Exploring the near-kernel sector of quantum reduced loop gravity

We study the near-kernel sector of the Hamiltonian constraint operator in the one-vertex model of quantum reduced loop gravity using exact diagonalisation methods and variational Monte Carlo methods with neural quantum states. The analysis is based on the symmetric Hamiltonian containing both Euclidean and Lorentzian contributions, and on the variational minimization of the positive quadratic operator Q̂=Ĉ², where Ĉ is the symmetric Hamilton constraint, in truncated Hilbert spaces with spin cutoff up to jₘₐₓ=1001. In the cases where exact diagonalization is possible, the eigenspace of lowest eigenvalue is one- or two-dimensional, and the eigenvalue, as well as the structure of the eigenstates depends on the cutoff. These ground states are entangled across the three edges. The variational method does not find the lowest eigenspace, but near kernel states, i.e., states with components in a number of low lying eigenspaces. For low cutoffs, they also do not factorize across the three edge degrees of freedom. At larger cutoffs, it increasingly finds near kernel states that factorize over the edges, matched with near-unit fidelity by reduced Thiemann coherent states. We discuss some indication that these states are the optimal near-kernel states if factorization is assumed.