Remarks Concerning the Geometries of Gravity and Gauge Fields

Summary

Abstract

An important limitation is shown in the analogy between the Aharonov-Bohm effect and the parallel transport on a cone. It illustrates a basic difference between gravity and gauge fields due to the existence of the solder form for the space-time geometry. This difference is further shown by the observability of the gravitational phase for open paths. This reinforces a previous suggestion that the fundamental variables for quantizing the gravitational field are the solder form and the connection, and not the metric.

INTRODUCTION

I recall with great pleasure the discussions which I had with Charles Misner on fundamental aspects of physics, such as the geometry of gravity, gauge fields, and quantum theory. In particular, I remember the encouragement he gave to my somewhat unorthodox attempts to understand the similarities and differences between gauge fields and gravity from their effects on quantum interference, and their implications to physical geometry. It therefore seems appropriate to present here for his Festschrift some observations which came out of this investigation.

Geometry is a part of mathematics which can be visualized, and is intimately related to symmetries. This may explain the tremendous usefulness of geometry in physics. In section 2, I shall make some basic remarks about the similarities and differences between the geometries of gravity and gauge field. Then I shall illustrate, in section 3, an important difference between them that arises due to the existence of the solder form for gravity, using the Aharonov-Bohm (AB) effect and parallel transport on a cone.