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arXiv · 1805.08644

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

Cong Zhang·Jerzy Lewandowski·Yongge Ma
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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.

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Cong Zhang, Jerzy Lewandowski, Yongge Ma. 2019-12-15. Towards the self-adjointness of a Hamiltonian operator in loop quantum gravity. https://doi.org/10.1103/physrevd.98.086014

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