The conserved probability current of the Schrodinger field admits a dimension-dependent gauge representation that provides a convenient setting for comparing hydrodynamic, topological and infrared structures. We use this representation to develop a unified treatment of several deformations of the Madelung momentum. BF couplings incorporate electromagnetic interactions, Berry and spin connections, projected non-abelian adiabatic data, and intrinsic phase holonomy within the same framework. In $(2+1)$ dimensions, the Chern-Simons deformation can be reduced to a nonlocal functional of the wavefunction. Its Coulomb-gauge form separates into a phase-sensitive geometric contribution and a density-dependent dynamical contribution; we show explicitly how this separation changes under a general gauge transformation and recover the topological braiding phase in the well-separated adiabatic regime. For systems with a spatial boundary, we derive the equal-time symplectic structure, the associated surface charges, and their boundary algebra. Finally, after a nonlinear interaction produces the Bogoliubov acoustic branch, the radiative memory can be expressed in the dual two-form variables. Its zero-average component is generated by a residual large gauge transformation, whereas the monopole component remains independent global data. This provides a gauge-theoretic description of the memory/asymptotic-symmetry sector of the acoustic infrared structure.
Comments
v1: 58 pages; v2: 59 pages, text modified, misprints corrected, results improved; v3: 64 pages, major revision of sections 3.3 and 4, details added, new references added, comments and references are welcome
The conserved probability current of the Schrodinger field admits a dimension-dependent gauge representation that provides a convenient setting for comparing hydrodynamic, topological and infrared structures. We use this representation to develop a unified treatment of several deformations of the Madelung momentum. BF couplings incorporate electromagnetic interactions, Berry and spin connections, projected non-abelian adiabatic data, and intrinsic phase holonomy within the same framework. In (2+1) dimensions, the Chern-Simons deformation can be reduced to a nonlocal functional of the wavefunction. Its Coulomb-gauge form separates into a phase-sensitive geometric contribution and a density-dependent dynamical contribution; we show explicitly how this separation changes under a general gauge transformation and recover the topological braiding phase in the well-separated adiabatic regime. For systems with a spatial boundary, we derive the equal-time symplectic structure, the associated surface charges, and their boundary algebra. Finally, after a nonlinear interaction produces the Bogoliubov acoustic branch, the radiative memory can be expressed in the dual two-form variables. Its zero-average component is generated by a residual large gauge transformation, whereas the monopole component remains independent global data. This provides a gauge-theoretic description of the memory/asymptotic-symmetry sector of the acoustic infrared structure.
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.
We investigate a nonlinear extension of the Madelung transformation based on a hyperbolic phase--amplitude coupling of the form Ψ=ReθcothR, 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.
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.
The completed Chronoflux Atlas recovers the quantum regime as the coherent phase branch of conserved temporal transport. Continuity supplies the current, temporal order and positive kinetic response recovered in DPL-00 through DPL-04. Coherence adds a non-vanishing complex state with a physical phase and action scale. Once that descendant is admitted, variation returnsphase transport, topology quantises circulation, the Fisher-gradient term returns the Bohm quantum potential, and the density-phase equations recombine exactly into the Schrödinger equation. Quantum behaviour is therefore not attached to continuity by analogy. It is recovered as the lawful dynamics of a coherent continuity branch. The coherent action gives a conserved density current and a quantum Hamilton-Jacobi equation. On the domain where the amplitude does not vanish, the phase one-form is locally exact. Its global periods are integer winding numbers, giving circulation in units of ℎ/𝑚. Those integers persist under every continuous admissible evolution and can change only when a defect crosses the return path or the coherent amplitude vanishes. The apparent discontinuity of a quantum phase slip is thus the topological consequence of continuous evolution through a configuration where the phase ceases to be defined. The same structure returns the exact Madelung representation of a selected Schrödinger branch. On configuration space it yields the Born current, current velocity and quantum potential. Adding an actual configuration following that velocity is the Bohmian ontological completion, not an unstated consequence of continuity. Equivariance then proves persistence of quantum equilibrium. At field level, a shift-symmetric action closes a classical current, its quadratic perturbations recover a characteristic metric, and stationary canonical data produce a recoverable bosonic quantum state. Gauge selection, spin, fermionic statistics, outcome selection and Standard Model promotion remain separately indexed descendants. The paper also closes the non-ideal boundary. Multiplying a density by an arbitrary positive weight generates an exact continuity source and an exact transformed quantum potential; it cannot be presented as a free admissibility correction. Any departure from the closed branch requires a declared self-adjoint operator or trace-preserving open-system law. Admissibility, Recoverability and Persistence then distinguish a genuine quantum branch from incomplete phase data, singular nodal flow or an unreturned measurement model. The result closes coherence, quantised circulation and quantum response within the recovered tree of nature as such. Keywords: coherence; phase; circulation; quantum potential; Madelung transform; Schrödinger equation; Bohmian guidance; equivariance; quantum field reconstruction; Chronoflux Atlas