Classical and Quantum Dilaton Gravity in Two Dimensions with Fermions

TL;DR

This thesis develops a quantum formulation of two-dimensional dilaton gravity coupled to fermions, revealing exact path integral solutions, scattering vertices, and the formation of virtual black holes, with implications for bosonization.

Contribution

It introduces a Hamiltonian and path integral quantization of dilaton gravity with fermions, including exact evaluation of the gravity sector and analysis of fermionic scattering processes.

Findings

Exact path integral over dilaton gravity sector

Identification of gravitationally induced four-fermi vertices

Formation of virtual black holes in scattering processes

Abstract

In this thesis the first order formulation of generalized dilaton gravities in two dimensions coupled to a Dirac fermion is considered. After a Hamiltonian analysis of the gauge symmetries and constraints of the theory and fixing Eddington-Finkelstein gauge by use of the Batalin-Vilkovisky-Fradkin method, the system is quantized in the Feynman path integral approach. It turns out that the path integral over the dilaton gravity sector can be evaluated exactly, while in the matter sector perturbative methods are applied. The gravitationally induced four-fermi scattering vertices as well as asymptotic states are calculated, and -- as for dilaton gravities coupled to scalar fields -- a ``virtual black hole'' is found to form as an intermediary geometric state in scattering processes. The results are compared to the well-known scalar case and evidence for bosonization in this context is found.