Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.26016 · Physical review. D/Physical review. D.

The Schrodinger Equation as a Gauge Theory

Dmitry S. Ageev, Vladimir A. Bykov

hep-thcond-mat.str-elmath-phquant-phSubmitted Apr 28, 2026• Announced Apr 30, 2026• 0 citations
v1 · Apr 28, 2026v2 · Jun 20, 2026v3 · Sep 25, 2026
arXivHTMLPDF

Abstract

In this paper, we reformulate the Schrodinger equation in gauge-theoretic terms. Starting from the Madelung representation, we rewrite the conserved probability-current using gauge fields, namely a one-form gauge field in the $(2+1)$-dimensional theory and a two-form gauge field in the $(3+1)$-dimensional theory. This gives a local equivalence between the Schrodinger equation, quantum hydrodynamics, and a gauge formulation, while the global information is carried by the quantization of phase winding around zeros of the wavefunction. We then explore how this correspondence organizes structures on both sides of the duality. On the gauge side, BF couplings to additional one-forms describe electromagnetic coupling, Berry connections, spinor dynamics, projected non-abelian adiabatic connections, and intrinsic holonomy, while Chern--Simons term generate Hopf functionals, charge-flux attachment, and anyonic sectors. In the presence of boundaries, the topological terms produce edge degrees of freedom and boundary charge algebras. Finally, in the nonlinear regime with a Bogoliubov sound mode, the dual two-form description relates acoustic memory, large gauge transformations and the corresponding soft theorem, organizing them into an infrared triangle.

Comments
58 pages, comments and references are welcome

Related papers

From General-Relativistic Boundary Symplectic Structure to the Bell Singlet: A Conditional Spin-Endpoint Reconstruction

Guojun Pan

This work presents a logically closed theoretical framework that unifies the geometric structure of General Relativity (GR) with the axiomatic foundations of quantum mechanics and the emergence of observable forces, organized in three interconnected modules. I. GR / CQG Geometric Backbone The framework is rooted in GR/CQG fields. A primary chain defines topological invariants via connection A and curvature F: Connection A and curvature F → Chern-Weil invariant, defined as the trace of the wedge product of F with itself (tr(F wedge F)) This leads to the Chern-Simons functional CS(A), with the topological current defined as the Hodge dual of CS (K = *CS) These constructs yield a projected current / force seat, which provides the topological basis for force interactions. A parallel boundary chain establishes the geometric basis for quantization: The boundary symplectic form induces a period lattice, defined as Per(Omega) = 2 pi hbar Z This lattice structure supports the construction of a prequantum line bundle The prequantum bundle then defines Hilbert space sectors and their associated projection operators, forming the stage for quantum mechanics. II. Quantum Axiomatization and Bell Branch From the geometric backbone, we derive the quantum structure of spin and entanglement: A GR boundary spin endpoint carries an SU(2) residual surface charge J, with a fixed Casimir invariant |J| = s This charge defines a KKS orbit SU(2)/U(1) = S2, which selects the minimal spin branch s = hbar/2, corresponding to the V1/2 representation The tensor product of two V1/2 endpoints yields a total trivial SU(2) charge, forming a singlet sector with invariant space Inv = V0 Applying the Born projection measure to this singlet sector recovers the Bell/Tsirelson correlation function E(a,b) = -a dot b, with no-signaling conditions automatically preserved, as marginal distributions are independent of remote measurement settings. III. Response, Force, Normalization, and Data Anchors The framework connects topological structure to observable physics and experimental data: A thermodynamic free-energy route leads to a resonance / linear response route, which feeds into the CS transgression current route These yield the projected equivalence relation P_top J = lambda K_CS, where the projected current J is proportional to the Chern-Simons current K_CS via the coupling constant lambda This relation defines a rank-two force/current seat, providing a concrete realization of topological forces. The coupling lambda is normalized via physical gauge matching, linking to the electromagnetic U(1) branch defined by e^2 = 4 pi alpha, where alpha is the fine-structure constant. This connection enables low-energy Hamiltonian tests, anchoring the framework to experimental data including neutrino holonomy, SPARC measurements, and Bell/geometric phase tests. The program achieves conditional closure across all branches. Remaining hard tasks include full parent-action endpoint selection, minimal Casimir selection, and likelihood-level experimental constraints.

•May 28, 2026•0 citations

Conserved Currents on Multiply Connected Domains: Covering-Space Constraints in Quantum and Gravitational Dynamics

Scott Broock

The relationship between quantum mechanics and general relativity is typically pursued via quantizing the spacetime metric or modifying the Standard Model. This paper proposes an alternative structural approach: treating both frameworks as local differential theories whose admissible global solutions are restricted by the geometry of their domains. The Schrödinger equation, expressed in Madelung form, makes this explicit: probability is conserved, and the resulting current must route around regions where the wavefunction is zero. This same structure of current routing around obstructions explains double-slit interference, Bell correlations, gravitational lensing, and the geometric role of phase in coherent evolution. Here, "phase" is used in its geometric sense as a connection on the space of states, while "density" is the conserved quantity whose zeros define the topological obstructions. The incompatibility therefore dissolves not by quantizing gravity or modifying quantum mechanics, but by recognizing their common structure. This paper initiates a series (Unitary Scale Theory, UST) examining scale-independent consequences of covering-space constraints on conserved currents. Version history v1 (January 2026): Initial publication. v2c (February 2026): Title refined to clearly reflect scope. Holonomy language updated throughout for consistency with covering-space / lift-projection framework developed in subsequent papers in the series. No changes to mathematical content or results.

•Jan 1, 2026•2 citations

The Fisher Information Action for Quantum Dynamics

Jesper Lyng Jensen

This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.

•Dec 16, 2025•1 citation

Operator Gauge Transformations in Nonrelativistic Quantum Electrodynamics

Raymond Dale Gray

A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.

•Dec 1, 1982•0 citations

The symplectic geometrical formulation of quantum hydrodynamics

Andika Kusuma Wijaya, Arief Hermanto, Muhammad Farchani Rosyid

Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.

International Journal of Geometric Methods in Modern Physics•Jul 1, 2022•3 citations

Amplitude-dependent quantum hydrodynamics from a coth⁡\cothcoth-Madelung ansatz

C. Dedes

We investigate a nonlinear extension of the Madelung transformation based on a hyperbolic phase--amplitude coupling of the form \[ Ψ= R e^{\imathθ\coth R}, \] where \(R\) is a real amplitude field and \(θ\) is an auxiliary phase coordinate governed by Schrödinger's equation. In contrast to the conventional polar decomposition, this construction imposes a singular hyperbolic relation between amplitude and phase, thereby endowing the Bohmian or hydrodynamic description with an intrinsically geometric structure. We show that the associated velocity field acquires a density-gradient contribution, producing generalized continuity equations and modified quantum force terms. When interpreted as a complex macroscopic order parameter, this generalized phase structure leads to modified superconducting electrodynamics; in particular, the London equations acquire additional contributions that render the Meissner response sensitive to spatial density gradients. The proposed framework is motivated by broader developments involving complex group velocities, dissipative wave propagation, and amplitude-sensitive transport in quantum systems.

quant-pharXiv:2606.06561•Jun 5, 2026•0 citations
In this paper, we reformulate the Schrodinger equation in gauge-theoretic terms. Starting from the Madelung representation, we rewrite the conserved probability-current using gauge fields, namely a one-form gauge field in the (2+1)(2+1)(2+1)-dimensional theory and a two-form gauge field in the (3+1)(3+1)(3+1)-dimensional theory. This gives a local equivalence between the Schrodinger equation, quantum hydrodynamics, and a gauge formulation, while the global information is carried by the quantization of phase winding around zeros of the wavefunction. We then explore how this correspondence organizes structures on both sides of the duality. On the gauge side, BF couplings to additional one-forms describe electromagnetic coupling, Berry connections, spinor dynamics, projected non-abelian adiabatic connections, and intrinsic holonomy, while Chern--Simons term generate Hopf functionals, charge-flux attachment, and anyonic sectors. In the presence of boundaries, the topological terms produce edge degrees of freedom and boundary charge algebras. Finally, in the nonlinear regime with a Bogoliubov sound mode, the dual two-form description relates acoustic memory, large gauge transformations and the corresponding soft theorem, organizing them into an infrared triangle.
This work presents a logically closed theoretical framework that unifies the geometric structure of General Relativity (GR) with the axiomatic foundations of quantum mechanics and the emergence of observable forces, organized in three interconnected modules. I. GR / CQG Geometric Backbone The framework is rooted in GR/CQG fields. A primary chain defines topological invariants via connection A and curvature F: Connection A and curvature F → Chern-Weil invariant, defined as the trace of the wedge product of F with itself (tr(F wedge F)) This leads to the Chern-Simons functional CS(A), with the topological current defined as the Hodge dual of CS (K = *CS) These constructs yield a projected current / force seat, which provides the topological basis for force interactions. A parallel boundary chain establishes the geometric basis for quantization: The boundary symplectic form induces a period lattice, defined as Per(Omega) = 2 pi hbar Z This lattice structure supports the construction of a prequantum line bundle The prequantum bundle then defines Hilbert space sectors and their associated projection operators, forming the stage for quantum mechanics. II. Quantum Axiomatization and Bell Branch From the geometric backbone, we derive the quantum structure of spin and entanglement: A GR boundary spin endpoint carries an SU(2) residual surface charge J, with a fixed Casimir invariant |J| = s This charge defines a KKS orbit SU(2)/U(1) = S2, which selects the minimal spin branch s = hbar/2, corresponding to the V1/2 representation The tensor product of two V1/2 endpoints yields a total trivial SU(2) charge, forming a singlet sector with invariant space Inv = V0 Applying the Born projection measure to this singlet sector recovers the Bell/Tsirelson correlation function E(a,b) = -a dot b, with no-signaling conditions automatically preserved, as marginal distributions are independent of remote measurement settings. III. Response, Force, Normalization, and Data Anchors The framework connects topological structure to observable physics and experimental data: A thermodynamic free-energy route leads to a resonance / linear response route, which feeds into the CS transgression current route These yield the projected equivalence relation P_top J = lambda K_CS, where the projected current J is proportional to the Chern-Simons current K_CS via the coupling constant lambda This relation defines a rank-two force/current seat, providing a concrete realization of topological forces. The coupling lambda is normalized via physical gauge matching, linking to the electromagnetic U(1) branch defined by e^2 = 4 pi alpha, where alpha is the fine-structure constant. This connection enables low-energy Hamiltonian tests, anchoring the framework to experimental data including neutrino holonomy, SPARC measurements, and Bell/geometric phase tests. The program achieves conditional closure across all branches. Remaining hard tasks include full parent-action endpoint selection, minimal Casimir selection, and likelihood-level experimental constraints.
The relationship between quantum mechanics and general relativity is typically pursued via quantizing the spacetime metric or modifying the Standard Model. This paper proposes an alternative structural approach: treating both frameworks as local differential theories whose admissible global solutions are restricted by the geometry of their domains. The Schrödinger equation, expressed in Madelung form, makes this explicit: probability is conserved, and the resulting current must route around regions where the wavefunction is zero. This same structure of current routing around obstructions explains double-slit interference, Bell correlations, gravitational lensing, and the geometric role of phase in coherent evolution. Here, "phase" is used in its geometric sense as a connection on the space of states, while "density" is the conserved quantity whose zeros define the topological obstructions. The incompatibility therefore dissolves not by quantizing gravity or modifying quantum mechanics, but by recognizing their common structure. This paper initiates a series (Unitary Scale Theory, UST) examining scale-independent consequences of covering-space constraints on conserved currents. Version history v1 (January 2026): Initial publication. v2c (February 2026): Title refined to clearly reflect scope. Holonomy language updated throughout for consistency with covering-space / lift-projection framework developed in subsequent papers in the series. No changes to mathematical content or results.
This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.
Quantum hydrodynamics described by Madelung equations is analyzed in the framework of symplectic geometry i.e. in covariant phase space approach to geometric field theory. The pre-symplectic manifolds providing the phase spaces describing the Hamiltonian dynamics of quantum fluid are constructed from the set of all solutions of Madelung equations and their corresponding Lagrangian densities. The Madelung equations under consideration are the Madelung equations associated to Schroedinger equations (in the nonrelativistic case) and Madelung equations associated to Klein–Gordon equations (in the relativistic case). The cases where the coupling with electromagnetic fields is present are also considered here. Our symplectic formulation is different from that of [M. Spera, Moment map and gauge geometric aspects of the Schrodinger and Pauli equations, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1–36] in the choice of fundamental fields or variables. Here we regard density function [Formula: see text] and phase function [Formula: see text] not as canonical pair but as the fundamental fields of the theory. The Hamiltonian vector fields corresponding to an observable are obtained from the Hamiltonian equation generated by the observable. The Poisson bracket of two observables then is determined by the Hamiltonian vector fields associated to each observable. In general, the Poisson bracket of two observables is not unique due to the fact that every observable has more than one corresponding Hamiltonian vector field. It is pointed out that the Poisson bracket has a unique value over a certain subset of the set of all observables defined on the pre-symplectic manifold of the Madelung equation under consideration.
We investigate a nonlinear extension of the Madelung transformation based on a hyperbolic phase--amplitude coupling of the form Ψ=Reıθcoth⁡R, Ψ= R e^{\imathθ\coth R}, Ψ=ReθcothR, where RRR is a real amplitude field and θθθ is an auxiliary phase coordinate governed by Schrödinger's equation. In contrast to the conventional polar decomposition, this construction imposes a singular hyperbolic relation between amplitude and phase, thereby endowing the Bohmian or hydrodynamic description with an intrinsically geometric structure. We show that the associated velocity field acquires a density-gradient contribution, producing generalized continuity equations and modified quantum force terms. When interpreted as a complex macroscopic order parameter, this generalized phase structure leads to modified superconducting electrodynamics; in particular, the London equations acquire additional contributions that render the Meissner response sensitive to spatial density gradients. The proposed framework is motivated by broader developments involving complex group velocities, dissipative wave propagation, and amplitude-sensitive transport in quantum systems.