Poincaré–Plebański formulation of GR and dual simplicity constraints

Abstract

We revise the classical continuum formulation behind the Spin Foam approach to the quantization of gravity. Based on the recent applications of the current EPRL-FK model beyond triangulations, we identify the tension with the implementation of the 'volume' part of simplicity constraints, required for the passage from the topological BF theory to gravity. The crucial role, played by 4d normals in the linear version of constraints, necessitates the extension of the configuration space, and we argue to switch from normal 3-forms directly to tetrads. The requirement of vanishing torsion leads to consider first an unconstrained extended Poincaré BF theory, which we characterize fully both at the Lagrangian and Hamiltonian levels, paying special attention to its gauge symmetries. The simplicity constraints are introduced naturally, in the spirit of Plebański formulation, and we give their tetradic version, dual to that of using 3-forms. This brings us much closer to the geometric content of General Relativity.

Simplicity constraints: A 3D toy model for loop quantum gravity

“…The construction follows some developments in [18] by enlarging the phase space with independent triad variables. Similar techniques have recently been developed in [19]. It should be noted however that the focus is way different.…”

Volume simplicity constraint in the Engle-Livine-Pereira-Rovelli spin foam model

“…Another possibility to be mentioned is to consider all constraints linearly, along the lines proposed in [6,11]. This would probably necessitate first to enlarge the space of variables of the theory, and then quantize the respective topological model, which is elusive as of yet.…”

“…Since the 4-volume constraint is decidedly quadratic in nature, this discrepancy may pose an additional question: which form of volume simplicity is an appropriate one to impose? One such alternative, formulated linearly, appeared in [12], and in [11] we gave its dual reinterpretation, actually, as prescribing unambiguous 3-volume.…”

Deformation of the Engle-Livine-Pereira-Rovelli spin foam model by a cosmological constant

“…The original EPRL amplitude was defined on a complete graph K 5 , corresponding to the boundary of a 4-simplex. The model has been generalized to arbitrary graphs [10], although it can be argued that the model would have to be amended to include the correct implementation of the volume simplicity constraints [11,12].…”

“…were ω ∈ R is the deformation parameter, σ(C) = ±1 is the type of crossing (ove-or under-crossing, see figure 2), and with (11) where the X + I (X − I ) are an orthonormal basis of the selfdual (anti-self-dual) su (2). The operator R C acts as en- (5).…”