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arXiv:2201.02770 · Universe

Qubit Clock in Quantum Cosmology

Yasusada Nambu

Published Feb 4, 2022• 1 citation
arXivHTMLPDF

Abstract

We investigate the emergent time scenario in quantum cosmology based on the Page–Wotters approach. Using a quantum cosmological model with a qubit clock, it is demonstrated how the entanglement between the qubit clock and the geometry derives emergence of a time parameter, which defines evolution of the timeless quantum state of the universe. We show that the universe wave function conditioned by a qubit clock obeys the standard Schrödinger equation and the Fisher information for the clock state, which quantifies entanglement between the universe and the clock, and contributes as a negative energy density.

DOI
10.3390/universe8020099

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We investigate the emergent time scenario in quantum cosmology based on the Page–Wotters approach. Using a quantum cosmological model with a qubit clock, it is demonstrated how the entanglement between the qubit clock and the geometry derives emergence of a time parameter, which defines evolution of the timeless quantum state of the universe. We show that the universe wave function conditioned by a qubit clock obeys the standard Schrödinger equation and the Fisher information for the clock state, which quantifies entanglement between the universe and the clock, and contributes as a negative energy density.
This note studies the emergence of time in a timeless quantum universe using a Wheeler–DeWitt-style constrained model with quantum geometry, clocks, and matter records. The model contains no external time parameter. The total state satisfies a timeless constraint while the geometry register, clock, and matter systems remain fully quantum and entangled. Despite the absence of fundamental time, conditioning on the clock and geometry recovers branch-dependent Schrödinger evolution with the correct redshifted proper-time rates. The work shows that quantum geometry can act as a reference frame for time. An observer without access to the geometry register carries a geometric information deficit, while conditioning on the geometry restores the individual geometric branches exactly. Superposed geometries generate proper-time interference, with matter records determining the transition between quantum time-history interference and classical branch evolution. The central result is an exact emergence-of-time identity: the observable coherence between different geometry branches is equal to the product of geometry coherence and matter-record overlap. When geometry information is fully recorded by matter, the interference between different time histories disappears, and each branch exhibits its own classical Schrödinger evolution. The note also derives a continuous crossover law: the emergence of classical time depends quantitatively on how strongly matter measures the quantum geometry. This connects the relational-time mechanism of Page–Wootters with the semiclassical time program of Kiefer and Zeh in an exactly solvable setting. The results are obtained in finite Wheeler–DeWitt-style toy models with declared assumptions: quantum geometry registers, simplified redshift structure, and no claim of a complete theory of dynamical quantum gravity. Verification and reproducibility materials are included.
We consider the conditioning of the timeless solution to the Wheeler–DeWitt equation by a predefined matter clock state in the simple scenario of an FLRW universe. The resulting evolution of the geometrodynamical degree of freedom with respect to clock time is characterized by the “Berry connection” of the reduced geometrodynamical space, which relies on the coupling of the clock with the geometry. When the connection vanishes, the standard Schrödinger equation is obtained for the geometry with respect to clock time. When one considers environment-induced decoherence in the semiclassical limit, this condition is satisfied, and clock time coincides with cosmic time. Explicit results for the conditioned wave functions for minimal clocks made up of two quantum harmonic oscillator eigenstates are shown.
Nonrelativistic model quantum cosmologies are studied in which the basic time variable is the position of a clock indicator and the time parameter of the Schr\"odinger equation is an unobservable label. Familiar Schr\"odinger-Heisenberg quantum mechanics emerges if the clock is ideal---arbitrarily accurate for arbitrarily long times. More realistically, however, the usual formulation emerges only as an approximation appropriate to states of this model universe in which part of the system functions approximately as an ideal clock. It is suggested that the quantum kinematics of spacetime theories such as general relativity may be analogous to those of this model. In particular it is suggested that our familiar notion of time in quantum mechanics is not an inevitable property of a general quantum framework but an approximate feature of specific initial conditions.
Entanglement is at the core of quantum physics, playing a central role in quantum phenomena involving composite systems. According to the timeless picture of quantum dynamics, entanglement may also be essential for understanding the very origins of dynamical evolution and the flow of time. Within this point of view, the Universe is regarded as a bipartite entity comprising a clock CCC and a system RRR (or ``rest of the Universe'') jointly described by a global stationary state, and the dynamical evolution of RRR is construed as an emergent phenomena arising from the entanglement between CCC and RRR. In spite of substantial recent efforts, many aspects of this approach remain unexplored, particularly those involving mixed states. In the present contribution we investigate the timeless picture of quantum dynamics for mixed states of the clock-system composite, focusing on quantitative relations linking the clock-system entanglement with the emerging dynamical evolution experienced by the system.
Abstract Dynamical evolution can be reconstructed within stationary, closed quantum systems by employing the Page-Wootters “timeless approach”. When conditioning upon the state of a “clock” subsystem, the rest of the system regains its time dependence. This mechanism, involving entanglement between the above subsystems has gained much attention during the last few years. After a brief introduction to the topic we will elaborate on a few recent results: The derivation of new time-energy uncertainty relations, emergence of non-Hermitian dynamics when utilizing non-inertial quantum clocks and dynamical nonlocality in quantum time.
The `problem of time' remains an unresolved issue in all known physical descriptions of the Universe. One aspect of this problem is the conspicuous absence of time in the Wheeler-Dewitt equation, which is the analogue of the Schrodinger equation for the Universal wavefunction. Page and Wootters famously addressed this problem by providing a mechanism for effectively introducing time evolution into this timeless cosmological picture. Their method, which is sometimes called the conditional probability interpretation (CPI), requires the identification of an internal clock system that is meant to keep time for the remainder of the Universe. Most investigations into this idea employ the idealized limit of a non-interacting clock system, the so-called ideal clock. However, by allowing for interactions, we have found the counter-intuitive result that a non-interacting clock is not necessarily the optimal choice, even if it is `ideal'. In particular, the uncertainty that is associated with the physical measurement of an atomic clock is found to decrease monotonically as the interactions grow stronger. This observation, which is reinforced by a previous study using a semi-classical clock, paves the way to an independent argument that is based on the energy conservation of any isolated system. Our conclusion is that ideal clocks must be prohibited from the CPI when recovering cosmological time evolution. Interactions are necessary for describing time evolution as a strict matter of principle. Lastly, we also consider the implications of this result for the experience of time in the evolution of the Universe.