We develop a structural bridge between relativistic Hamilton–Jacobi theory and the relativistic Schrödinger equation within the framework of tempered distributions and Schwartz linear algebra. For translation-invariant Hamiltonians, the principal functions \( S_p(x)=\langle p,x\rangle \) restricted to the mass shell form a complete integral of the Hamilton–Jacobi equation, while their exponential images \( \eta_p=\exp\!\left(\frac{i}{\hbar}S_p\right) \) constitute a Schwartz basis of the tempered state space. On each spectral fiber, both classical and quantum equations reduce to the same Einstein dispersion relation. We prove that the relativistic Schrödinger equation is precisely the Schwartz–von Neumann S–linear extension of the classical energy relation from certainty momentum states to arbitrary tempered superpositions. In the presence of scalar potentials, the Hamiltonian arises as a mixed (momentum-diagonal and position-diagonal) extension, showing that the extension principle is not restricted to the free case. We further demonstrate that exact quantum dynamics cannot, in general, be represented by a single exponential phase \( \exp\!\left(\frac{i}{\hbar}S\right) \) unless \( S \) is affine in space. Instead, quantum evolution is obtained by S–superpositions of the principal exponential family associated with a complete integral of the Hamilton–Jacobi equation. In this sense, classical elimination of parameters is replaced by linear spectral superposition. Geometrically, the exponential mapping transforms the flat affine space of Minkowski generators into a curved manifold of principal waves on which the nonlinear Hamilton–Jacobi flow pushes forward to a linear unitary Schrödinger flow. Through de Broglie–Maxwell isomorphisms, the construction extends to complex electromagnetic-like fields, preserving translation representation, dispersion relations, and polarization geometry. The results suggest that, for translation-invariant systems, quantization may be understood as an infinite-dimensional complex linearization of a classical certainty space rather than as a semiclassical approximation. Within the tempered-distribution setting, relativistic quantum dynamics emerges as the superpositional completion of a classical complete integral.
We develop a structural bridge between relativistic Hamilton–Jacobi theory and the relativistic Schrödinger equation within the framework of tempered distributions and Schwartz linear algebra. For translation-invariant Hamiltonians, the principal functions ( S_p(x)=\langle p,x\rangle ) restricted to the mass shell form a complete integral of the Hamilton–Jacobi equation, while their exponential images ( \eta_p=\exp!\left(\frac{i}{\hbar}S_p\right) ) constitute a Schwartz basis of the tempered state space. On each spectral fiber, both classical and quantum equations reduce to the same Einstein dispersion relation. We prove that the relativistic Schrödinger equation is precisely the Schwartz–von Neumann S–linear extension of the classical energy relation from certainty momentum states to arbitrary tempered superpositions. In the presence of scalar potentials, the Hamiltonian arises as a mixed (momentum-diagonal and position-diagonal) extension, showing that the extension principle is not restricted to the free case. We further demonstrate that exact quantum dynamics cannot, in general, be represented by a single exponential phase ( \exp!\left(\frac{i}{\hbar}S\right) ) unless ( S ) is affine in space. Instead, quantum evolution is obtained by S–superpositions of the principal exponential family associated with a complete integral of the Hamilton–Jacobi equation. In this sense, classical elimination of parameters is replaced by linear spectral superposition. Geometrically, the exponential mapping transforms the flat affine space of Minkowski generators into a curved manifold of principal waves on which the nonlinear Hamilton–Jacobi flow pushes forward to a linear unitary Schrödinger flow. Through de Broglie–Maxwell isomorphisms, the construction extends to complex electromagnetic-like fields, preserving translation representation, dispersion relations, and polarization geometry. The results suggest that, for translation-invariant systems, quantization may be understood as an infinite-dimensional complex linearization of a classical certainty space rather than as a semiclassical approximation. Within the tempered-distribution setting, relativistic quantum dynamics emerges as the superpositional completion of a classical complete integral.
Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density ρ=R2 coupled to Hamilton's principal function S. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair (R,S) into a single complex field. Starting from a completely general complex ansatz ψ=f(R,S)eig(R,S), and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose ∣κ∣→0 limit reproduces the HJ formulation exactly. Remarkably, when Re(κ)=0, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.
In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the SchrÅNodinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′ - represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.
Quantum mechanics is conventionally formulated in terms of complex Hilbert spaces, linear operators, and the Schrödinger equation. Although geometric structures play an increasingly important role in modern formulations of quantum theory, the wavefunction itself is generally regarded as a fundamental object. In this work, we investigate an alternative formulation in which the fundamental object is a coherence coordinate , where Omega represents radial coherence and Theta represents phase coherence associated with geometric transport. Starting from three assumptions—a coherence coordinate, a probability density generated by radial coherence, and a geometric Hamiltonian—we reconstruct the continuity equation, the quantum Hamilton–Jacobi equation, and ultimately the Schrödinger equation. Within this framework, the wavefunction appears as a derived quantity, while Hilbert spaces emerge naturally as spaces of square-integrable functions on a coherence manifold. The resulting construction establishes a differential-geometric representation of quantum mechanics and provides a common geometric language connecting quantum states, probability flow, geometric phases, holonomies, and gauge transport.
We develop a Maxwell–Schwartz field-theoretic formulation of spin-1/2 quantum systems within the framework of tempered distribution spaces and Schwartz linear algebra. In this approach, the fundamental objects of the theory are Maxwellian fields, namely elements of S′(M4,C3), while scalar quantum states arise as derived projections carrying the spectral structure of the theory. Relativistic dynamics is formulated through square-root Hamiltonian operators defined via spectral calculus. On a distinguished subspace of Maxwellian fields, the curl operator realizes the momentum magnitude operator, allowing a direct correspondence between scalar and vectorial relativistic dynamics. This leads to a Schrödinger-type formulation of relativistic evolution without requiring Dirac’s matrix factorization as a primitive assumption. Spin-1/2 structures emerge naturally from two-component Cartesian configurations of Maxwellian fields. The Pauli algebra is realized as an internal symmetry acting on such pairs, and spin observables correspond to explicit transformations of field components. A first-order Dirac-type dynamics is obtained through operator factorization, yielding a Maxwellian realization of relativistic two-component equations, including the massless Weyl regime. The tensorial structure of the theory provides a natural setting for entanglement. Bell-type correlations are derived intrinsically from Maxwellian field configurations, reproducing the standard quantum predictions and their violation of Bell inequalities. A central result of the present work is the energetic interpretation of quantum probability. The spectral coefficients of a state determine the amplitudes of its Maxwellian realization, whose energy density is quadratic in these amplitudes. The Born rule is thus recovered as a normalized energy distribution law. Moreover, the electromagnetic continuity equation induces a probability continuity equation, with the probability current identified as the normalized Poynting flux. This transport interpretation extends to entangled states, where Bell correlations are interpreted as correlations in the transport of energy across field channels. Finally, the framework is extended to massive Maxwellian fields, where probability currents are expressed through nonlocal spectral kernels associated with square-root Hamiltonians, yielding a unified continuity structure for both massless and massive regimes. These results suggest that spinorial quantum structures, probabilistic interpretation, and quantum correlations may be understood as emerging from the internal organization, spectral structure, and energy transport of Maxwellian fields in the Schwartz distributional setting.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.