A Immirzi-like parameter for 3D quantum gravity

Abstract

We study an Immirzi-like ambiguity in three-dimensional quantum gravity. It shares some features with the Immirzi parameter of four-dimensional loop quantum gravity: it does not affect the equations of motion, but modifies the Poisson brackets and the constraint algebra at the canonical level. We focus on the length operator and show how to define it through non-commuting fluxes. We compute its spectrum and show the effect of this Immirzi-like ambiguity. Finally, we extend these considerations to 4d gravity and show how the different topological modifications of the action affect the canonical structure of loop quantum gravity.

Hamiltonian dynamics and Faddeev–Jackiw formulation of 3D gravity with a Barbero–Immirzi like parameter

“…It is important to comment that our results have not been reported in the literature, and as a special case those results reported in [4,5] are reproduced. In addition, we would also remark that for BL theory we shall construct the Dirac brackets by eliminating the second class constraints and there will remain first class ones.…”

“…However, in this work the structure of the constraints is not complete, thus, in order to compare the FJ method with the Dirac one it is mandatory to perform the Dirac analysis on the full phase space by following all Dirac steps. It is well known that three-dimensional gravity with a cosmological constant can be written as a Chern-Simons theory [1][2][3][4][5]. In fact, if the principal gauge bundle G over M is given by G = SU(2) for 3d Euclidean gravity, then we can enlarge the group G toG, whereG could be SO(4), ISO(3) or SO(3, 1), depending on the sign of the cosmological constant , it being positive, zero or negative respectively.…”

“…In this respect, it is common to obtain in three dimensions a relation between the Palatini and the Chern-Simons theories, which are equivalent at the Lagrangian level up to a total derivative [2][3][4]. Such a relation is not the only one, since the so-called exotic action with a Barbero-Immirzi like parameter [we call it from now on the Bonzom-Livine action (BL)] turns out to be classically equivalent to Palatini's theory as well [5]. In fact, the BL model describes a set of actions sharing the equations of motion with Palatini's theory; however, the symplectic structure is different.…”

“…In fact, the equivalence is not given only with the presence of a γ parameter, but at the classical level if we perform a partial gauge fixing in the canonical description of BL, it is possible obtain a full Ashtekar connection dynamics [4]. In this respect, the analysis of the symmetries of the BL action has been performed in [4,5]; in this work the canonical analysis was performed by using the Dirac method. However, the analysis was developed on a smaller phase space and the complete structure of the constraints on the full phase space was not reported.…”

Group field theory and simplicial quantum gravity

“…This defines the discrete BF action. Then, we have simplicity constraints at the level of each tetrahedron of the simplicial complex [22] (see also [29]). These constraints impose the restriction on the classical discrete Lie algebra B variables constraining the set of B f variables to be determined by a discrete tetrad field [28] and thus giving gravity from BF theory.…”