A possible interpretation of the Barbero–Immirzi parameter and its consequences

Motivations: The Barbero–Immirzi (BI) parameter is a free constant that appears in the spectra of area and volume operators of LQG. Many authors have contributed to shed light on the origin of this ambiguity. a Here we present an extension of the idea originally proposed by Gambini, Obregon and Pullin. Idea: The BI parameter is a quantization ambiguity analogous to the θ-angle of Yang–Mills gauge theories. Further extension: We present a motivation to promote the BI parameter to be a field rather than a constant, digressing on the dynamics determined by the coupling of the BI field with the Nieh–Yan topological density. By the way, an extension of the Holst action to spacetimes with torsion is also discussed. Outline General remarks: brief review of the role of the BI parameter in classical and quantum gravity. Einstein-Cartan theory: description of the interaction between spinor matter fields and gravity. Torsion and its geometrical features. Generalization of the Holst action: brief description of the Nieh–Yan topological density and its relation to the Pontryagin densities. Generalization of the Holst theory to spacetimes with torsion. Comparison between the BI parameter and the vacuum angle of Yang–Mills gauge theories. Outline Promoting the BI parameter to be a field: motivation to consider the hypothesis that the BI parameter is actually a field rather than a constant. Proposal on the nature of the BI parameter. Dynamics and comparison with previously proposed models. Interaction with fermions and Peccei–Quinn mechanism: introduction of fermion fields and dynamical consequences related to the presence of the BI field. Description of the PQ mechanism for determining the value of the BI parameter. Canonical theory: constraints of the new theory with the BI field. Proposal for the Ashtekar–Barbero formulation. Concluding remarks and discussion: summary and discussion of the described results, proposals for future research. The BI parameter is a free constant appearing in the spectra of length, area and volume operators of LQG, e.g., the area spectrum is: A γ = 8πγℓ 2 P l k j k (j k + 1) .