin, 1983) and Volume II, (1985)
An Axiomatic Basis of Quantum Mechanics, Volume I, (Springer-Verlag, Berlin, 1983) and Volume II, (1987)
K. Kraus, State, Effects and Operations, Springer Lecture Notes in Physics 190, (Springer-Verlag, Berlin, 1983).
M. Gell-Mann and J.B. Hartle, in Complexity, Entropy and the Physics of Information, SFI Studies in Science and Complexity Vol. VIII, W. Zurek, ed., (1990)
M. Gell-Mann and J.B. Hartle in ref. [1], p. 311
R.B. Griffiths, J. Stat. Phys., 36 (1984) 219
R.B. Griffiths in Symposium on the Foundation of Modern Physics 1994, K.V. Laurikainen, et al., eds., (Editions Frontières, 1984), p. 85.
I. Prigogine, Non-Equilibrium Statistical Mechanics (Wiley, New York, 1962)
E.B. Davis, Quantum Theory of Open Systems, (Academic Press, London, 1976) where detailed references to the original papers can be found.
K. Kraus, Ann. Phys. 64 (1971) 311
A. Kossakowski, Rep. Math. Phys. 3 (1972) 247
G. Lindblad, Commun. Math. Phys. 40 (1975) 147
48 (1976) 119
V. Gorini, A. Kossakowski and E.C.G. Sudarshan, J. Math. Phys. 17 (1976) 821, A. Kossakowski, On Dynamical Semigroups and Open Systems, this volume and references thereof.
A special case of this irreversible time evolution is obtained if one chooses for the reservoir R the measuring apparatus. It has been shown that the collapse axiom (2) together with the Schrödinger equation (1c) leads to semigroup evolutions (10) generated by a Liouvillian L
G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34 (1986) 470
I. Antoniou and S. Tasaki, Int. J. Quant. Chem. 46 (1993) 425.
P. Huet and M.E. Peskin, Nucl. Phys. B 434 (1995) 3
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
CERN-TH-6755/92 (1992)
Fabio Benatti, Complete Positivity and Neutral Kaon Decay, this volume.
C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics: Volume II (Wiley, New York, 1977), p. 1345.
A. Bohm, S. Maxson, Mark Loewe and M. Gadella, Physica A 236 (1997) 485.
P.A.M. Dirac, The Principles of Quantum Mechanics, (Clarendon Press, Oxford, 1930).
N. Bourbaki, Élements de Mathématique, (Hermann, Paris, 1953).
L. Schwartz, Théorie des Distributions, (Hermann, Paris, 1950).
I.M. Gel'fand and N. Ya. Vilenkin, Generalized Functions, Volume 4, (Moscow, 1961) (English trans. Academic Press, New York, 1964)
K. Maurin, Generalized Eigenfunction Expansions and Unitary Representations of Topological Groups. (Polish Scientific Publishers, Warzawa, 1968).
H. Weyl, Gruppentheorie und Quantenmechanik, (S. Hirzel, Leipzig, 1928).
J. von Neumann, Mathematische Grundlagen der Quantentheorie, (Springer, Berlin, 1931) (English translation by R.T. Beyer) (Princeton University Press, Princeton, 1955).
K. Maurin, Analysis, Part II, (Polish Scientific, Warsaw, 1980), Chap. XIII
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1, (Academic Press, San Diego, 1980).
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, Singapore, 1993), p. 458.
N. van Kampen, Physica A 153 (1988) 97
R. Omnès, Reviews of Modern Physics 64 (1992) 339.
A. Bohm, Quantum Mechanics, 1st Ed. (Springer, New York, 1979)
3rd Ed. (1993).
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; see also O. Melsheimer, J. Math. Phys., 15 (1974) 902; 917.
A. Bohm and M. Gadella. Dirac Kets, Gamow Vectors and Gel'fand Triplets, Lecture Notes in Physics, Volume 348, (Springer-Verlag, Berlin, 1989).
L.A. Khalfin, JETP Lett. 15 (1972) 388; see also L. A. Khalfin, this volume.
L. Fonda, G.C. Ghirardi and A. Rimini, Repts. on Prog. in Phys., 41 (1978) 587, and references thereof.
G. Gamow, Z. Phys. 51 (1928) 204.
H. Frauenfelder and E. M. Henley, Subatomic Physics (Prentice Hall, Englewood Cliffs, N.J., 1991).
T.D. Lee, R. Oehme, and C.N. Yang, Phys. Rev. 106 (1957) 340.
Particle Data Group, Review of Partical Physics, Phys. Rev. D 54 (1996) 1.
G.C. Hegerfeldt, Phys. Rev. Lett. 72 (1994) 596.
D. Buchholz and J. Yugvason, Phys. Rev. Lett. 73 (1994) 613
P.W. Milonni, D.F.V. James and H. Fearn, Phys. Rev. A 52 (1995) 1525.
R. Peierls, More Surprises in Theoretical Physics, (Princeton University Press, 1992). Gamow's complex energy state have been defined as eigenstates of the Schrödinger equation with purely out-going boundary conditions in the following: P.L. Kapur and R. Peierls, Proc. Roy. Soc. A166 (1938) 277
R. Peierls, Proceedings of the 1954 Glasgow Conference on Nuclear and Meson Physics, E.M. Bellamy, et al., editors, (Pergamon Press, New York, 1955)
G. Garcia-Calderon and R. Peierls, Nuclear Physics A 265 (1976) 443
E. Hernandez and A. Mondragon, Phys. Rev. C 29 (1984) 722
A. Mondragon and E. Hernandez, Annual der Physik 48 (1991) 503
G. Garcia-Calderon, Symmetries in Physics (Moshinsky Symposium), Eds. A. Frank and K.B. Wolf, (Springer-Verlag, Berlin, 1992).
T.D. Lee, Particle Physics and Introduction to Field Theory, Chapter 13, (Harwood Academic, New York, 1981). In this reference the quantum mechanical time reversed state is called complicated and improbable.
A. Bohm, J. Math. Phys. 22 (1981) 2813
Lett. Math. Phys. 3 (1978) 455 (1978).
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. Pertosky, I. Prigogine and S. Tasaki, Physica A 173 (1991) 175
T. Pertosky and I. Prigogine, Physica A 147 (1988) 439
T. Pertosky and I. Prigogine, Physica A 175 (1991) 146
M. de Haan, C. George, and F. Mayné, Physica A 92 (1978) 584.
I. Antoniou, Proc. 2nd Internat. Wigner Symposium, Gosla 1991, Eds. H.D. Doebner, et al., (World Scientific, Singapore, 1992). I. Antoniou and I. Prigogine, Physica Ia, 192 (1993) 443. I antoniou and S. Tasaki, Intern. Journ. Quantum Chemistry 46 (1993) 425.
K. Napiorkowski, Bulletin of the Polish Academy of Sciences 22 (1974) 1215
23 (1975) 251
Not much is gained with this change (except that Dirac's bra-ket formalism can be made rigorous).
E.P. Wigner, Symmetries and Reflections, (Indiana University Press, Bloomington, 1967), page 38.
P.L. Duren, H p Spaces, (Academic Press, New York, 1970)
K. Hoffman, Banach Space of Analytic Functions, (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962; Dover Publications, Mineola, NY, 1988).
M. Gadella, J. Math. Phys. 24 (1983) 1462.
A. Bohm, Proceedings, Group Theoretical Methods in Physics, Springer Lecture Notes in Physics, Vol. 94 (Springer, Berlin, 1978) 245; and also [7], 1st Ed., Chap. XXI. H. Baumgartel mentioned the Hardy class functions around 1977 in a private communication.
A. Bohm, I. Antoniou, P. Kielanowski, Phys. Lett. A189 (1994) 442
A. Bohm, I. Antoniou, P. Kielanowski, J. Math. Phys. 36 (1995) 2593.
S. Weinberg, The Quantum Theory of Fields, Vol. 1, Chapter 2, Appendix C, (Cambridge University Press, Cambridge, 1995); also in [7], chapter XIX.
E.P. Wigner, Group Theoretical Concepts and Methods in Elementary Particle Physics, Ed. F. Gürsey (Gordon and Breach, New York, 1994), p. 37.
A. Bohm, Phys. Rev. A 51 (1995) 1978
A. Bohm and Sujeewa Wickramasekara, Found. of Phys. 27 (1997) 969.
I. Antoniou, Z. Suchanecki, and S. Tasaki, to appear in Generalized Functions, Operator Theory and Dynamical Systems, eds. I. Antoniou and G. Lumer, (Addison Wesley Longman, London, 1998).
I. Antoniou, L. Dmetrieva, Yu. Kuperin and Yu. Melnikov, Comp. Math. Appl., 34 (1997) 399; see also I. Antoniou and Yu. Melnikov, Quantum Scattering Resonances: Poles of a continued S-matrix and Poles of an extended Resolvant, this volume.
I. Antoniou and M. Gadella, preprint (Intern. Solvay Institute, Brussels, 1995). Results of this preprint were published in: A. Bohm et al., Rep. Math. Phys. 36 (1995) 245.
A. Bohm, M. Loewe, S. Maxson, P. Patuleanu, C. Püntmann, and M. Gadella, on WWW at http://www.ph.utexas.edu/~bohmwww, JMP (1997) to be published.
M.L. Goldberger and K.M. Watson, Phys. Rev. B 136 (1964) 1472
M.L. Goldberger and K.M. Watson, Collision Theory, (Wiley, New York, 1964)
R.G. Newton, Scattering Theory of Waves and Particles, 2nd ed. (Springer-Verlag, 1982).
C. van Winter, Trans. Am. Math. Soc. 162 (1972) 103
J. Math. Anal. and Appl. 47 (1974) 633.
E. Fermi, Nuclear Physics, (University of Chicago Press, 1950).
An exponential law for transition probabilities in atoms was first obtained by V.E. Weisskopf and E.P. Wigner, Zeitschr. f. Physik 63 (1930) 54
65 (1930) 18, in the Weisskopf-Wigner approximation, cf. also Chap. 8 of the first reference of [53].
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag
About this paper
Cite this paper
Bohm, A., Harshman, N.L. (1998). Quantum theory in the rigged hilbert space — Irreversibility from causality. In: Bohm, A., Doebner, HD., Kielanowski, P. (eds) Irreversibility and Causality Semigroups and Rigged Hilbert Spaces. Lecture Notes in Physics, vol 504-504. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0106783
Download citation
DOI: https://doi.org/10.1007/BFb0106783
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64305-0
Online ISBN: 978-3-540-69725-1
eBook Packages: Springer Book Archive