Spinor representation of the Hamiltonian constraint in 3D loop quantum gravity with a nonzero cosmological constant

We develop in a companion paper the kinematics of three-dimensional loop quantum gravity in Euclidean signature and with a negative cosmological constant, focusing in particular on the spinorial representation that is well known at zero cosmological constant. In this paper, we put this formalism to the test by quantizing the Hamiltonian constraint on the dual of a triangulation. The Hamiltonian constraints are obtained by projecting the flatness constraints onto spinors, as done in the flat case by the first author and Livine. Quantization then relies on $q$-deformed spinors. The quantum Hamiltonian constraint acts in the $q$-deformed spin network basis as difference equations on physical states, which are thus the Wheeler-DeWitt equations in this framework. Moreover, we study how physical states transform under Pachner moves of the canonical surface. We find that those transformations are in fact $q$ deformations of the transition amplitudes of the flat case as found by Noui and Perez. Our quantum Hamiltonian constraints, therefore, build a Turaev-Viro model at real $q$.

Figure 1

A ribbon edge $R(e)$. The variables $\ell ,u,\tilde{\ell },\tilde{u}$ are assigned to the four sides of the ribbon edge and they are subject to the ribbon constraint $\ell u{\tilde{\ell }}^{-1}{\tilde{u}}^{-1}$ represented as the trivialization of the loop around $R(e)$. The positions of these variables are fixed such that the directions of $u$ and $\tilde{u}$ are opposite to that of the edge $e$ (in gray).

Figure 2

A vertex $v$ on which edges meet becomes a ribbon vertex $R(v)$ incident to ribbon edges. The ribbon vertex is here depicted with a dashed boundary. Due to the clockwise orientation of the short edges of every ribbon edge, the matrix ${\ell }_{ev}$ around $R(v)$ are all oriented counterclockwise.

Figure 3

When considering ribbon edges and ribbon vertices, the face on the left bounded by edges $e_1,\dots ,e_6$ becomes bounded by long edges on the right. All matrices $u_{e_i{v}_{i+1}},i=1,\dots ,6$ are oriented counterclockwise around the face.

Figure 4

The ribbon edge with the holonomies on its long edges, fluxes on its short edges and spinors on its corners.

Figure 5

Two edges meet at a vertex and share a corner. There are four configurations of orientations and we indicate the spinors at the common corner.

Figure 6

A sunny graph with edges $e_1,\dots ,e_d$ counterclockwise oriented around the face $f$. Each triple of edges $(e_i,e_{i-1},e_i^{\prime })$ are incident to a vertex $v_i$.

Figure 7

On the left, a triangular face with its adjacent edges. On the right, the ribbon graph it gives rise to.

Figure 8

The schematic representation of the quantum constraint (67) with its $A$ terms and $B$ terms associated to the corners around the face.

Figure 9

A graphical representation of Eq. (81) on the lhs and (82) on the rhs.