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On time as a quantum observable canonically conjugate to energy

V. S. Olkhovsky

Published Aug 31, 2011• 14 citations
Publisher

Abstract

A number of studies on time as a quantum observable canonically conjugate to energy, with reference to systems with continuous and discrete energy spectra, are briefly reviewed in this minimal-mathematics methodological note. The results of and prospects for the time analysis of quantum processes using time as a quantum observable are examined.

DOI
10.3367/ufne.0181.201108d.0859

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A number of studies on time as a quantum observable canonically conjugate to energy, with reference to systems with continuous and discrete energy spectra, are briefly reviewed in this minimal-mathematics methodological note. The results of and prospects for the time analysis of quantum processes using time as a quantum observable are examined.
Analytical review of developments in researching timetimetime as a quantum-physical observable, which is canonically conjugated to energy, is presented. A special attention is devoted to the duality of energy and time operators and also to the verification of time-energy uncertainty relations for continuous and discrete energy spectra.
Some new results are presented and recent developments are reviewed on the study of Time in quantum physics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the Hermitian (more precisely, maximal Hermitian, but non-self-adjoint) operator for Time which appears: (i) for particles, in ordinary nonrelativistic quantum mechanics; and (ii) for photons (i.e. in first-quantization quantum electrodynamics). In conclusion, various recent and possible future applications of the time quantum analysis for tunnelling processes, nuclear collisions and systems with (quasi)discrete energy spectra are indicated.
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.
Some results are briefly reviewed and developments are presented on the study of Time in quantum mechanics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the hermitian (more precisely, maximal hermitian, but non-selfadjoint) operator for Time which appears: (i) for particles, in ordinary non-relativistic quantum mechanics; and (ii) for photons (i.e., in first-quantization quantum electrodynamics).
It is shown that a canonical time observable may be defined for any quantum system having a discrete set of energy eigenvalues, thus significantly generalizing the known case of time observables for periodic quantum systems (such as the harmonic oscillator). The general case requires the introduction of almost-periodic probability operator measures (POMs), which allow the expectation value of any almost-periodic function to be calculated. An entropic uncertainty relation for energy and time is obtained which generalizes the known uncertainty relation for periodic quantum systems. While non-periodic quantum systems with discrete energy spectra, such as hydrogen atoms, typically make poor clocks that yield no more than 1 bit of time information, the anisotropic oscillator provides an interesting exception. More generally, a canonically conjugate observable may be defined for any Hermitian operator having a discrete spectrum.