Chataignier, Leonardo 
ORCID: 0000-0001-6691-3695
(2021).
Timeless Quantum Mechanics and the Early Universe.
PhD thesis, Universität zu Köln.
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Abstract
We discuss the construction and interpretation of observables in quantum theories with worldline diffeomorphism invariance, in which a preferred or absolute time parameter is absent. These theories are also called time-reparametrization invariant, and they can be seen as mechanical toy models of quantum gravity. The interest in these models stems from the necessity of understanding the so-called problem of time in a theory of quantum gravitation: how can the dynamics of quantum states of matter and geometry be defined in a diffeomorphism-invariant way? What is the relevant space of physical states and which operators act on it? How are the quantum states related to probabilities in the absence of a preferred time? The restriction to the mechanical case allows us to focus on this problem without further issues that accompany field-theoretical treatments. We first analyze the consequences of diffeomorphism invariance in the classical theory, and we emphasize that it warrants a relational ontology of spacetime. Observers record the evolution of physical fields in generalized reference frames that are defined from the readings of generalized clocks and rods, which are themselves physical fields. As the only physical information that is available in an experiment are the values of the fields and not the spacetime points, one concludes that observables are relational: the outcomes of experiments can be described or predicted by determining the values of the fields relative to (or conditioned on) the values of the generalized clocks and rods (the reference fields that define a generalized reference frame). The description of the dynamics in terms of relational observables is diffeomorphism invariant, as it does not refer to the underlying abstract spacetime but rather solely to the physical fields. Technically, relational observables can be seen as diffeomorphism-invariant extensions of geometrical objects in analogy to gauge-invariant extensions of noninvariant quantities in the usual gauge (Yang-Mills) theories. We take this analogy seriously and use it as a basis of a method of construction of invariant operators in the quantum theory. These operators act solely in the space of solutions to the quantum constraints (i.e., on the space of physical states) and, as such, they are defined solely in terms of the physical states. Furthermore, we discuss how the notion of a physical propagator may be used to define a unitary evolution in the quantum theory, which is to be understood in terms of a generalized clock, as is the classical theory. We then put forth a set of tentative postulates that dictate how an observer is to make use of probabilities in the description of the quantum dynamics in a quantum generalized reference frame. In this way, we emphasize that the dynamics is relational also in the quantum theory, and we define a notion of relative initial data, which determine the quantum evolution of the relational observables. We also discuss under which circumstances the above mentioned formalism can be related to the use of conditional probabilities in the quantum theory. These probabilities are defined from the physical states, and we argue that our formalism can be regarded as a generalization of the well-known Page-Wootters approach. On this subject, we show that the quantum averages of relational observables can be related to conditional expectation values of worldline tensor fields. We discuss how our formalism is related to the earlier literature. We also illustrate the method presented here with conceptually useful examples, such as the free quantum relativistic particle, the Kasner model, and a closed, recollapsing Friedmann- Lamaitre-Robertson-Walker model. We construct the quantum relational observables for these models and discuss their quantum evolution. In the context of cosmology, we also mention how the notion of relative initial data may be used to establish a criterion for quantum singularity avoidance, which we refer to as the conditional DeWitt criterion. In the interest of making contact with observations, we also examine how our formalism can be adapted to calculations of quantum-gravitational effects in the early Universe. To this end, we show that the usual weak-coupling expansion used in the Born-Oppenheimer approach to quantum gravity leads to a perturbative definition of the inner product on the space of physical states, with respect to which the dynamics is unitary. This is important because the issue of unitarity in the Born-Oppenheimer approach has been controversial. Interestingly, we also show how this perturbatively defined physical inner product corresponds to a quantization of the classical Faddeev-Popov determinant associated with the choice of background clock that is used in the weak-coupling expansion. In this way, the usual results of the Born-Oppenheimer approach coincide with a ‘choice of gauge’, and they can be extended beyond the semiclassical level of the gravitational field. Time is to be understood relationally in the exact quantum theory. We apply these results to the calculation of quantum-gravitational corrections to the primordial power spectra in (quasi-)de Sitter space, comparing the results to the ones previously obtained in the literature, and discussing the physical interpretation of such corrections. Lastly, we conclude with some remarks about the relevance and usefulness of the approach presented here, as well as its limitations. In particular, we mention how the approach may be useful for the definition and interpretation of observables as diffeomorphism invariants in a full quantum theory of gravitation, and we offer some comments on possible future directions of research.
| Item Type: | Thesis (PhD thesis) | ||||||||
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| URN: | urn:nbn:de:hbz:38-525576 | ||||||||
| Date: | 2021 | ||||||||
| Language: | English | ||||||||
| Faculty: | Faculty of Mathematics and Natural Sciences | ||||||||
| Divisions: | Faculty of Mathematics and Natural Sciences > Department of Physics > Institute for Theoretical Physics | ||||||||
| Subjects: | Physics | ||||||||
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| Date of oral exam: | 21 April 2021 | ||||||||
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| Refereed: | Yes | ||||||||
| URI: | http://kups.ub.uni-koeln.de/id/eprint/52557 |
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