Advances in Mathematical Physics
Volume 2009 (2009), Article ID 859710, 83 pages
Review Article

Time as a Quantum Observable, Canonically Conjugated to Energy, and Foundations of Self-Consistent Time Analysis of Quantum Processes

Laboratory of Time Analysis of Nuclear Processes, Institute for Nuclear Research, National Academy of Sciences of Ukrain (NASU), 03028 Kiev, Ukraine

Received 4 July 2008; Revised 28 October 2008; Accepted 24 November 2008

Academic Editor: Ricardo Weder

Copyright © 2009 V. S. Olkhovsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Recent developments are reviewed and some new results are presented in the study of time in quantum mechanics and quantum electrodynamics as an observable, canonically conjugate to energy. This paper deals with the maximal Hermitian (but nonself-adjoint) operator for time which appears in nonrelativistic quantum mechanics and in quantum electrodynamics for systems with continuous energy spectra and also, briefly, with the four-momentum and four-position operators, for relativistic spin-zero particles. Two measures of averaging over time and connection between them are analyzed. The results of the study of time as a quantum observable in the cases of the discrete energy spectra are also presented, and in this case the quasi-self-adjoint time operator appears. Then, the general foundations of time analysis of quantum processes (collisions and decays) are developed on the base of time operator with the proper measures of averaging over time. Finally, some applications of time analysis of quantum processes (concretely, tunneling phenomena and nuclear processes) are reviewed.