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

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