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

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 3-surface and it does not have a natural spacetime interpretation, this does not mean that the Barbero connection is not a suitable variable for quantization; it appears naturally in the formalism even in the 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.