On the dynamics of single-vertex states in quantum-reduced loop gravity

TL;DR

This paper investigates the dynamics of single-vertex states in quantum-reduced loop gravity, deriving the Hamiltonian constraint's action and revealing similarities to loop quantum cosmology, with implications for the Lorentzian part of the Hamiltonian.

Contribution

It derives the Hamiltonian constraint's action on single-vertex states and proposes a modified approach for the Lorentzian part in loop quantum cosmology.

Findings

Hamiltonian constraint action on single-vertex states derived

Euclidean part resembles Bianchi I models in loop quantum cosmology

Suggests a non-trivial operator for the Lorentzian part in quantum theory

Abstract

In this article we examine a Hamiltonian constraint operator governing the dynamics of simple quantum states, whose graph consists of a single six-valent vertex, in quantum-reduced loop gravity. To this end, we first derive the action of the Hamiltonian constraint on generic basis states in the Hilbert space of quantum-reduced loop gravity. Specializing to the example of the single-vertex states, we find that the Euclidean part of the Hamiltonian bears a close formal similarity to the Hamiltonian constraint of Bianchi I models in loop quantum cosmology. Extending the formal analogy to the Lorentzian part of the Hamiltonian suggests a possible modified definition of the Hamiltonian constraint for loop quantum cosmology, in which the Lorentzian part, corresponding to the scalar curvature of the spatial surfaces, is not assumed to be identically vanishing, and is represented by a non-trivial operator in the quantum theory.