Time as a Quantum Observable

Abstract

Some results are 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).

Time operator in QFT with Virasoro constraints

“…For example, the non-self-adjoint (but hermitian, or better maximal hermitian) operator proposed in Ref. [1,2,[4][5][6][7][8][9], as an operator for the observable time in quantum mechanics, is fully acceptable, mathematically and physically, without 3 having to modify or split the ordinary Hilbert space.…”

Time as Quantum Observable, Canonical Conjugated to Energy

“…The list of papers devoted to the problem of time in quantum mechanics is extremely large (see, for instance, Refs. (Aharonov et al, 1998;Atmanspacher & Amann, 1998;Blanchard P & Jadczyk, 1996;Busch et al, 1994;Delgado, 1999;Egusquiza & Muga, 1999;Giannitrapani, 1997;GóźdźA& Dȩbicki, 2007;Grot et al, 1996;Holevo, 1978;Kijowski, 1997;Kobe et al, 1994;Kocha'nski & Wo'dkievicz, 1999;Leo'n, 1997;Muga et al, 1999;Olkhovsky & Recami, 1968;1970;Olkhovsky, 1973;Olkhovsky et al, 1974;Olkhovsky, 1984;1990;1992;Olkhovsky & Recami, 1992;Olkhovsky et al, 1995;Olkhovsky & Agresti, 1997;Olkhovsky, 1998;Olkhovsky et al, 2004;Olkhovsky & Recami, 2007;Olkhovsky, 2009;2011;Recami, 1976;Srinivas & Vijayalakshmi, 1981;Toller, 1999;Wang & Xiong, 2007), and references therein). The same situation had to be faced also in quantum electrodynamics and, more in general, in relativistic quantum field theory (see, for instance, Refs.…”

“…The same situation had to be faced also in quantum electrodynamics and, more in general, in relativistic quantum field theory (see, for instance, Refs. (Olkhovsky & Recami, 1968;Olkhovsky et al, 2004;Olkhovsky & Recami, 2007)). As to quantum mechanics, the very first relevant articles are probably Refs.…”

“…therein. A second set of papers on time in quantum physics (Aharonov et al, 1998;Atmanspacher & Amann, 1998;Blanchard P & Jadczyk, 1996;Busch et al, 1994;Delgado, 1999;Egusquiza & Muga, 1999;Giannitrapani, 1997;GóźdźA&D ȩbicki, 2007;Grot et al, 1996;Kijowski, 1997;Kobe et al, 1994;Kocha'nski & Wo'dkievicz, 1999;Leo'n, 1997;Muga et al, 1999;Olkhovsky & Recami, 1992;Olkhovsky et al, 1995;Olkhovsky & Agresti, 1997;Olkhovsky et al, 2004;Olkhovsky & Recami, 2007;Srinivas & Vijayalakshmi, 1981;Toller, 1999;Wang & Xiong, 2007) appeared in the nineties, stimulated partially by the need of a consistent definition for the tunneling time. It is noticeable, and let us stress it right now, that this second set of papers seems however to have ignored Naimark's theorem (Naimark, 1940), which had previously constituted (directly or indirectly) an important basis for the results in Refs.…”

“…The main goal of the first part of the present paper is to justify the use of time as a quantum observable, basing ourselves on the properties of the hermitian (or, rather, maximal hermitian) operators for the case of continuous energy spectra: cf., e.g., the Refs. (Olkhovsky & Recami, 1992;Olkhovsky et al, 1995;2004;Olkhovsky & Recami, 2007)). The question of time as a quantum-theoretical observable is conceptually connected with the much more general problem of the four-position operator and of the canonically conjugate four-momentum operator, both endowed with an hermitian and an anti-hermitian part, for relativistic spin-zero particles: This problem is analyzed in the second part of the present paper.…”