Time as an observable

Miura, Hiroaki

This study presents a minimal model demonstrating how the structure of time emerges from observational operations alone, without assuming geometry, metrics, or physical laws. A countably infinite sequence of selections of generative bases determines observation, defining an irreversible arrow of internal time. Since the entire history of these selections has the cardinality of the real numbers, the completion points of observations lie on the real line. A discrete set of observation moments embedded in continuous time defines a reversible arrow of external time. Furthermore, the finiteness of entropy, defined as the finite change in the cardinality of generative bases in observation, is a necessary condition to ensure the continuity and consistency of the observable world. The continuity and discreteness of time, as well as its reversibility and irreversibility, naturally arise from selection operations in observation.

Don Nelson Page

G. Gabrielse

Time-resolved measurements of atomic observables are analyzed using a Liouville space formulation and a Hermitian unit tensor base. This approach makes it possible to distinguish cleanly the symmetries of a formation and/or excitation process completed by time $t=0$, a time evolution under experimental control between $t=0$ and $t=t$, and a measurement at $t=t$, even for hydrogenic observables. Each observable is labeled by a time-reversal quantum number, allowing exploration for the first time of the close relationship between time-reversal symmetry and the time evolution of atomic observables. The experimental reconstruction of atomic observables (at $t=0$) from subsequent time-resolved measurements of the anisotropy and polarization of emitted electric dipole photons is discussed. Hydrogenic observables are stressed and the use of strong fields is included, thus generalizing Fano and Macek's approach.

Bryan Wilson Roberts

Gertjan J. Burghouts, Jan‐Mark Geusebroek, Arnold W. M. Smeulders

Marek Kanter

In this paper time is indexed by a “time group” G that may be either the real line or the integers. A quantum process is a homomorphic representation of G via unitary operators on a Hilbert space L, with the usual continuity restrictions if time is indexed by the reals. Consider a quantum process subject to nonselective observation at all points in time via an observation operator with purely discrete eigenvalues and corresponding simple eigenvectors. Suppose that the process evolves in such a way that the results of observation at a given time are the same, whether or not there have been previous observations. Suppose further that L is spanned by the set of vectors invariant modulo phase under the quantum evolution. If the time group is the integers, then the process is observed to evolve in a classical manner. If the time group is the real line, then the process is never observed to undergo a change of state. The results of this paper are essentially a corollary of von Neumann’s mean ergodic theorem.