In quantum theory, physical systems are usually assumed to evolve relative to a c-number time. This c-number time is unphysical and has turned out to be unnecessary for explaining dynamics: in the timeless approach to quantum theory developed by Page & Wootters 1983 Phys. Rev. D 27 , 2885. ( doi:10.1103/PhysRevD.27.2885 ), subsystems of a stationary universe can instead evolve relative to a ‘clock', which is a quantum system with a q-number time observable. Page & Wootters formulated their construction in the Schrödinger picture, which left open the possibility that the c-number time still plays an explanatory role in the Heisenberg picture. I formulate their construction in the Heisenberg picture and demonstrate how to eliminate c-number time from that picture, too. When the Page–Wootters construction is formulated in the Heisenberg picture, the descriptors of physical systems are functions of the clock's q-number time, and derivatives with respect to this q-number time can be defined in terms of the clock's algebra of observables, which results in a calculus for q-numbers.
In quantum theory, physical systems are usually assumed to evolve relative to a c-number time. This c-number time is unphysical and has turned out to be unnecessary for explaining dynamics: in the timeless approach to quantum theory developed by Page & Wootters 1983 Phys. Rev. D 27 , 2885. ( doi:10.1103/PhysRevD.27.2885 ), subsystems of a stationary universe can instead evolve relative to a ‘clock', which is a quantum system with a q-number time observable. Page & Wootters formulated their construction in the Schrödinger picture, which left open the possibility that the c-number time still plays an explanatory role in the Heisenberg picture. I formulate their construction in the Heisenberg picture and demonstrate how to eliminate c-number time from that picture, too. When the Page–Wootters construction is formulated in the Heisenberg picture, the descriptors of physical systems are functions of the clock's q-number time, and derivatives with respect to this q-number time can be defined in terms of the clock's algebra of observables, which results in a calculus for q-numbers.
In quantum theory it is possible to explain time, and dynamics, in terms of entanglement. This is the timeless approach to time, which assumes that the universe is in a stationary state, where two noninteracting subsystems, the ``clock'' and the ``rest,'' are entangled. As a consequence, by choosing a suitable observable of the clock, the relative state of the rest of the universe evolves unitarily with respect to the variable labeling the clock observable's eigenstates, which is then interpreted as time. This model for an ``evolution without evolution'' (Page and Wootters, 1983), albeit elegant, has never been developed further, because it was criticized for generating severe ambiguities in the dynamics of the rest of the universe. In this paper we show that there are no such ambiguities; we also update the model, making it amenable to possible new applications.
Physical systems are usually assumed to evolve relative to an external time parameter, which is problematic because in quantum theory that parameter is not a physical observable. Page and Wootters (1984) solved this by proposing that the Universe is in a stationary state, eliminating the need for the external time parameter. Instead, their model contains an isolated subsystem, a clock, with which other subsystems are entangled, making the latter appear to evolve relative to different states of the clock. While this resolves the problem of the time parameter, the assumption that the clock is isolated prevents it from being measured, as this requires an interaction with another system. We prove that the clock can be measured while preserving the core features of the Page-Wootters construction. We also discuss clock synchronization.
We draw a picture of physical systems that allows us to recognize what "time" is by requiring consistency with the way that time enters the fundamental laws of Physics. Elements of the picture are two non-interacting and yet entangled quantum systems, one of which acting as a clock. The setting is based on the Page and Wootters mechanism, with tools from large-N quantum approaches. Starting from an overall quantum description, we first take the classical limit of the clock only, and then of the clock and the evolving system altogether; we thus derive the Schrödinger equation in the first case, and the Hamilton equations of motion in the second. This work shows that there is not a "quantum time", possibly opposed to a "classical" one; there is only one time, and it is a manifestation of entanglement.
Time in relativity theory has a status different from that adopted by standard quantum mechanics, where time is considered as a parameter measured with reference to an external absolute Newtonian frame. This status strongly restricts its role in the dynamics of systems and hinders any formulation to merge quantum mechanics with general relativity, specifically when considering quantum gravity. To overcome those limitations, several authors tried to construct an operator which is conjugate to the Hamiltonian of quantum systems implementing some essential features of the relativistic time. These formulations use the concept of internal or intrinsic time instead of the universal coordinate time used in textbooks. Furthermore, recently it is remarked that the consideration of time with relativistic features could enhance the analysis techniques in quantum information processing and have an impact on its status in causal orders and causal structures of quantum information. The role of clocks, their accuracy and stability has become an important issue in quantum information processing. This article present a substantiative review of recent works which reflect the possibility of utilizing quantum time, measured by quantum clock devised according to Page-Wootters scheme, to stand as a resource for quantum information processing.
We present an implementation of a recently proposed procedure for defining time, based on the description of the evolving system and its clock as noninteracting, entangled systems, according to the Page and Wootters approach. We study how the quantum dynamics transforms into a classical-like behavior when conditions related to macroscopicity are met by the clock alone, or by both the clock and the evolving system. In the description of this emerging behavior finds its place the classical notion of time, as well as that of phase-space and trajectories on it. This allows us to analyze and discuss the relations that must hold between quantities that characterize the system and clock separately, in order for the resulting overall picture to be that of a physical dynamics as we mean it.
In quantum theory, the concept of time rests on shaky ground. One way to address this problem is to remove the usual background time parameter as a primitive entity and explain its emergence via correlations between physical systems. This approach was adopted by Page and Wootters [Phys. Rev. D 27, 2885 (1983)], who showed how time can emerge in a stationary quantum universe from the correlations between two of its subsystems, one of them acting as a clock for the other. In this work, I study the robustness of the Page-Wootters construction across different pictures, states of the universe, and clock interactions, clarifying the role and the nature of the correlations between the subsystems of the universe. I start by showing how to formulate the Page-Wootters construction in the Heisenberg picture via a unitary change of basis. I consider both pure and mixed states of the universe and extend the analysis to include interactions between the clock and the other subsystem of the universe. The study reveals what kind of correlations are necessary for the construction to work. Interestingly, entanglement is not required as long as there are no interactions with the clock. The study also shows that these interactions can lead to a nonunitary evolution for some mixed states of the universe. In a simple two-level system, this aspect becomes relevant at scales where one would expect strong relativistic effects. At these scales, I also observe an inversion in the system's direction of time.