Quantum time

Abstract

We give a consistent quantum description of time, based on Page and Wootters's conditional probabilities mechanism, which overcomes the criticisms that were raised against similar previous proposals. In particular we show how the model allows one to reproduce the correct statistics of sequential measurements performed on a system at different times. DOI: 10.1103/PhysRevD.92.045033 PACS numbers: 03.65.Ta, 06.30.Ft, 03.65.Ud, 03.67.-a Time in quantum mechanics appears as a classical parameter in the Schrödinger equation. Physically it represents the time shown by a "classical" clock in the laboratory. Even though this is acceptable for all practical purposes, it is important to be able to give a fully quantum description of time. Many such proposals have appeared in the literature (e.g. [1][2][3][4][5][6][7][8][9][10][11]), but none seem entirely satisfactory [6,[12][13][14][15][16]. One of these is the Page and Wootters (PaW) mechanism [5] (see also [2,[17][18][19][20]), which considers "time" as a quantum degree of freedom by assigning to it a Hilbert space H T . The "flow" of time then consists simply in the correlation (entanglement) between this quantum degree of freedom and the rest of the system, a correlation present in a global, time-independent state jΨii. An internal observer will see such a state as describing normal time evolution: the familiar system state jψðtÞi at time t arises by conditioning (via projection) the state jΨii to a time t (Fig. 1), it is a conditioned state. The PaW mechanism was criticized in [6,12] and a proposal that overcomes these criticisms [21,22] used Rovelli's evolving constants of motion [3,23] parametrized by an arbitrary parameter that is then averaged over to yield the correct propagators. Although the end result matches the quantum predictions [24], the averaging used there amounts to a statistical averaging which is typically reserved to unknown physical degrees of freedom rather than to parameters with no physical significance. (A different way of averaging over time was also presented in [25] to account for some fundamental decoherence mechanism.)Here we use a different strategy: we show that the same criticisms can be overcome by carefully formalizing measurements through the von Neumann prescription [26] (which we extend to generalized observables, positive operator valued measures [POVMs]). We show how this implies that all quantum predictions can be obtained by conditioning the global, timeless state jΨii: this procedure gives the correct quantum propagators and the correct quantum statistic for measurements performed at different times, features that were absent in the original PaW mechanism [6,16]. We also show how the PaW mechanism can be extended to give the time-independent Schrödinger equation and give a physical interpretation of the mechanism.What is the physical significance of the quantized time in the PaW representation? One is free to consider the time quantum degree of freedom either as an abstract purification space without any physical signific...

Space-time-symmetric extension of nonrelativistic quantum mechanics

“…We will show that the standard arrival-time distribution, obtained by various approaches that usually resort to ad hoc assumptions, is a natural, built-in part of the formalism presented here. Recent works [12,17] building on a proposal by Page and Wootters [24], present specific similarities to this letter. As in reference [12], we also consider a time variable t with a Hilbert space H T isomorphic to that of a spinless particle in one dimension, H X .…”

“…Recent works [12,17] building on a proposal by Page and Wootters [24], present specific similarities to this letter. As in reference [12], we also consider a time variable t with a Hilbert space H T isomorphic to that of a spinless particle in one dimension, H X .…”

“…Moreover, differently from Ref. [12], what we propose to be extended is the set of possible statistical inferences that QM is able to deal with.Some symmetry between time and position in QM can be found, although often concealed by the standard presentation of the theory. An example is the pair of equations:…”

“…Moreover, differently from Ref. [12], what we propose to be extended is the set of possible statistical inferences that QM is able to deal with.…”

“…As in reference [12], we also consider a time variable t with a Hilbert space H T isomorphic to that of a spinless particle in one dimension, H X . However, our approach does not assume that it is related to an external clock, or that it is a formal extension of a physical system.…”