Gauge reduction with respect to simplicity constraint in all dimensional loop quantum gravity

In this paper, we discuss the gauge reduction with respect to the simplicity constraint in both classical and quantum theory of all dimensional loop quantum gravity. With the gauge reduction with respect to the edge-simplicity constraint being processed and the anomalous vertex simplicity constraint being imposed weakly in holonomy-flux phase space, the simplicity reduced holonomy can be established. However, we find that the simplicity reduced holonomy cannot capture the degrees of freedom of intrinsic curvature, which leads to its failure to construct a correct scalar constraint operator in all dimensional loop quantum gravity (LQG) following the standard strategy. To tackle this problem, we establish a new type of holonomy corresponding to the simplicity reduced connection, which captures the degrees of freedom of both intrinsic and extrinsic curvature properly. Based on this new type of holonomy, we propose three new strategies to construct the scalar constraint operators, which serve as valuable candidates to study the dynamics of all dimensional LQG in the future.

Automated summary

In this paper, we explore the gauge reduction of the simplicity constraint in both classical and quantum theory of loop quantum gravity (LQG) in all dimensions. We find that the simplicity reduced holonomy fails to capture the degrees of freedom of intrinsic curvature, preventing the construction of a correct scalar constraint operator. To address this issue, we introduce a new type of holonomy corresponding to the simplicity reduced connection, which properly captures the degrees of freedom of both intrinsic and extrinsic curvature. Based on this new holonomy, we propose three new strategies to construct scalar constraint operators, which could serve as valuable candidates for studying the dynamics of all dimensional LQG in the future.