Thesis
Quantum Thomas-Whitehead projective gravity in two dimensions
University of Iowa
Master of Science (MS), University of Iowa
Spring 2024
DOI: 10.25820/etd.007454
Abstract
There are many different approaches to the quantization of gravity; however, a successful theory of quantum gravity has yet to be found. Paul Dirac’s Lectures on Quantum Mechanics [1] offer up canonical quantization as a process that has been shown to correctly quantize theories such as electrodynamics. We will begin with a brief review of canonical quantization through electrodynamics. Highlighting the importance of Dirac brackets and constraint analysis along the way.
Applying canonical quantization to general relativity, however, produces hard to interpret results. Bryce DeWitt shows in [2] that through canonical quantization of the Einstein-Hilbert action one can pass general relativity from a classical theory to a quantum mechanical one. However, in doing so they find two constraints present in general relativity; the diffeomorphism constraints and the Hamiltonian constraint. Although the Wheeler-DeWitt equation (formally known as the Einstein-Schrödinger equation) is a quantum Hamiltonian many issues arise in the theory. Notably these issues are operator ordering ambiguity, the problem of time, and having a wave functional dependent on an infinite dimensional manifold. As a way to interpret later results, we will introduce the derivation of the Wheeler-DeWitt equation.
One leading attempt at the quantization of gravity is string theory. Polyakov offers [3] an alternate action to the well-known Nambu-Goto action. From the generating functional of the Polyakov action, an effective action may be established [3, 4]. Brief discussions will be held on the interesting results of the quantization of the Polyakov action in the light-cone gauge as well as the quantization of the effective Polyakov action with a dynamical light-cone metric. It has been shown by A. Alekseev and S. Shatashvili [5] and later by G. Delius, P. Nieuwenhuizen, and V.G.J. Rodgers [6] that a geometric action constructed from the coadjoint orbits of the Virasoro algebra recovers the effective Polyakov action coupled to the diffeomorphism field. Utilizing projective geometry [7] the diffeomorphism field’s exact nature has been identified as part of a larger connection coefficient. Furthermore, a projective gravitational action has been produced giving dynamics to the diffeomorphism field. The construction and importance of these key concepts will be reviewed.
Quantization of this novel projective theory of gravity will be our main objective. We will produce the quantum Hamiltonian for the effective Polyakov action coupled to a background diffeomorphism field and the dynamical diffeomorphism field coupled to a background light-cone metric. We will also explore the quantization of the projective gravitational action with a two-dimensional and four-dimensional Minkowksi metric. From the derived classical Hamiltonians wave solutions will be found. Lastly, we will consider quantization of the projective gravitational action with dynamical metric, dynamical diffeomorphism field, and an independent dynamical connection.
Details
- Title: Subtitle
- Quantum Thomas-Whitehead projective gravity in two dimensions
- Creators
- Salvatore M. Quaid
- Contributors
- Vincent G. J. Rodgers (Advisor)Kory Stiffler (Committee Member)Wayne N. Polyzou (Committee Member)Benjamin Cooper (Committee Member)
- Resource Type
- Thesis
- Degree Awarded
- Master of Science (MS), University of Iowa
- Degree in
- Physics
- Date degree season
- Spring 2024
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.007454
- Number of pages
- x, 99 pages
- Copyright
- Copyright 2024 Salvatore M. Quaid
- Language
- English
- Date submitted
- 04/20/2024
- Description bibliographic
- Includes bibliographical references (pages 95-99).
- Public Abstract (ETD)
There has long been interest in the interaction of gravity with particles. However, attempts to bring together general relativity and quantum mechanics have so far been unsuccessful. Thomas-Whitehead projective gravity is a novel theory of gravity that has its sources in string theory. The presence of the diffeomorphism field as well as the projectively invariant connection serve to modify the current approach to quantum gravity. Here I offer reviews of how one quantizes a theory, the historical investigations of quantum gravity, and string theories’ relevance to the larger investigation. I provide reviews of the origin of the Thomas-Whitehead projective theory in string theory as well as the Thomas-Whitehead projective gravity action. Next, I investigate the relevant coordinate transformation which will explore the new fields in terms of a dynamical light-cone metric favored by Polyakov. Finally, I investigate the effective Polyakov action coupled to the diffeomorphism field and then the projective gravitational action. Exploring these actions through the Minkowski metric and the dynamical light-cone metric.
- Academic Unit
- Physics and Astronomy
- Record Identifier
- 9984647150002771
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