Towards the self-adjointness of a Hamiltonian operator in loop quantum gravity

Abstract

Although the physical Hamiltonian operator can be constructed in the deparameterized model of loop quantum gravity coupled to a scalar field, its property is still unknown. This open issue is attacked in this paper by considering an operator ${\hat{H}}_v$ representing the square of the physical Hamiltonian operator acting nontrivially on two-valent spin networks. The Hilbert space ${\mathcal{H}}_v$ preserved by the graphing changing operator ${\hat{H}}_v$ is consist of spin networks with a single two-valent non-degenerate vertex. The matrix element of ${\hat{H}}_v$ are explicitly worked out in a suitable basis. It turns out that the operator ${\hat{H}}_v$ is essentially self-adjoint, which implies a well-defined physical Hamiltonian operator in ${\mathcal{H}}_v$ for the deparameterized model.