The Immirzi parameter in quantum general relativity

Abstract

Barbero has generalized the Ashtekar canonical transformation to a one-parameter scale transformation $U(ι)$ on the phase space of general relativity. Immirzi has noticed that in loop quantum gravity this transformation alters the spectra of geometrical quantities. We show that $U(ι)$ is a canonical transformation that cannot be implement unitarily in the quantum theory. This implies that there exists a one-parameter quantization ambiguity in quantum gravity, namely a free parameter that enters the construction of the quantum theory. The purpose of this letter is to elucidate the origin and the role of this free parameter.