Quantum-Elastic Geometry I: From a Single Background-Free Carrier to Lorentzian Geometry and Quantum Kinematics

Quantum geometry is conventionally understood in parameter or momentum space and need not be related to a geometry in configuration space. Here we show for Dirac fermions in a background electromagnetic field that an induced Lorentzian geometry is generated from the fermionic quantum metric. We study a Lorentz covariant phase space worldline path integral construction, whose application of the adiabatic theorem onto spin degrees of freedom leads to the appearance of the Berry phase at $O(\hbar)$ , and the quantum metric at $O(\hbar^2)$ in the worldline action. Both the Feynman and retarded/advanced propagators are examined in 3+1 and 2+1 dimensions, whose different on-shell implementations lead to related quantum metrics; QED/QED ${}_3$ and condensed matter Dirac theory are considered. An effective theory is found by integrating out the momenta degrees of freedom, which gives rise to an emergent Lorentzian metric in configuration space, whose perturbation to the flat metric is dependent on the quantum metric and the background field strength. Apart from the spin degrees of freedom encoded into the Berry phase, the effective theory describes a scalar particle moving in the induced quantum geometry. Phenomenological applications to the covariant geometry are further studied using spectral holonomy in a gapped 3-dimensional Dirac material.Quantum geometry is conventionally understood in parameter or momentum space and need not be related to a geometry in configuration space. Here we show for Dirac fermions in a background electromagnetic field that an induced Lorentzian geometry is generated from the fermionic quantum metric. We study a Lorentz covariant phase space worldline path integral construction, whose application of the adiabatic theorem onto spin degrees of freedom leads to the appearance of the Berry phase at $O(\hbar)$, and the quantum metric at $O(\hbar^2)$ in the worldline action. Both the Feynman and retarded/advanced propagators are examined in 3+1 and 2+1 dimensions, whose different on-shell implementations lead to related quantum metrics; QED/QED${}_3$ and condensed matter Dirac theory are considered. An effective theory is found by integrating out the momenta degrees of freedom, which gives rise to an emergent Lorentzian metric in configuration space, whose perturbation to the flat metric is dependent on the quantum metric and the background field strength. Apart from the spin degrees of freedom encoded into the Berry phase, the effective theory describes a scalar particle moving in the induced quantum geometry. Phenomenological applications to the covariant geometry are further studied using spectral holonomy in a gapped 3-dimensional Dirac material.

hep-thCross-listedarXiv:2610.41138Nov 2, 2026

Hydrodynamic Mechanics of Wave-Particle Duality and the Kinematic Derivation of Fermionic Spin

V.L. Fernandez-Alonso

This paper provides a deterministic, local mechanical framework for two of the most elusive phenomena in the microcosm: wave-particle duality and fermionic spin-½ . Under the framework of Quantum Continuum Mechanics, we demonstrate that the probabilistic formalism of the wave function can be physically understood as the real tensional perturbation of a pilot wave propagating through a viscoelastic background termed the Closed String Matrix (CSM). By defining the free electron as an autonomous micro-vortex, the double-slit experiment is resolved via deterministic hydrodynamic guiding. Furthermore, the 720° rotation paradox of fermions is modeled as a three-dimensional elastic holonomy, allowing us to derive the Dirac Equation kinematically as the Navier-Stokes equivalent for shear stress within a continuous medium. Low-energy experimental forecasts are provided.This paper provides a deterministic, local mechanical framework for two of the most elusive phenomena in the microcosm: wave-particle duality and fermionic spin-½ . Under the framework of Quantum Continuum Mechanics, we demonstrate that the probabilistic formalism of the wave function can be physically understood as the real tensional perturbation of a pilot wave propagating through a viscoelastic background termed the Closed String Matrix (CSM). By defining the free electron as an autonomous micro-vortex, the double-slit experiment is resolved via deterministic hydrodynamic guiding. Furthermore, the 720° rotation paradox of fermions is modeled as a three-dimensional elastic holonomy, allowing us to derive the Dirac Equation kinematically as the Navier-Stokes equivalent for shear stress within a continuous medium. Low-energy experimental forecasts are provided.

Jul 15, 20260 citations

Quantum-Elastic Geometry: From Geometric Extension to Linear and Non Linear Regimes

Juan Moreno Borrallo

Jan 1, 20260 citations

Quantum kinematics and geometric quantization

Zhao Qiang

Journal of Geometry and PhysicsDec 1, 19962 citations

Configuration-Space Schrödinger from Capacity Geometry: Many-body Extension, Wallstrom Obstruction, and Conditional Closure

Nala Trees

This paper studies the fixed-background many-body extension and the Wallstrom obstruction within the scalar density-phase framework of capacity geometry. Within the same local first-order homogeneous density-phase action class as the general scalar forcing theorem, the forcing argument extends immediately to declared configuration-space backgrounds and yields the standard spinless many-body Schrödinger equation without any factorisation ansatz. The bare density-phase equations do not force Wallstrom quantisation: a punctured-disc family gives explicit finite-action counterexamples, and the sharpened locallocking relation reduces that family only continuously, not integrally. The strongest continuum closure theorem is a domain-selection statement: on the intrinsic nonvanishing complex-field domain the forced action is the Schrödinger action, Wallstrom counterexamples are excluded, and circulation quantisation is automatic by degree theory. What remains open is whether the SGOC construction selects that single-valued complex domain rather than only the larger density-phase representation. A second contribution is a precise theorem map for the discrete carrier-transport approach. Native phase-sensitive SGOC data live on the Fourier/triad skeleton, while the discrete Wallstrom problem needs carrier data on a spatial triangulation of a nodal-complement component. The missing theorem is a genuine native-to-carrier transport theorem. On closed orientable surface carriers the remaining cases split into three transport regimes. Once a global carrier edge-phase field exists, the total-curvature scalar obstruction vanishes automatically by simplicial Stokes; once non-flat support is confined to a proper carrier,homology-killing enlargement and relative H^1 complement globalisation handle the rest of the downstream topology. Selector-side reductions are also consolidated. Before localisation, the remaining selector-side existence content is one universal H^2 transgression class. After localisation to a proper carrier, existence of a strong completion is automatic, and the only surviving noncanonical part is the ambient surface period torsor — an affine A^{2g}-torsor on genus g — with complement residues canonically separated off. Exchange sectors and spin are additional topological or multicomponent input, not outputs of the present scalar variational route. Keywords: - Schrödinger equation - Capacity geometry - Wallstrom obstruction - Many-body quantum mechanics - Madelung equations - Circulation quantisation - Domain selection - Carrier transport - SGOC programme Related identifiers (references cited in this paper): - 10.5281/zenodo.19156067 (cites) — Capacity Geometry: REG, Dirichlet Tension, and Quasi-Local Mass (SGOC Volume 2, Part 1) - 10.5281/zenodo.19156316 (cites) — The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge (Part 1A) - 10.5281/zenodo.19189901 (cites) — The Schrödinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background - 10.5281/zenodo.19259771 (cites) — Radial Surcharge Inversion and Free-Boundary Host Selection - 10.5281/zenodo.19267944 (cites) — Free-Boundary Surface Selection for a First-Order Nonlinear Profile Equation in Three Dimensions Related identifiers (external references): - 10.1103/PhysRevA.49.1613 (cites) — Wallstrom, T. C., "Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations," Physical Review A 49(3) (1994), 1613–1617This paper studies the fixed-background many-body extension and the Wallstrom obstruction within the scalar density-phase framework of capacity geometry. Within the same local first-order homogeneous density-phase action class as the general scalar forcing theorem, the forcing argument extends immediately to declared configuration-space backgrounds and yields the standard spinless many-body Schrödinger equation without any factorisation ansatz. The bare density-phase equations do not force Wallstrom quantisation: a punctured-disc family gives explicit finite-action counterexamples, and the sharpened locallocking relation reduces that family only continuously, not integrally. The strongest continuum closure theorem is a domain-selection statement: on the intrinsic nonvanishing complex-field domain the forced action is the Schrödinger action, Wallstrom counterexamples are excluded, and circulation quantisation is automatic by degree theory. What remains open is whether the SGOC construction selects that single-valued complex domain rather than only the larger density-phase representation. A second contribution is a precise theorem map for the discrete carrier-transport approach. Native phase-sensitive SGOC data live on the Fourier/triad skeleton, while the discrete Wallstrom problem needs carrier data on a spatial triangulation of a nodal-complement component. The missing theorem is a genuine native-to-carrier transport theorem. On closed orientable surface carriers the remaining cases split into three transport regimes. Once a global carrier edge-phase field exists, the total-curvature scalar obstruction vanishes automatically by simplicial Stokes; once non-flat support is confined to a proper carrier,homology-killing enlargement and relative H^1 complement globalisation handle the rest of the downstream topology. Selector-side reductions are also consolidated. Before localisation, the remaining selector-side existence content is one universal H^2 transgression class. After localisation to a proper carrier, existence of a strong completion is automatic, and the only surviving noncanonical part is the ambient surface period torsor — an affine A^{2g}-torsor on genus g — with complement residues canonically separated off. Exchange sectors and spin are additional topological or multicomponent input, not outputs of the present scalar variational route. Keywords: - Schrödinger equation - Capacity geometry - Wallstrom obstruction - Many-body quantum mechanics - Madelung equations - Circulation quantisation - Domain selection - Carrier transport - SGOC programme Related identifiers (references cited in this paper): - 10.5281/zenodo.19156067 (cites) — Capacity Geometry: REG, Dirichlet Tension, and Quasi-Local Mass (SGOC Volume 2, Part 1) - 10.5281/zenodo.19156316 (cites) — The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge (Part 1A) - 10.5281/zenodo.19189901 (cites) — The Schrödinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background - 10.5281/zenodo.19259771 (cites) — Radial Surcharge Inversion and Free-Boundary Host Selection - 10.5281/zenodo.19267944 (cites) — Free-Boundary Surface Selection for a First-Order Nonlinear Profile Equation in Three Dimensions Related identifiers (external references): - 10.1103/PhysRevA.49.1613 (cites) — Wallstrom, T. C., "Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations," Physical Review A 49(3) (1994), 1613–1617

Mar 28, 20260 citations

O(\hbar)

, and the quantum metric at

O(\hbar^2)

in the worldline action. Both the Feynman and retarded/advanced propagators are examined in 3+1 and 2+1 dimensions, whose different on-shell implementations lead to related quantum metrics; QED/QED

{}_3

and condensed matter Dirac theory are considered. An effective theory is found by integrating out the momenta degrees of freedom, which gives rise to an emergent Lorentzian metric in configuration space, whose perturbation to the flat metric is dependent on the quantum metric and the background field strength. Apart from the spin degrees of freedom encoded into the Berry phase, the effective theory describes a scalar particle moving in the induced quantum geometry. Phenomenological applications to the covariant geometry are further studied using spectral holonomy in a gapped 3-dimensional Dirac material.