Time in Quantum Cosmology

Time in quantum gravity is not a well-defined notion despite its central role in the very definition of dynamics. Using the formalism of quantum geometrodynamics, we briefly review the problem and illustrate it with two proposed solutions. Our main application is quantum cosmology—the application of quantum gravity to the Universe as a whole.

Hawking Radiation and Black Hole Gravitational Back Reaction—A Quantum Geometrodynamical Simplified Model

The purpose of this paper is to analyse the back reaction problem, between Hawking radiation and the black hole, in a simplified model for the black hole evaporation in the quantum geometrodynamics context. The idea is to transcribe the most important characteristics of the Wheeler-DeWitt equation into a Schrödinger’s type of equation. Subsequently, we consider Hawking radiation and black hole quantum states evolution under the influence of a potential that includes back reaction. Finally, entropy is estimated as a measure of the entanglement between the black hole and Hawking radiation states in this model.

Classical and Quantum f(R) Cosmology: The Big Rip, the Little Rip and the Little Sibling of the Big Rip

The big rip, the little rip and the little sibling of the big rip are cosmological doomsdays predicted by some phantom dark-energy models that could describe the future evolution of our universe. When the universe evolves towards either of these future cosmic events, all bounded structures and, ultimately, space–time itself are ripped apart. Nevertheless, it is commonly believed that quantum gravity effects may smooth or even avoid these classically predicted singularities. In this review, we discuss the classical and quantum occurrence of these riplike events in the scheme of metric f(R) theories of gravity. The quantum analysis is performed in the framework of f(R) quantum geometrodynamics. In this context, we analyze the fulfilment of the DeWitt criterion for the avoidance of these singular fates. This review contains as well new unpublished work (the analysis of the equation of state for the phantom fluid and a new quantum treatment of the big rip and the little sibling of the big rip events).

Quantization of Gravitationally Bound Systems

Some of the approaches to quantization in gravity theory concerning gravitationally bound systems are considered. Grades of quantization applicable to these systems have been classified in terms of quantum mechanics, quantum field theory, and quantum geometrodynamics. Energy levels for the graviatom, Lemaître’s atom, quantum gravitational collapse have been calculated, and relationships for the masses of bound system components, as well as Universe’s birth probabilities, are presented to exemplify the properties of gravitationally bound systems. Objects and processes in them have been analyzed to construct quantum models of compact astrophysical objects and the early Universe.

The Functional Schrödinger Equation in the Semiclassical Limit of Quantum Gravity with a Gaussian Clock Field

We derive the functional Schrödinger equation for quantum fields in curved spacetime in the semiclassical limit of quantum geometrodynamics with a Gaussian incoherent dust acting as a clock field. We perform the semiclassical limit using a WKB-type expansion of the wave functional in powers of the squared Planck mass. The functional Schrödinger equation that we obtain exhibits a functional time derivative that completes the usual definition of WKB time for curved spacetime, and the usual Schrödinger-type evolution is recovered in Minkowski spacetime.

Geon Wheeler

This chapter focuses on John Wheeler’s work on geons and geometrodynamics which would lead to many concepts and results that would be of importance to quantum gravity research - these projects, initially, were rather old fashioned, harking back to the classical ‘unified field theory’ work of Einstein. Moreover, we find that this work that we now tend to think of as foundational in quantum gravity---e.g., we often think of ‘quantum geometrodynamics’ as just another phrase for ‘quantum gravity’---had its roots firmly embedded in the quest for understanding the elementary particles. It wasn’t until after 1957 that Wheeler began to look seriously at general relativity and quantum gravity independently from concerns in particle physics, and this shift in fact coincides with a more general trend to treat gravitational physics as a worthwhile field in its own right.

On the meaning of the wave function of the Universe

The meaning of the wave function of the Universe was actively discussed in 1980s. In most works on quantum cosmology, it is accepted that the wave function is a probability amplitude for the Universe to have some space geometry, or to be found in some point of the Wheeler superspace. It seems that the wave function gives maximally objective description compatible with quantum theory. However, the probability distribution does not depend on time and does not take into account the existing of our macroscopic evolving Universe. What we wish to know is how quantum processes in the Early Universe determined the state of the present Universe in which we are able to observe macroscopic consequences of these quantum processes. As an alternative to the Wheeler–DeWitt quantum geometrodynamics, we consider the picture that can be obtained in the extended phase space approach to quantization of gravity. The wave function in this approach describes different states of the Universe which correspond to different stages of its evolution.

Quantum mechanics with geometric constraints of Friedmann type

The paper presents the study of quantum mechanics of a free particle with the constraints in the phase space, the canonical equations, which are the geometrical constraints of Friedmann type. It has been proved that the constraints can imitate force. As well as in quantum geometrodynamics with Logunov constraints, in quantum mechanics with constraints time does not vanish