Fermions in the Ashtekar-Barbero connection formalism for arbitrary values of the Immirzi parameter

Abstract

The Ashtekar-Barbero-Immirzi formulation of General Relativity is extended to include spinor matter fields. Our formulation applies to generic values of the Immirzi parameter and reduces to the Ashtekar-Romano-Tate approach when the Immirzi parameter is taken equal to the imaginary unit. The dynamics of the gravity-fermions coupled system is described by the Holst plus Dirac action with a non-minimal coupling term. The non-minimal interaction together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant, so that the effective action coming out is the one of Einstein-Cartan theory with a typical Fermi-like interaction term: in spite of the presence of spinor matter fields, the Immirzi parameter plays no role in the classical effective dynamics and results to be only a multiplicative factor in front of a total divergence.We reduce the total action of the theory to the sum of dynamically independent Ashtekar-RomanoTate actions for self and anti-self dual connections, with different weights depending on the Immirzi parameter. This allows to calculate the constraints of the complete theory in a simple way, it is only necessary to realize that the Barbero-Immirzi connection is a weighted sum of the self and anti-self dual Ashtekar connections. Finally the obtained constraints for the separated action result to be polynomial in terms of the self and anti-self dual connections, this could have implications in the inclusion of spinor matter in the framework of non-perturbative quantum gravity.

Erratum: Cosmological Bardeen-Cooper-Schrieffer condensate as dark energy [Phys. Rev. D81, 043511 (2010)]

“…However, when torsion is present (for instance, when it is generated by fermions or a spacetime-dependent Barbero-Immirzi field [18][19][20][21][22]), it is natural to include it explicitly in the fundamental action, so that the Holst term is completed by a torsion-torsion piece to form the Nieh-Yan invariant [17,23]. There are several other reasons why to prefer the latter alternative.…”

“…having two contributions [15][16][17], the torsion-free spin connection and a torsion term related to the axial fermion current:…”

Fermions in loop quantum cosmology and the role of parity

“…For α → ∞ we have minimal coupling from the viewpoint of the Holst action, while α = γ corresponds to minimal coupling from the viewpoint of Einstein-Cartan theory [23]. In this article, we allow all possible real values of α as introduced in [23] (see also [53]).…”

“…In this article, we allow all possible real values of α as introduced in [23] (see also [53]). We emphasize that we have parity invariance for all real α, even though some torsion components such as those written below consist of contributions of different parity behavior.…”

“…We follow the symmetry reduction of torsion-free Bianchi class A models [20,21,22], combined with the canonical formulation of gravity with fermions [14,15,16,23]; the general formulation without symmetry is summarized in the Appendix. Here, we combine these research lines and explore the symmetry reduction of gravity coupled to fermions in a first-order formalism, implying a theory with torsion.…”