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

TL;DR

This paper demonstrates that a specific operator representing the square of the physical Hamiltonian in loop quantum gravity is essentially self-adjoint, ensuring a well-defined Hamiltonian in a simplified model.

Contribution

It proves the essential self-adjointness of the Hamiltonian operator acting on two-valent spin networks in loop quantum gravity.

Findings

The operator $ ilde{H}_v$ is explicitly constructed.

$ ilde{H}_v$ is shown to be essentially self-adjoint.

Results support a well-defined physical Hamiltonian in the model.

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.