The Immirzi parameter in quantum general relativity

TL;DR

This paper investigates the Immirzi parameter in loop quantum gravity, showing it introduces a quantization ambiguity that affects the spectra of geometric operators, and clarifies its origin and significance.

Contribution

It demonstrates that the Immirzi parameter corresponds to a canonical transformation not implementable unitarily, revealing a fundamental quantization ambiguity in quantum gravity.

Findings

The transformation U(ι) is canonical but not unitarily implementable.

The Immirzi parameter introduces a free quantization ambiguity.

This ambiguity affects the spectra of geometric quantities in loop quantum gravity.

Abstract

Barbero has generalized the Ashtekar canonical transformation to a one-parameter scale transformation $U(\iota)$ 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 $U(\iota)$ 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.