$\mathcal{N}$-bein formalism for degenerate states in the parameter space of quantum geometry

Quantum Physics arXiv:2609.01752 (quant-ph) [Submitted on 1 Sep 2026] Title:$\mathcal{N}$-bein formalism for degenerate states in the parameter space of quantum geometry Authors:Jorge Romero, Carlos A Velasquez, J David Vergara View a PDF of the paper titled $\mathcal{N}$-bein formalism for degenerate states in the parameter space of quantum geometry, by Jorge Romero and 2 other authors View PDF HTML (experimental) Abstract:Recently, we introduced a geometric object analogous to an orthonormal frame in the Cartan formalism to study the parameter space of quantum systems; we called it $\mathcal{N}$-bein, with $\mathcal{N}$ being the number of parameters that characterize the quantum system. Acting as the ``square root'' of the quantum geometric tensor (QGT), the $\mathcal{N}$-bein allows us to define new tensors to improve our understanding of the structure beneath the parameter space of quantum mechanics. In this work, we extend this mathematical framework surrounding the $\mathcal{N}$-bein to analyze the parameter space of quantum systems with degenerate spectra. As in the non-degenerate case, we define a non-Abelian two-state QGT to identify possible transitions between degenerate states after two consecutive parameter variations. Additionally, using the Wilczek-Zee connection, we introduce a torsion-like tensor as the covariant derivative of the $\mathcal{N}$-bein. This torsion captures the noncommutativity of successive parameter variations and coincides with the antisymmetric part of the two-state QGT. We also present a geometrical formulation using differential forms and discuss the physical implications of the newly defined tensors. Furthermore, we construct several gauge-invariant observables from the $\mathcal{N}$-bein and its derivatives to highlight the utility of the new tensors. Finally, to illustrate the convenience and applications of this formalism, we apply the theoretical framework to a system of coupled harmonic oscillators immersed in an electric field. The coupling between the oscillators results in a degenerate system. Thus, using the new formalism, we found correlations among the quantum states quantified by the new invariants. Comments: Subjects: Quantum Physics (quant-ph); Other Condensed Matter (cond-mat.other); General Relativity and Quantum Cosmology (gr-qc) Cite as: arXiv:2609.01752 [quant-ph] (or arXiv:2609.01752v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2609.01752 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Journal reference: Jorge Romero et al 2026 J. Phys. A: Math. Theor. 59 355305 Related DOI: https://doi.org/10.1088/1751-8121/ae9c7a Focus to learn more DOI(s) linking to related resources Submission history From: Jorge Romero [view email] [v1] Tue, 1 Sep 2026 18:22:09 UTC (129 KB) Full-text links: Access Paper: View a PDF of the paper titled $\mathcal{N}$-bein formalism for degenerate states in the parameter space of quantum geometry, by Jorge Romero and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-09 Change to browse by: cond-mat cond-mat.other gr-qc References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) 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