The Immirzi parameter in quantum general relativity

TL;DR

The paper identifies a fundamental one-parameter quantization ambiguity in loop quantum gravity, arising from Barbero's scale transformation $U(\iota)$ that is canonical but not unitarily implementable in the quantum theory. By analyzing the scaled variables $A_\iota$ and $E_\iota$ and their effect on geometric spectra (e.g., area $A(S)=\iota\,f(S,E_\iota)$), the authors show that different $\iota$ yield unitarily inequivalent quantum theories with spectra scaled by $\iota$. They debunk several incorrect interpretations and illustrate the phenomenon with toy models, including a simple system where eigenvalues scale as $\iota^{2}$. The result clarifies the status and physical significance of the Immirzi parameter, indicating it as a potentially measurable quantum gravity parameter that introduces a second fundamental length scale alongside the Planck length.

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.