On Loop States in Loop Quantum Gravity

TL;DR

This paper constructs and characterizes all independent loop states in 3+1 dimensional loop quantum gravity on a lattice, using Schwinger bosons and angular momentum quantum numbers, and computes volume operator matrix elements.

Contribution

It provides a complete explicit construction of loop states in 3+1D loop quantum gravity using a lattice regularization and Schwinger bosons, including volume operator analysis.

Findings

Explicit loop state basis constructed on a 3D lattice.

Volume operator matrix elements computed in the loop basis.

Some eigenstates of the volume operator explicitly constructed.

Abstract

We explicitly construct and characterize all possible independent loop states in 3+1 dimensional loop quantum gravity by regulating it on a 3-d regular lattice in the Hamiltonian formalism. These loop states, characterized by the (dual) angular momentum quantum numbers, describe SU(2) rigid rotators on the links of the lattice. The loop states are constructed using the Schwinger bosons which are harmonic oscillators in the fundamental (spin half) representation of SU(2). Using generalized Wigner Eckart theorem, we compute the matrix elements of the volume operator in the loop basis. Some simple loop eigenstates of the volume operator are explicitly constructed.