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.
Highlights
In the canonical approach to quantum mechanics, one supposes existence of classical time as an external parameter used by the observer
We demonstrate the Page–Wotters mechanism in quantum cosmology, which explains emergence of time in timeless stationary quantum systems
The essence of this mechanism is entanglement between a clock degrees of freedom and the target system; the information of the clock system is mediated by entanglement and transferred to the timeless universe
Summary
In the canonical approach to quantum mechanics, one supposes existence of classical time as an external parameter used by the observer. Regarding the energy eigenstate of the scalar field as a system of quantum clock, the wave function of the universe conditioned by the clock is obtained. We investigated the condition for the emergence of the standard Schrödinger equation for the universe wave function conditioned by the scalar clock. We revisit this problem using a qubit as a quantum clock in cosmology This model is simpler than the scalar field model and captures an essential feature of the Page–Wooters mechanism. We can explicitly evaluate entanglement between the qubit clock and the universe, and discuss relation between the emergence of time and the clock-system entanglement in the timeless quantum cosmology. From the behavior of phase factors of the solution, the wave function represents superposition of expanding and contracting universes with the qubit energy ±ω/2. From the WDW Equation (10), the contribution of the qubit to the total matter energy density is proportional to ρ−1, which is the same behavior as a dust fluid
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