The ‘time of occurrence’ in quantum mechanics

Time and quantum theory: A history and a prospectus

“…19 In effect, what this means is that the experimental results (depending on relative times) will be independent of the time at which the experiment begins, and so this expresses the homogeneity of time. However, 'Pauli's theorem' (Srinivas & Vijayalakshmi, 1981, §2) entails that there is no time translation covariant Projection Valued Measure (PVM) mapping intervals of time to orthogonal projections, and, since the PVMs and self-adjoint operators are in one-toone correspondence, there is thus no self-adjoint operator corresponding to the time of an event. 20 Nonetheless, there are time translation covariant Positive Operator Valued Measures (POVMs) which map intervals of time to positive operators with spectrum [0,1].…”

“…I will argue that the conceptual confusion is somewhat less severe, and the motivations somewhat more subtle, than Hilgevoord alleges. Hilgevoord claims that the expectation of the authors of quantum mechanics that time should be an 1 See Srinivas & Vijayalakshmi (1981) for a rigorous derivation of this result.…”

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.…”