On the implementation of the canonical quantum simplicity constraint

Abstract

In this paper, we are going to discuss several approaches to solve the quadratic and linear simplicity constraints in the context of the canonical formulations of higher dimensional General Relativity and Supergravity developed in [1,2,3,4,5,6]. Since the canonical quadratic simplicity constraint operators have been shown to be anomalous in any dimension D ≥ 3 in [3], nonstandard methods have to be employed to avoid inconsistencies in the quantum theory. We show that one can choose a subset of quadratic simplicity constraint operators which are nonanomalous among themselves and allow for a natural unitary map of the spin networks in the kernel of these simplicity constraint operators to the SU(2)-based Ashtekar-Lewandowski Hilbert space in D = 3. The linear constraint operators on the other hand are non-anomalous by themselves, however their solution space will be shown to differ in D = 3 from the expected Ashtekar-Lewandowski Hilbert space. We comment on possible strategies to make a connection to the quadratic theory. Also, we comment on the relation of our proposals to existing work in the spin foam literature and how these works could be used in the canonical theory. We emphasise that many ideas developed in this paper are certainly incomplete and should be considered as suggestions for possible starting points for more satisfactory treatments in the future.

Black hole entropy from loop quantum gravity in higher dimensions

“…In this section, we will recall the isolated horizon degrees of freedom using the dimension-independent connection variables as derived in [9,10]. We will neglect the detailed treatment of spatial infinity in this paper, as it is not relevant for the discussion, see e.g.…”

“…(5) is the analogue of the isolated horizon boundary condition F IJ (A) ∝ e I ∧ e J familiar from the 3 + 1-dimensional treatment [8]. In fact, (5) can be rewritten in the form F IJ (A) ∝ e I ∧ e J in 3 + 1 dimensions [9]. Since n Is I = 0 classically by construction, the information contained in these variables associated to the boundary is the (D − 1)-area-density √ h = s Is I of H. We consider the densitized bi-normals L IJ := 2/β n [IsJ] since they already contain this information.…”

“…We demand ∆ to be a spherical non-rotating isolated horizon, see [9] for details. From this, it follows that the null normal l µ , µ = 0, .…”

“…For this, we choose a maximally commuting subset by imagining the edges puncturing H in each of the two charts to be connected to a spin network vertex and following the recipe in [32]. It is easy to see that the subset is identical to a subset that would have been obtained from imposing the off-diagonal simplicity constraints on all of H [33].…”

“…The classical part of this derivation was extended to higher dimensions in [9] and to LanczosLovelock gravity in [10], where the recently introduced connection variables for higher-dimensional general relativity [11][12][13][14] were employed. While the horizon degrees of freedom can be rewritten in a form similar to a higherdimensional Chern-Simons theory, it turns out to be more economical to use a canonically conjugate pair of normals n I ands I as horizon degrees of freedom [9]. In this paper, we will use the latter formulation in order to generalize the entropy calculation to higher dimensions.…”