Quantum Thomas-Whitehead projective gravity in two dimensions
Creators: name: Quaid, Salvatore M.
Publisher: name: University of Iowa
Year: 2024
Subjects: Canonical Quantization
Projective Geometry
Quantum Gravity
String Theory

description: 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.
description: 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.
description: 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.
description: 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.