Holonomy Operator and Quantization Ambiguities on Spinor Space

TL;DR

This paper develops the holonomy-flux operator algebra within the spinor formulation of loop gravity, highlighting quantization ambiguities and their implications for the Hamiltonian constraint operator.

Contribution

It introduces a new algebraic framework using spinor variables and generalized ladder operators, and analyzes quantization ambiguities in the context of loop quantum gravity.

Findings

Holonomy-flux algebra constructed in spinor formulation

Revealed quantization ambiguities affecting operator definitions

Provided insights into Hamiltonian constraint quantization issues

Abstract

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 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.