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

Abstract

Although the physical Hamiltonian operator can be constructed in the deparametrized 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 ${\stackrel{^}{H}}_{v}$ representing the square of the physical Hamiltonian operator acting nontrivially on a two-valent vertex of spin networks. The Hilbert space ${\mathcal{H}}_{v}$ preserved by the graphing changing operator ${\stackrel{^}{H}}_{v}$ is consist of spin networks with a single two-valent nondegenerate vertex. The matrix element of ${\stackrel{^}{H}}_{v}$ are explicitly worked out in a suitable basis. It turns out that the operator ${\stackrel{^}{H}}_{v}$ is essentially self-adjoint, which implies a well-defined physical Hamiltonian operator in ${\mathcal{H}}_{v}$ for the deparametrized model.