The Barbero connection and its relation to the histories connection formalism without gauge fixing

TL;DR

This paper develops a histories-based spacetime formalism for general relativity, exploring the properties of the Barbero connection without gauge fixing and its implications for loop quantum gravity.

Contribution

It introduces a histories approach to the connection formalism of general relativity, analyzing the Barbero connection's role without gauge fixing and its potential for quantization.

Findings

Barbero connection can be defined without gauge fixing

The loop variables can be used without a natural spacetime interpretation

Loop algebra requires augmentation with additional variables

Abstract

We present a histories version of the connection formalism of general relativity. Such an approach introduces a spacetime description--a characteristic feature of the histories approach--and we discuss the extent to which the usual loop variables are compatible with a spacetime description. In particular, we discuss the definability of the Barbero connection without any gauge fixing. Although it is not the pullback of a spacetime connection onto the three-surface and it does not have a natural spacetime interpretation, this does not mean that the Barbero connection is not suitable variable for quantisation; it appears naturally in the formalism even in absence of gauge fixing. It may be employed therefore, to define loop variables similar to those employed in loop quantum gravity. However, the loop algebra would have to be augmented by the introduction of additional variables.