Quantum Dilaton Gravity in Two Dimensions with Matter

TL;DR

This thesis explores the quantization of a spherically reduced Einstein-massless-Klein-Gordon model in two dimensions, revealing finite scattering amplitudes and a novel gravitational decay phenomenon of spherical waves.

Contribution

It introduces a first order quantization approach for the model, derives a non-local effective action, and demonstrates finite scattering amplitudes with the discovery of gravitational decay of spherical waves.

Findings

Finite S-matrix elements due to cancellation of divergences.

Existence of a virtual black hole in the model.

Prediction of gravitational decay of spherical waves.

Abstract

In this thesis special emphasis is put on the quantization of the spherically reduced Einstein-massless-Klein-Gordon model using a first order approach for geometric quantities, because phenomenologically it is probably the most relevant of all dilaton models with matter. After a Hamiltonian BRST analysis path integral quantization is performed using temporal gauge for the Cartan variables. Retrospectively, the simpler Faddeev-Popov approach turns out to be sufficient. It is possible to eliminate all unphysical and geometric quantities establishing a non-local and non-polynomial action depending solely on the scalar field and on some integration constants, fixed by suitable boundary conditions on the asymptotic effective line element. Then, attention is turned to the evaluation of the (two) lowest order tree vertices, explicitly assuming a perturbative expansion in the scalar field being valid. Each of them diverges, but unexpected cancellations yield a finite S-matrix element when both contributions are summed. The phenomenon of a "virtual black hole" -- already encountered in the simpler case of minimally coupled scalars in two dimensions -- occurs, as the study of the (matter dependent) metric reveals. A discussion of the scattering amplitude leads to the prediction of gravitational decay of spherical waves, a novel physical phenomenon. Several possible extensions conclude this dissertation.