Holonomy operator and quantization ambiguities on spinor space

We construct the holonomy-flux operator algebra in the recently developed spinor formulation of loop gravity. We show that, when restricting to SU(2)-gauge invariant operators, the familiar grasping and Wilson loop operators are written as composite operators built from the gauge-invariant “generalized ladder operators” recently introduced in the $\mathrm{U}(N)$ approach to intertwiners and spin networks. We comment on quantization ambiguities that appear in the definition of the holonomy operator and use these ambiguities as a toy model to test a class of quantization ambiguities which is present in the standard regularization and definition of the Hamiltonian constraint operator in loop quantum gravity.

Figure 1

On the left, an oriented edge of a spinor network carrying the holonomy $g$ which maps the spinor $\tilde{z}$ at its source vertex onto the spinor $z$ at its target vertex. On the right, a vertex $v$ of a spinor network with each of the attached edges carrying a spinor $z_i$

Figure 2

The loop $\mathcal{L}=\{e_1,e_2,\dots ,e_n\}$ on the graph $\Gamma$.

Figure 3

The simplest case: $\Gamma$ contains just one vertex $v$ and a loop attached to it. The associated Hilbert space ${\mathcal{H}}_{\Gamma }$ consists of holomorphic square integrable functions in two spinors $|z_1⟩$ and $|z{⟩}_2$. Thus, there are four sets of ladder operators $[a_1,{\overline{a}}_1]=[b_1,\overline{b_1}]=[a_2,{\overline{a}}_2]=[b_2,\overline{b_2}]=1$ out of which Grasping—and Wilson loop-operators, which provide a complete set of SU(2) invariants, are constructed.

Figure 4

A loop within $\Gamma$ that goes through only two vertices. The doublet of harmonic oscillator operators $(a,b)$ is attached to the vertex $v$ while The doublet of harmonic oscillator operators $(\tilde{a},\tilde{b})$ is attached to the vertex $w$.