Abstract
We analyze the quantization of dynamical systems that do not involve any background notion of space and time. We give a set of conditions for the introduction of an intrinsic time in quantum mechanics. We show that these conditions are a generalization of the usual procedure of deparametrization of relational theories with Hamiltonian constraint that allow one to include systems with an evolving Hilbert space. We apply our quantization procedure to the parametrized free particle and to some explicit examples of dynamical systems with an evolving Hilbert space. Finally, we conclude with some considerations concerning the quantum gravity case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ashtekar, A. (1986).Physical Review Letters,57, 2244.
Dirac, P. A. M. (1964).Lectures on Quantum Mechanics, Yeshiva University Press, New York.
Hartle, J. (1991). Time and predictions in quantum cosmology, inConceptual Problems of Quantum Gravity, A. Ashtekaret al., eds., Birkhäuser, Boston.
Isham, C. (1992). Canonical quantum gravity and the problem of time, Imperial College preprint Imperial/TP/91–92/25.
Jacobson, T. (1991). Black hole thermodynamics and the space time discontinuum, inConceptual Problems of Quantum Gravity, A. Ashtekaret al., eds., Birkhäuser, Boston.
Kuchar, K. (1992a). Time and interpretations of quantum gravity, inProceedings of the 4th Canadian Conference on General Relativity and Astrophysics, G. Kunstatteret al., eds., World Scientific, Singapore.
Kuchar, K. (1992b). Canonical quantum gravity, inProceedings of the 13th International Conference on General Relativity and Gravitation, R. Gleiseret al., eds., Institute of Physics, Bristol and Philadelphia.
Rovelli, C., and Smolin, L. (1990).Nuclear Physics B,133, 80.
Smolin, L. (1991). Space and time in the quantum universe, inConceptual Problems of Quantum Gravity, A. Ashtekaret al., eds., Birkhäuser, Boston.
Sundermeyer, K. (1982).Constrained Dynamics, Springer-Verlag, Berlin.
Unruh, W. (1993). Time in quantum gravity and quantum mechanics, British Columbia University preprint.
Unruh, W., and Wald, R. (1989).Physical Review D,40, 2598.
Von Neumann, J. (1955).Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, New Jersey.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Doldán, R., Gambini, R. & Mora, P. Quantum mechanics for totally constrained dynamical systems and evolving hilbert spaces. Int J Theor Phys 35, 2057–2074 (1996). https://doi.org/10.1007/BF02302226
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02302226