Real and complex connections for canonical gravity

TL;DR

This paper explores the relationship between real and complex connections in canonical gravity, highlighting the role of a parameter that influences quantum spectra and discussing potential ways to determine its value.

Contribution

It demonstrates the arbitrary nature of the Barbero-Immirzi parameter in real connections and its impact on quantum geometric spectra, linking it to the Hamiltonian constraint and complex connections.

Findings

The parameter β affects the scale of quantum spectra.

The value of β can be determined by the Hamiltonian constraint.

Rotation to complex Ashtekar connection influences the parameter.

Abstract

Both real and complex connections have been used for canonical gravity: the complex connection has SL(2,C) as gauge group, while the real connection has SU(2) as gauge group. We show that there is an arbitrary parameter $\beta$ which enters in the definition of the real connection, in the Poisson brackets, and therefore in the scale of the discrete spectra one finds for areas and volumes in the corresponding quantum theory. A value for $\beta$ could be could be singled out in the quantum theory by the Hamiltonian constraint, or by the rotation to the complex Ashtekar connection.