Regularization ambiguities in loop quantum gravity

One of the main achievements of loop quantum gravity is the consistent quantization of the analog of the Wheeler-DeWitt equation which is free of ultraviolet divergences. However, ambiguities associated to the intermediate regularization procedure lead to an apparently infinite set of possible theories. The absence of an UV problem—the existence of well-behaved regularization of the constraints—is intimately linked with the ambiguities arising in the quantum theory. Among these ambiguities is the one associated to the $SU(2)$ unitary representation used in the diffeomorphism covariant “point-splitting” regularization of the nonlinear functionals of the connection. This ambiguity is labeled by a half-integer $m$ and, here, it is referred to as the $m$ ambiguity. The aim of this paper is to investigate the important implications of this ambiguity. We first study $2+1$ gravity (and more generally BF theory) quantized in the canonical formulation of loop quantum gravity. Only when the regularization of the quantum constraints is performed in terms of the fundamental representation of the gauge group does one obtain the usual topological quantum field theory as a result. In all other cases unphysical local degrees of freedom arise at the level of the regulated theory that conspire against the existence of the continuum limit. This shows that there is a clear-cut choice in the quantization of the constraints in $2+1$ loop quantum gravity. We then analyze the effects of the ambiguity in $3+1$ gravity exhibiting the existence of spurious solutions for higher representation quantizations of the Hamiltonian constraint. Although the analysis is not complete in $3+1$ dimensions—due to the difficulties associated to the definition of the physical inner product—it provides evidence supporting the definitions quantum dynamics of loop quantum gravity in terms of the fundamental representation of the gauge group as the only consistent possibilities. If the gauge group is $SO(3)$ we find physical solutions associated to spin-two local excitations.

Figure 1

Infinitesimal plaquette delta distributions can be integrated and fusioned with the corresponding modification of the face amplitude. The lines in the previous figure represent the holonomy around plaquettes of the regulator in the representation denoted by the Latin index $k$ and $j$ in this case. The dark boxes denote integration of the generalized connection associated to the corresponding edge. The equation represented by the figure is a trivial consequence of the orthogonality of unitary irreducible representations of $SU(2)$. The plaquettes in the picture are square as a matter of simplicity.

Figure 2

Interpretation of the solutions (45) for $m=1$ as graviton excitations. Starting from a solution to the constraints given by a diff-invariant spin network with a vertex with no exceptional edge we can construct a new solution as explained in Sec. 3c and illustrated here. The solution space is parametrized by the quantum numbers $\alpha$ and $\beta$ in this figure. The dotted region corresponds to a single point in the spin-network graph.