Time observables in quantum theory

On the coexistence of position and momentum observables

“…If the range of E contains only projections, then E is called a sharp observable. For more about observables as normalized positive operator measures, the reader may refer to the monographs [4], [7], [8], and [9].…”

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.…”

“…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.…”

“…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. (Holevo, 1978;Olkhovsky & Recami, 1968;1970;Olkhovsky, 1973;Olkhovsky et al, 1974;Olkhovsky, 1984;1990;1992;1998;Recami, 1976;, moreover, all the papers (Aharonov et al, 1998;Atmanspacher & Amann, 1998;Blanchard P & Jadczyk, 1996;Busch et al, 1994;Grot et al, 1996;Kobe et al, 1994;Leo'n, 1997;Srinivas & Vijayalakshmi, 1981) attempted at solving the problem of time as a quantum observable by means of formal mathematical operations performed outside the usual Hilbert space of conventional quantum mechanics. Let us recall that Naimark's theorem states (Naimark, 1940) that the non-orthogonal spectral decomposition of a hermitian operator can be approximated by an orthogonal spectral function (which corresponds to a selfadjoint operator), in a weak convergence, with any desired accuracy.…”

Position and momentum observables on R and on R3

“…This generalization allows one, among many other things, to describe measurements with limited accuracy. (For a review of positive operator measures in quantum mechanics, see [4], [10], [15]. )…”