Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 086, 13 pages      arXiv:1109.0598
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Time Asymmetric Quantum Mechanics

Arno R. Bohm a, Manuel Gadella b and Piotr Kielanowski c
a) Department of Physics, University of Texas at Austin, Austin, TX 78712, USA
b) Departamento de FTAO, Universidad de Valladolid, 47071 Valladolid, Spain
c) Cinvestav, Dept Fis, Mexico City 07000, DF Mexico

Received January 30, 2011, in final form August 22, 2011; Published online September 03, 2011

The meaning of time asymmetry in quantum physics is discussed. On the basis of a mathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group. Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution. Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)). If one requires that scattering resonances of width Γ and exponentially decaying states of lifetime τ=h/Γ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution t0t<∞, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system. The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry. From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics.

Key words: resonances; arrow of time; Hardy spaces.

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  1. Lee T.D., Particle physics and introduction to field theory, Chapter 13, Contemporary Concepts in Physics, Vol. 1, Harwood Academic Publishers, Chur, 1981.
  2. Bohm A., Harshman N.L., Quantum theory in rigged Hilbert space: irreversibility from causality, in Irreversibility and Causality, Semigroups and Rigged Hilbert Spaces (Goslar, 1996), Lecture Notes in Phys., Vol. 504, Editors A. Bohm, H.D. Doebner and P. Kielanowski, Springer, Berlin, 1998, 181-237, quant-ph/9805063.
  3. Bohm A., Time-asymmetric quantum physics, Phys. Rev. A 60 (1999), 861-876, quant-ph/9902085.
  4. Bohm A., Loewe M., Van de Ven B., Time asymmetric quantum theory. I. Modifying an axiom of quantum physics, Fortschr. Phys. 51 (2003), 551-568, quant-ph/0212130.
  5. Bohm A., Kaldass H., Wickramasekara S., Time asymmetric quantum theory. II. Relativistic resonances from S-matrix poles, Fortschr. Phys. 51 (2003), 569-603, hep-th/0212280.
  6. Bohm A., Kaldass H., Wickramasekara S., Time asymmetric quantum theory. III. Decaying states and the causal Poincaré semigroup, Fortschr. Phys. 51 (2003), 604-634, hep-th/0212282.
  7. Bohm A., Sato Y., Relativistic resonances: their masses, widths, lifetimes, superposition, and causal evolution, Phys. Rev. D 71 (2005), 085018, 22 pages, hep-ph/0412106.
  8. Bohm A., Bryant P.W., From Hardy spaces to quantum jumps: a quantum mechanical beginning of time, Internat. J. Theoret. Phys. 50 (2011), 2094-2105, arXiv:1011.4954.
  9. Gel'fand I.M., Vilenkin N.Ya., Generalized functions, Vol. 4, Applications of harmonic analysis, Academic Press, New York, 1964.
  10. Maurin K., Generalized eigenfunction expansions and unitary representations of topological groups, Polish Scientific Publishers, Warszawa, 1968.
  11. Dirac P.A.M., The principles of quantum mechanics, Clarendon Press, Oxford, 1930.
  12. Bohm A., Rigged Hilbert space and the mathematical description of physical systems, Boulder Lecture Notes in Theoretical Physics, Vol. 9A, Gordon and Breach Science Publishers, New York, 1967, 255-317.
  13. Bohm A., Gadella M., Dirac kets, Gamow vectors and Gel'fand triplets, Lecture Notes in Physics, Vol. 348, Springer-Verlag, Berlin, 1989.
  14. Gadella M., Gómez F., A unified mathematical formalism for the Dirac formulation of quantum mechanics, Found. Phys. 32 (2002), 815-869.
  15. Civitarese O., Gadella M., Physical and mathematical aspects of Gamow states, Phys. Rep. 396 (2004), 41-113.
  16. Baumgärtel H., Time asymmetry in quantum mechanics: a pure mathematical point of view, J. Phys. A: Math. Theor. 41 (2008), 304017, 7 pages.
  17. Baumgärtel H., Resonances of quantum mechanical scattering systems and Lax-Phillips scattering theory, J. Math. Phys. 51 (2010), 113508, 20 pages.
  18. Paley R., Wiener N., Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, Vol. 19, American Mathematical Society, Providence, RI, 1934.
  19. van Winter C., Complex dynamical variables for multiparticle systems with analytic interactions. I, J. Math. Anal. Appl. 47 (1974), 633-670.
  20. Lippmannn B.A., Schwinger J., Variational principles for scattering processes. I, Phys. Rev. 79 (1950), 469-480.
  21. Gell-Mann M., Goldberger H.L., The formal theory of scattering, Phys. Rev. 91 (1953), 398-408.
  22. Goldberger M.L., Watson K.M., Collision theory, John Wiley & Sons, Inc., New York - London - Sydney, 1964.
  23. Gadella M., Gómez F., The Lippmann-Schwinger equations in the rigged Hilbert space, J. Phys. A: Math. Gen. 35 (2002), 8505-8511.
  24. Bohm A., Quantum mechanics, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1986, Chapter XXI.
  25. Baumgärtel H., A private communication to A. Bohm, 1978.
  26. Baumgärtel H., Generalized eigenvectors for resonances in the Friedrichs model and their associated Gamow vectors, Rev. Math. Phys. 16 (2006), 61-78, Addendum, Rev. Math. Phys. 19 (2007), 227-229, math-ph/0509062.
  27. Baumgärtel H., Time asymmetry in quantum mechanics: a pure mathematical point of view, J. Phys. A: Math. Theor. 41 (2008), 304017, 7 pages.
  28. Duren P.L., Theory of Hp spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York - London, 1970.

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