Euclidean LQG Dynamics: An Electric Shift in Perspective

TL;DR

This paper introduces a new quantum dynamics framework for Euclidean Loop Quantum Gravity based on Electric Shift, which clarifies the action of the Hamiltonian constraint and resolves the spin j-ambiguity, with implications for Lorentzian LQG.

Contribution

It derives a physically motivated Euclidean LQG quantum dynamics using Electric Shift, resolving the spin j-ambiguity and informing Lorentzian quantum gravity construction.

Findings

Derived a new Euclidean LQG quantum dynamics based on Electric Shift.

Resolved the spin j-ambiguity in holonomy labels.

Implications for constructing Lorentzian LQG dynamics.

Abstract

Loop Quantum Gravity (LQG) is a non-perturbative attempt at quantization of a classical phase space description of gravity in terms of $SU(2)$ connections and electric fields. As emphasized recently [1], on this phase space, classical gravitational evolution in $time$ can be understood in terms of certain gauge covariant generalizations of Lie derivatives with respect to a $spatial$ $SU(2)$ Lie algebra valued vector field called the Electric Shift. We present a derivation of a quantum dynamics for Euclidean LQG which is informed by this understanding. In addition to the physically motivated nature of the action of the Euclidean Hamiltonian constraint so derived, the derivation implies that the spin labels of regulating holonomies are determined by corresponding labels of the spin network state being acted upon thus eliminating the `spin $j$-ambiguity' pointed out by Perez. By virtue of Thiemann's seminal work, the Euclidean quantum dynamics plays a crucial role in the construction of the Lorentzian quantum dynamics so that our considerations also have application to Lorentzian LQG.