| _version_ | 1866901845585166336 |
|---|---|
| author | Jonatan P. Camargo |
| author_facet | Jonatan P. Camargo |
| contents | <p>Continuing the program in which the action constitutes the primary structure of dynamics and the phase \(\phi=S/\hbar\) plays the role of the operational variable, we investigate the relationship between global variational frustration and the emergence of gauge connections in the structural operator of the network. In previous works, it was shown that local extremizing trajectories of the action are only physically realizable when compatible with a global spectral condition, formalized by the Laplacian operator associated with the admissible network, and that electromagnetism can be reinterpreted as a local modulation of the phase by a \(U(1)\) connection. The present article unifies these two directions.</p> <p>We show, first, that the Geometric Spectral Stability Principle (GSSP) does not replace the action as a fundamental structure, but acts as a superselection principle: it restricts the set of locally admissible variational configurations to those that remain globally stable. Next, we demonstrate that the incompatibility between local extrema and global coherence --- expressed by the inequality between the global minimum and the naive sum of local minima --- can be encoded as phase holonomy along cycles of the graph. This holonomy naturally leads to the promotion of the real adjacency matrix \(A_{ij}\) to a complex Hermitian matrix \(\widetilde{A}_{ij}=A_{ij}e^{i\theta_{ij}}\), in which \(\theta_{ij}\) represents a discrete \(U(1)\) connection.</p> <p>This results in a magnetic Laplacian</p> <p>\[ \widetilde{K}=D-\widetilde{A}, \] whose global quadratic functional \[ \widetilde{S}_{\mathrm{global}}[\Psi] = \frac12\sum_{(i,j)\in E}A_{ij}\left|\psi_i-e^{i\theta_{ij}}\psi_j\right|^2 \]</p> <p>provides the minimal form compatible with gauge, self-adjointness, and spectral stability. We show that the spectral response of the eigenvalues \(\widetilde{\lambda}_n\) to the holonomy defines the structural route for the emergence of charged states. Spin is not derived in this work; however, we argue that local frustrations incompatible with a scalar representation naturally point to a multicomponent extension of the formalism.</p> <p>The central result is that topological frustration should not be treated as a defect of the theory, but as the structural source that promotes the real combinatorial operator to a gauge covariant operator. Thus, mass and charge come to be seen as complementary manifestations of the same variational-spectral problem: mass associated with the spectrum of the admissible structural operator, and charge associated with the response of this spectrum to the holonomy connection.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19151462 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Variational Frustration, Gauge Connection, and Magnetic Laplacian: A Structural Extension of the Phase Operator Jonatan P. Camargo <p>Continuing the program in which the action constitutes the primary structure of dynamics and the phase \(\phi=S/\hbar\) plays the role of the operational variable, we investigate the relationship between global variational frustration and the emergence of gauge connections in the structural operator of the network. In previous works, it was shown that local extremizing trajectories of the action are only physically realizable when compatible with a global spectral condition, formalized by the Laplacian operator associated with the admissible network, and that electromagnetism can be reinterpreted as a local modulation of the phase by a \(U(1)\) connection. The present article unifies these two directions.</p> <p>We show, first, that the Geometric Spectral Stability Principle (GSSP) does not replace the action as a fundamental structure, but acts as a superselection principle: it restricts the set of locally admissible variational configurations to those that remain globally stable. Next, we demonstrate that the incompatibility between local extrema and global coherence --- expressed by the inequality between the global minimum and the naive sum of local minima --- can be encoded as phase holonomy along cycles of the graph. This holonomy naturally leads to the promotion of the real adjacency matrix \(A_{ij}\) to a complex Hermitian matrix \(\widetilde{A}_{ij}=A_{ij}e^{i\theta_{ij}}\), in which \(\theta_{ij}\) represents a discrete \(U(1)\) connection.</p> <p>This results in a magnetic Laplacian</p> <p>\[ \widetilde{K}=D-\widetilde{A}, \] whose global quadratic functional \[ \widetilde{S}_{\mathrm{global}}[\Psi] = \frac12\sum_{(i,j)\in E}A_{ij}\left|\psi_i-e^{i\theta_{ij}}\psi_j\right|^2 \]</p> <p>provides the minimal form compatible with gauge, self-adjointness, and spectral stability. We show that the spectral response of the eigenvalues \(\widetilde{\lambda}_n\) to the holonomy defines the structural route for the emergence of charged states. Spin is not derived in this work; however, we argue that local frustrations incompatible with a scalar representation naturally point to a multicomponent extension of the formalism.</p> <p>The central result is that topological frustration should not be treated as a defect of the theory, but as the structural source that promotes the real combinatorial operator to a gauge covariant operator. Thus, mass and charge come to be seen as complementary manifestations of the same variational-spectral problem: mass associated with the spectrum of the admissible structural operator, and charge associated with the response of this spectrum to the holonomy connection.</p> |
| title | Variational Frustration, Gauge Connection, and Magnetic Laplacian: A Structural Extension of the Phase Operator |
| url | https://doi.org/10.5281/zenodo.19151462 |