Abstract
The upside-down simple harmonic oscillator system is studied in the contexts of quantum mechanics and classical statistical mechanics. It is shown that in order to study in a simple manner the creation and decay of a physical system by way of Gamow vectors we must formulate the theory in a time-asymmetric fashion, namely using two different rigged Hilbert spaces to describe states evolving toward the past and the future. The spaces defined in the contexts of quantum and classical statistical mechanics are shown to be directly related by the Wigner function.
Similar content being viewed by others
References
Aquilano, R., and Castagnino, M. (1996).Modern Physics Letters A, in press.
Antoniou, I. E., and Prigogine, I. (1993).Physica A,192, 443.
Antoniou, I., and Tasaki, S. (1992).Physica A,190, 303.
Antoniou, I., and Tasaki, S. (1993).Journal of Physics A,26, 73.
Balazs, N. L., and Voros, A. (1990).Annals of Physics,199, 123.
Barton, G. (1986).Annals of Physics,166, 322.
Bohm, A. (1986).Quantum Mechanics: Foundations and Applications, 2nd ed., Springer-Verlag, Berlin.
Bohm, A., and Gadella, M. (1989).Dirac Kets, Gamow Vectors and Gel'fand Triplets: The Rigged Hilbert Space Formulation of Quantum Mechanics, Springer-Verlag, Berlin.
Bollini, C. G., and Oxman, L. E. (1993).Physical Review A,47, 2339.
Cohen-Tannoudji, C., Diu, B., and Laloe, F. (1977).Quantum Mechanics, Vol. 1, Wiley, New York.
Davies, P. C. W. (1994). Stirring up trouble, inPhysical Origin of Time Asymmetry J. J. Haliwellet al., eds., Cambridge University Press, Cambridge.
Gel'fand, I. M., and Shilov, N. Y. (1964).Generalized Functions, Vol. 4,Applications of Harmonic Analysis, Academic Press, New York.
Hillery, M., O'Connell, R. F., Scully, M. O., and Wigner, E. P. (1984).Physics Reports,106, 121.
Merzbacher, E. (1970).Quantum Mechanics, 2nd ed., Wiley, New York.
Reichenbach, H. (1956).The Direction of Time, University of California Press, Berkeley.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Castagnino, M., Diener, R., Lara, L. et al. Rigged Hilbert spaces and time asymmetry: The case of the upside-down simple harmonic oscillator. Int J Theor Phys 36, 2349–2369 (1997). https://doi.org/10.1007/BF02768929
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02768929