Realistic Clocks for a Universe Without Time

Abstract

There are a number of problematic features within the current treatment of time in physical theories, including the "timelessness" of the Universe as encapsulated by the Wheeler-DeWitt equation. This paper considers one particular investigation into resolving this issue; a conditional probability interpretation that was first proposed by Page and Wooters. Those authors addressed the apparent timelessness by subdividing a faux Universe into two entangled parts, "the clock" and "the remainder of the Universe", and then synchronizing the effective dynamics of the two subsystems by way of conditional probabilities.The current treatment focuses on the possibility of using a (somewhat) realistic clock system; namely, a coherent-state description of a damped harmonic oscillator. This clock proves to be consistent with the conditional probability interpretation; in particular, a standard evolution operator is identified with the position of the clock playing the role of time for the rest of the Universe. Restrictions on the damping factor are determined and, perhaps contrary to expectations, the optimal choice of clock is not necessarily one of minimal damping.

Time and classical equations of motion from quantum entanglement via the Page and Wootters mechanism with generalized coherent states

“…This was introduced by D. N. Page and W. K. Wotters in 1983 1 to formalize the idea that the expression "at a certain time t" should be understood as "conditioned to a clock being in a state labeled by a certain value t." This proposal, to which we will refer as the "Page and Wootters (PaW) mechanism," is based on three assumptions: (i) the clock does not interact with the system to which it provides the parameter t, but (ii) it is entangled with it; moreover, (iii) clock and system together are in an eigenstate of the total Hamiltonian (with eigenvalue that can be set equal to zero, for the sake of simplicity and without loss of generality). The PaW mechanism has been extensively used, and its assumptions scrutinized, in the recent literature, both from the theoretical and the experimental viewpoint [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] . Discussing about the many questions and answers on the PaW mechanism is not the scope of this work; however, references throughout the paper should help the reader to navigate the relevant literature, and our viewpoint on the contribution that our results furnish to the overall discussion on the mechanism itself is described in the concluding section.…”

Time in quantum mechanics: A fresh look at the continuity equation

“…to complex conjugation. Taking, for example, the complexconjugate of the CDSE (26) yields an equation equivalent to (26), provided one realizes that A[φ]…”

“…However, the special status of time in quantum mechanics is currently investigated [10][11][12][13][14][15][16][17][18][19][20] and different approaches are developed that view time mostly as an emergent property, assuming that quantum mechanics is fundamentally timeless. One of these is the Page-Wootters approach [10,[21][22][23][24][25][26] where the timeless universe (a closed system containing all relevant degrees of freedom) is partitioned into a clock and a system of interest. This system has to be entangled with the clock, and there has to exist a good clock in the sense that it has many distinguishable states but little interaction with the system [25].…”

There is only one time

“…This proposal, to which we will refer as the "Page and Wootters (PaW) mechanism", is based upon three assumptions: i) the clock does not interact with the system to which it provides the parameter t, but ii) it is entangled with it; moreover, iii) clock and system together are in an eigenstate of the total Hamiltonian (with eigenvalue that can be set equal to zero, for the sake of simplicity and without loss of generality). The Paw mechanism has been extensively used, and its assumptions scrutinized, in the recent literature, both from the theoretical and the experimental viewpoint [2][3][4][11][12][13][14][15][16][17][18][19][20][21].…”