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A Minor Clinical Surgery to the Schr¨odinger Equation by Physicist

Sardar Dilbag Singh Khalsa

Published Feb 25, 2026• 0 citations
Publisher

Abstract

\begin{abstract} We propose a structural modification of the non-relativistic Schrödinger equation obtained by introducing an additional operator-valued potential term that couples nontrivially to both kinetic and potential contributions. The modified dynamical equation takes the form \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\Phi+ m\delta\Phi- \frac{\hbar^2}{2m}\,\delta\Phi\,\nabla^2\right]\psi,\end{equation} where $\delta\Phi$ represents an additional scalar field or structural correction that deforms the standard Hamiltonian operator. Unlike perturbative potential additions, this term modifies the kinetic structure itself, effectively generating a position-dependent renormalization of the Laplacian operator and inducing a nontrivial coupling between propagation and background geometry. We analyze the mathematical consistency of the modified Hamiltonian, deriving conditions for self-adjointness on $L^2(\mathbb{R}^3)$ and establishing criteria under which unitary time evolution and probability conservation are preserved. The deformation leads to a modified dispersion relation and alters the spectral structure of the system, suggesting an interpretation in terms of emergent or dynamically modified configuration-space geometry. Beyond its formal properties, this framework raises foundational questions regarding the ontological status of the kinetic operator in quantum mechanics. By altering the generator of time evolution itself, the proposal challenges the assumption that the Laplacian represents a fixed Euclidean background structure. Instead, quantum evolution may reflect a deeper geometric or structural layer, in which mass, potential energy, and propagation are relational rather than primitive quantities. This formulation provides a mathematically consistent extension of the Schrödinger framework and opens avenues for interpreting quantum dynamics as arising from modified geometric or higher-structural principles. \end{abstract} \noindent\textbf{Keywords:} Modified Schrödinger Equation; Operator Deformation; Quantum Geometry; Self-Adjoint Hamiltonians; Foundations of Quantum Mechanics. \section{Spacetime–Induced Modification of the Schrödinger Equation} \subsection{1. Introduction} In conventional quantum mechanics, uncertainty is treated as a fundamental property of wavefunctions. In this work we explore an alternative interpretation: that quantum uncertainty may emerge from small-scale spacetime moduli perturbations. We begin in standard 4-dimensional spacetime and then extend the formulation to a constrained multi-time geometry. \subsection{2. Quantum Dynamics in Curved Spacetime} \paragraph{Explanation of the Curved-Spacetime Klein–Gordon Equation.} The equation \[\left( \Box_g + \frac{m^2 c^2}{\hbar^2} \right)\Phi = 0\]describes the dynamics of a scalar quantum field $\Phi$ propagating in a curved spacetime background. Here, $\Box_g$ denotes the covariant d'Alembert operator associated with the metric tensor $g_{\mu\nu}$, defined as \[\Box_g \Phi = \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g}\, g^{\mu\nu} \partial_\nu \Phi \right),\] where $g = \det(g_{\mu\nu})$ is the determinant of the spacetime metric. The factor $\sqrt{-g}$ ensures covariance under general coordinate transformations and accounts for the local volume element of curved spacetime. Physically, this equation generalizes the flat-spacetime Klein–Gordon equation by replacing ordinary derivatives with covariant derivatives determined by the geometry. The metric tensor $g_{\mu\nu}$ encodes gravitational effects, meaning that spacetime curvature directly modifies wave propagation. Thus, particle dynamics are influenced not only by mass $m$ but also by geometric structure. When weak metric perturbations are assumed, \[g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},\qquad |h_{\mu\nu}| \ll 1,\]the geometry deviates slightly from Minkowski space $\eta_{\mu\nu}$. In this regime, gravitational effects enter as small corrections to kinetic terms, enabling a perturbative analysis of how spacetime moduli fluctuations influence quantum evolution. We begin from the relativistic Klein–Gordon equation in curved spacetime: \begin{equation}\left( \Box_g + \frac{m^2 c^2}{\hbar^2} \right)\Phi = 0\end{equation} where \begin{equation}\Box_g \Phi = \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g} g^{\mu\nu} \partial_\nu \Phi \right)\end{equation} Assume weak metric perturbation: \begin{equation}g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},\qquad |h_{\mu\nu}| \ll 1\end{equation} We separate fast rest energy oscillation: \begin{equation}\Phi(x,t) = e^{-i m c^2 t / \hbar} \psi(x,t)\end{equation} Taking the non-relativistic limit yields a modified Schrödinger equation: \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m} \left(1 + \frac{2\phi}{c^2}\right)\nabla^2+ m\phi\right] \psi\end{equation} where $\phi$ is the weak gravitational potential. \subsection{3. Introducing Spacetime Moduli Perturbations} We decompose: \begin{equation}\phi(x,t) = \bar{\phi}(x,t) + \delta\phi(x,t)\end{equation} where $\delta\phi$ represents small-scale spacetime moduli fluctuations. The Hamiltonian becomes: \begin{equation}H = H_0 + \delta H\end{equation} with \begin{equation}H_0 = -\frac{\hbar^2}{2m}\nabla^2 + m\bar{\phi}\end{equation} and \begin{equation}\delta H \approx m\,\delta\phi(x,t)-\frac{\hbar^2}{m c^2}\delta\phi(x,t)\nabla^2\end{equation} Thus the modified Schrödinger equation is: \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\bar{\phi}+ m\delta\phi-\frac{\hbar^2}{m c^2}\delta\phi\nabla^2\right]\psi\end{equation} \textbf{Interpretation:}Spacetime fluctuations enter quantum evolution as both an effective potential term and a modulation of kinetic energy. \paragraph{Physical Interpretation and Motivation.} The decomposition \[\phi(x,t) = \bar{\phi}(x,t) + \delta\phi(x,t)\]separates spacetime structure into a smooth background component $\bar{\phi}$ and a small fluctuating component $\delta\phi$. The background term represents the classical gravitational potential arising from large-scale geometry, while $\delta\phi$ encodes microscopic spacetime moduli perturbations. These perturbations may originate from quantum geometric effects, discrete spacetime structure, or Planck-scale fluctuations. Physically, this decomposition allows us to isolate how small-scale spacetime irregularities influence quantum dynamics. By writing the Hamiltonian as\[H = H_0 + \delta H,\]we distinguish between ordinary quantum evolution in a smooth geometry ($H_0$) and additional corrections induced by spacetime fluctuations ($\delta H$). The term\[\delta H \approx m\,\delta\phi - \frac{\hbar^2}{m c^2}\delta\phi \nabla^2\]shows that spacetime perturbations modify both the potential energy and the kinetic structure of the particle. \paragraph{Intuitive Picture for a Non-Expert.} A helpful analogy is to imagine a wave traveling across the surface of a calm lake. If the water surface is perfectly smooth, the wave propagates cleanly. However, if the surface contains tiny ripples, the wave becomes distorted and slightly scattered. In this analogy, the smooth lake corresponds to $\bar{\phi}$, while the small ripples correspond to $\delta\phi$. Similarly, a quantum particle is described by a wavefunction. If spacetime itself contains microscopic fluctuations, then the wavefunction does not evolve in a perfectly smooth background. Instead, it experiences tiny distortions in its phase and amplitude. Over time, these small distortions accumulate and can lead to spreading of momentum and position distributions. Thus, the modified Schrödinger equation \[i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\bar{\phi}+ m\delta\phi-\frac{\hbar^2}{m c^2}\delta\phi \nabla^2\right]\psi\] mathematically encodes the idea that spacetime itself actively influences quantum states. Rather than treating uncertainty as purely intrinsic, this framework allows us to explore the possibility that part of quantum indeterminacy arises from dynamical geometric perturbations at very small scales. \subsection{4. Stochastic Model of Spacetime Perturbation} \paragraph{Physical Motivation for the Stochastic Model.} In the previous section, spacetime perturbations $\delta\phi(x,t)$ were introduced as small deviations from a smooth geometric background. To proceed analytically, we model these perturbations as stochastic fluctuations. This assumption does not imply that spacetime is fundamentally random; rather, it reflects our effective description after coarse-graining microscopic geometric degrees of freedom. In approaches such as Loop Quantum Gravity, spacetime geometry is fundamentally discrete at the Planck scale. When averaged over many microscopic quantum geometric states, these discrete structures manifest as effective fluctuations at larger scales. The condition\[\langle \delta\phi(x,t) \rangle = 0\]states that there is no preferred bias in the fluctuations: on average, spacetime remains centered around the smooth background $\bar{\phi}$. The two-point correlation function\[\langle \delta\phi(x,t)\delta\phi(x',t') \rangle = C(x-x')\delta(t-t')\]encodes how strongly spacetime fluctuations at one location are related to those at another. The spatial function $C(x-x')$ determines the correlation length, while the temporal delta function represents rapid (short-time) fluctuations. We introduce the interaction Hamiltonian\[H_{\text{int}}(t)= m \int d^3x\, \delta\phi(x,t) |x\rangle\langle x|\]to capture how spacetime perturbations couple locally to the mass-energy of the particle. The use of a second-order cumulant expansion is justified when fluctuations are weak but rapidly varying; it allows us to derive an effective evolution equation for the ensemble-averaged density matrix $\bar{\rho}$. The resulting term\[\partial_t \bar{\rho}(x,x',t) = -\Gamma(x-x') \bar{\rho}(x,x',t)\]describes decoherence: of

DOI
10.5281/zenodo.18767429

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\begin{abstract} We propose a structural modification of the non-relativistic Schrödinger equation obtained by introducing an additional operator-valued potential term that couples nontrivially to both kinetic and potential contributions. The modified dynamical equation takes the form \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\Phi+ m\delta\Phi- \frac{\hbar^2}{2m},\delta\Phi,\nabla^2\right]\psi,\end{equation} where δΦ\delta\PhiδΦ represents an additional scalar field or structural correction that deforms the standard Hamiltonian operator. Unlike perturbative potential additions, this term modifies the kinetic structure itself, effectively generating a position-dependent renormalization of the Laplacian operator and inducing a nontrivial coupling between propagation and background geometry. We analyze the mathematical consistency of the modified Hamiltonian, deriving conditions for self-adjointness on L2(R3)L^2(\mathbb{R}^3)L2(R3) and establishing criteria under which unitary time evolution and probability conservation are preserved. The deformation leads to a modified dispersion relation and alters the spectral structure of the system, suggesting an interpretation in terms of emergent or dynamically modified configuration-space geometry. Beyond its formal properties, this framework raises foundational questions regarding the ontological status of the kinetic operator in quantum mechanics. By altering the generator of time evolution itself, the proposal challenges the assumption that the Laplacian represents a fixed Euclidean background structure. Instead, quantum evolution may reflect a deeper geometric or structural layer, in which mass, potential energy, and propagation are relational rather than primitive quantities. This formulation provides a mathematically consistent extension of the Schrödinger framework and opens avenues for interpreting quantum dynamics as arising from modified geometric or higher-structural principles. \end{abstract} \noindent\textbf{Keywords:} Modified Schrödinger Equation; Operator Deformation; Quantum Geometry; Self-Adjoint Hamiltonians; Foundations of Quantum Mechanics. \section{Spacetime–Induced Modification of the Schrödinger Equation} \subsection{1. Introduction} In conventional quantum mechanics, uncertainty is treated as a fundamental property of wavefunctions. In this work we explore an alternative interpretation: that quantum uncertainty may emerge from small-scale spacetime moduli perturbations. We begin in standard 4-dimensional spacetime and then extend the formulation to a constrained multi-time geometry. \subsection{2. Quantum Dynamics in Curved Spacetime} \paragraph{Explanation of the Curved-Spacetime Klein–Gordon Equation.} The equation [\left( \Box_g + \frac{m^2 c^2}{\hbar^2} \right)\Phi = 0]describes the dynamics of a scalar quantum field Φ\PhiΦ propagating in a curved spacetime background. Here, □g\Box_g□g​ denotes the covariant d'Alembert operator associated with the metric tensor gμνg_{\mu\nu}gμν​, defined as [\Box_g \Phi = \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g}, g^{\mu\nu} \partial_\nu \Phi \right),] where g=det⁡(gμν)g = \det(g_{\mu\nu})g=det(gμν​) is the determinant of the spacetime metric. The factor −g\sqrt{-g}−g​ ensures covariance under general coordinate transformations and accounts for the local volume element of curved spacetime. Physically, this equation generalizes the flat-spacetime Klein–Gordon equation by replacing ordinary derivatives with covariant derivatives determined by the geometry. The metric tensor gμνg_{\mu\nu}gμν​ encodes gravitational effects, meaning that spacetime curvature directly modifies wave propagation. Thus, particle dynamics are influenced not only by mass mmm but also by geometric structure. When weak metric perturbations are assumed, [g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},\qquad |h_{\mu\nu}| \ll 1,]the geometry deviates slightly from Minkowski space ημν\eta_{\mu\nu}ημν​. In this regime, gravitational effects enter as small corrections to kinetic terms, enabling a perturbative analysis of how spacetime moduli fluctuations influence quantum evolution. We begin from the relativistic Klein–Gordon equation in curved spacetime: \begin{equation}\left( \Box_g + \frac{m^2 c^2}{\hbar^2} \right)\Phi = 0\end{equation} where \begin{equation}\Box_g \Phi = \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g} g^{\mu\nu} \partial_\nu \Phi \right)\end{equation} Assume weak metric perturbation: \begin{equation}g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu},\qquad |h_{\mu\nu}| \ll 1\end{equation} We separate fast rest energy oscillation: \begin{equation}\Phi(x,t) = e^{-i m c^2 t / \hbar} \psi(x,t)\end{equation} Taking the non-relativistic limit yields a modified Schrödinger equation: \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m} \left(1 + \frac{2\phi}{c^2}\right)\nabla^2+ m\phi\right] \psi\end{equation} where ϕ\phiϕ is the weak gravitational potential. \subsection{3. Introducing Spacetime Moduli Perturbations} We decompose: \begin{equation}\phi(x,t) = \bar{\phi}(x,t) + \delta\phi(x,t)\end{equation} where δϕ\delta\phiδϕ represents small-scale spacetime moduli fluctuations. The Hamiltonian becomes: \begin{equation}H = H_0 + \delta H\end{equation} with \begin{equation}H_0 = -\frac{\hbar^2}{2m}\nabla^2 + m\bar{\phi}\end{equation} and \begin{equation}\delta H \approx m,\delta\phi(x,t)-\frac{\hbar^2}{m c^2}\delta\phi(x,t)\nabla^2\end{equation} Thus the modified Schrödinger equation is: \begin{equation}i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\bar{\phi}+ m\delta\phi-\frac{\hbar^2}{m c^2}\delta\phi\nabla^2\right]\psi\end{equation} \textbf{Interpretation:}Spacetime fluctuations enter quantum evolution as both an effective potential term and a modulation of kinetic energy. \paragraph{Physical Interpretation and Motivation.} The decomposition [\phi(x,t) = \bar{\phi}(x,t) + \delta\phi(x,t)]separates spacetime structure into a smooth background component ϕˉ\bar{\phi}ϕˉ​ and a small fluctuating component δϕ\delta\phiδϕ. The background term represents the classical gravitational potential arising from large-scale geometry, while δϕ\delta\phiδϕ encodes microscopic spacetime moduli perturbations. These perturbations may originate from quantum geometric effects, discrete spacetime structure, or Planck-scale fluctuations. Physically, this decomposition allows us to isolate how small-scale spacetime irregularities influence quantum dynamics. By writing the Hamiltonian as[H = H_0 + \delta H,]we distinguish between ordinary quantum evolution in a smooth geometry (H0H_0H0​) and additional corrections induced by spacetime fluctuations (δH\delta HδH). The term[\delta H \approx m,\delta\phi - \frac{\hbar^2}{m c^2}\delta\phi \nabla^2]shows that spacetime perturbations modify both the potential energy and the kinetic structure of the particle. \paragraph{Intuitive Picture for a Non-Expert.} A helpful analogy is to imagine a wave traveling across the surface of a calm lake. If the water surface is perfectly smooth, the wave propagates cleanly. However, if the surface contains tiny ripples, the wave becomes distorted and slightly scattered. In this analogy, the smooth lake corresponds to ϕˉ\bar{\phi}ϕˉ​, while the small ripples correspond to δϕ\delta\phiδϕ. Similarly, a quantum particle is described by a wavefunction. If spacetime itself contains microscopic fluctuations, then the wavefunction does not evolve in a perfectly smooth background. Instead, it experiences tiny distortions in its phase and amplitude. Over time, these small distortions accumulate and can lead to spreading of momentum and position distributions. Thus, the modified Schrödinger equation [i\hbar \partial_t \psi =\left[-\frac{\hbar^2}{2m}\nabla^2+ m\bar{\phi}+ m\delta\phi-\frac{\hbar^2}{m c^2}\delta\phi \nabla^2\right]\psi] mathematically encodes the idea that spacetime itself actively influences quantum states. Rather than treating uncertainty as purely intrinsic, this framework allows us to explore the possibility that part of quantum indeterminacy arises from dynamical geometric perturbations at very small scales. \subsection{4. Stochastic Model of Spacetime Perturbation} \paragraph{Physical Motivation for the Stochastic Model.} In the previous section, spacetime perturbations δϕ(x,t)\delta\phi(x,t)δϕ(x,t) were introduced as small deviations from a smooth geometric background. To proceed analytically, we model these perturbations as stochastic fluctuations. This assumption does not imply that spacetime is fundamentally random; rather, it reflects our effective description after coarse-graining microscopic geometric degrees of freedom. In approaches such as Loop Quantum Gravity, spacetime geometry is fundamentally discrete at the Planck scale. When averaged over many microscopic quantum geometric states, these discrete structures manifest as effective fluctuations at larger scales. The condition[\langle \delta\phi(x,t) \rangle = 0]states that there is no preferred bias in the fluctuations: on average, spacetime remains centered around the smooth background ϕˉ\bar{\phi}ϕˉ​. The two-point correlation function[\langle \delta\phi(x,t)\delta\phi(x',t') \rangle = C(x-x')\delta(t-t')]encodes how strongly spacetime fluctuations at one location are related to those at another. The spatial function C(x−x′)C(x-x')C(x−x′) determines the correlation length, while the temporal delta function represents rapid (short-time) fluctuations. We introduce the interaction Hamiltonian[H_{\text{int}}(t)= m \int d^3x, \delta\phi(x,t) |x\rangle\langle x|]to capture how spacetime perturbations couple locally to the mass-energy of the particle. The use of a second-order cumulant expansion is justified when fluctuations are weak but rapidly varying; it allows us to derive an effective evolution equation for the ensemble-averaged density matrix ρˉ\bar{\rho}ρˉ​. The resulting term[\partial_t \bar{\rho}(x,x',t) = -\Gamma(x-x') \bar{\rho}(x,x',t)]describes decoherence: of
Wave mechanics, geometrical optics, classical Hamilton--Jacobi dynamics, electron optics, and gauge-coupled quantum mechanics are usually introduced as conceptually distinct frameworks. Yet they share a common mathematical structure: an environment modifies a local phase relation, the phase defines a local propagation momentum, and the resulting momentum field determines characteristic curves along which propagation occurs. This paper develops a propagation-first synthesis of these structures for particles with nonzero rest mass. For a nonrelativistic spinless particle with wave functionΨ(r,t)=R(r,t)eiS(r,t)/ℏ,\Psi(\mathbf r,t)=R(\mathbf r,t)e^{iS(\mathbf r,t)/\hbar},Ψ(r,t)=R(r,t)eiS(r,t)/ℏ,the Schrödinger equation in scalar and electromagnetic potentials is exactly equivalent, wherever the amplitude is sufficiently regular, to a continuity equation and a quantum Hamilton--Jacobi equation,∂R2∂t+∇⋅[R2∇S−qAm]=0,\frac{\partial R^2}{\partial t}+\nabla\cdot\left[R^2\frac{\nabla S-q\mathbf A}{m}\right]=0,∂t∂R2​+∇⋅[R2m∇S−qA​]=0,∂S∂t+(∇S−qA)22m+qϕ+U+Q=0,\frac{\partial S}{\partial t}+\frac{(\nabla S-q\mathbf A)^2}{2m}+q\phi+U+Q=0,∂t∂S​+2m(∇S−qA)2​+qϕ+U+Q=0,whereQ=−ℏ22m∇2RR.Q=-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}.Q=−2mℏ2​R∇2R​. The gauge-invariant kinetic propagation momentum is thereforeΠ=∇S−qA,\boldsymbol{\Pi}=\nabla S-q\mathbf A,Π=∇S−qA,and the associated local propagation velocity isv=Πm.\mathbf v=\frac{\boldsymbol{\Pi}}{m}.v=mΠ​. We call this coupled amplitude--phase representation the \textbf{Unified Propagation System}. The name denotes an organizational framework rather than a new fundamental equation. Within this framework, external interactions can be interpreted as modifications of a local phase--dispersion structure. For static scalar potentials, spatial variation of the local propagation momentum defines an effective matter-wave refractive index and reproduces the optical--mechanical analogy familiar from Hamiltonian and electron optics. Electromagnetic vector potentials generalize this picture: they enter directly into the gauge-invariant phase gradient and therefore modify propagation direction in a manner that cannot in general be represented by a scalar refractive index. This motivates a generalized definition of refraction: \textbf{refraction is the bending of propagation characteristics caused by spatial or temporal variation of the local phase--dispersion relation}. In the semiclassical limit this definition reduces directly to geometrical ray bending. Beyond that limit, the full amplitude-dependent term QQQ provides an additional modification of the propagation structure. The same equations can be written in a second-order form,mdvdt=q(E+v×B)−∇U−∇Q,m\frac{d\mathbf v}{dt}=q(\mathbf E+\mathbf v\times\mathbf B)-\nabla U-\nabla Q,mdtdv​=q(E+v×B)−∇U−∇Q,showing that force-based and propagation-based descriptions can represent the same dynamics at different levels. We therefore propose that electromagnetic deflection, potential scattering, and matter-wave refraction may be regarded as different manifestations of a common propagation principle rather than as fundamentally unrelated mechanisms. The paper deliberately avoids a deeper ontological claim. It does not assert that the wave function, electromagnetic field, and particle arise from a common physical substrate. It instead identifies the propagation structure that any such deeper account would have to reproduce. A relativistic extension based on the gauge-covariant Hamilton--Jacobi equation suggests that the same architecture survives beyond the nonrelativistic Schrödinger regime.
This work proposes a structural theory of wave dynamics in which quantum, statistical, and cosmological phenomena emerge from a single underlying principle: whether a wave process achieves phase closure or remains phase-open. Within a discrete conservative framework, wave evolution is shown to admit two qualitatively distinct regimes. Phase-closed processes generate stationary normal modes and localized particle-like states, whereas phase-open processes produce dispersive envelope dynamics and persistent background fluctuations. In this framework, Schrödinger-type evolution is not postulated as a fundamental law but arises universally as the effective continuum description of structurally incomplete (phase-open) wave processes. A central result of this paper is that Heisenberg-type uncertainty relations emerge as a structural consequence of phase openness rather than measurement postulates or operator non-commutativity. Persistent background fluctuations, including vacuum noise and cosmological background phenomena, are likewise interpreted as unavoidable residuals of incomplete phase closure in conservative wave systems. The theory further introduces a probabilistic structural law governing phase closure, predicting that particle-like states are rare events in a dominantly phase-open universe. This leads to falsifiable scaling relations connecting microscopic phase dynamics to macroscopic cosmological matter fractions. The framework is integrated with the JS–SH discrete geometry and the SRCD ratio coordinate, providing a unified structural interpretation of quantum mechanics, particle formation, and cosmic background structure. Unlike conventional approaches, this work does not modify quantum mechanics by additional axioms or stochastic assumptions. Instead, it derives quantum and cosmological phenomena from structural properties of discrete wave dynamics, offering a unified conceptual and mathematical foundation linking Schrödinger evolution, uncertainty, and cosmic background fluctuations.
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.
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.
Dynamical Localized Wave Packet Theory (DLWPT) is a theoretical framework that extends conventional quantum mechanics by interpreting the wavefunction as a physically real coherence field rather than a purely probabilistic object. In this approach, quantum particles correspond to dynamically stable, localized wave packets whose evolution is governed by modified dynamical equations incorporating a fundamental coherence scale. In its relativistic formulation, DLWPT modifies the standard Dirac dynamics by introducing a nonlocal coherence operator. This operator captures the finite spacetime coherence region associated with localized fermionic wave packets. As a result, the theory naturally describes fermions as extended but localized structures with Planck-scale coherence tails, rather than strictly pointlike particles. The DLWPT framework provides a deterministic description of wave packet evolution while maintaining compatibility with relativistic covariance. At energies far below the Planck scale, the modified dynamics reduce smoothly to the standard Dirac equation, ensuring consistency with established quantum field theory. However, at ultra-high energies the coherence operator introduces Planck-suppressed corrections to the dispersion relation and particle propagation. Several conceptual implications follow from this formulation. The nonlocal structure of the coherence operator provides a natural interpretation of quantum non-separability through overlapping coherence regions. Localized fermions arise as stable eigenmodes of the coherence operator, and the theory suggests that internal orientation degrees of freedom of these coherence packets may underlie the emergence of gauge symmetries. Within this framework, gravity can be interpreted as an emergent phenomenon associated with gradients in the coherence energy density of localized wave packets. The Planck scale therefore appears not merely as a parameter of quantum gravity but as the fundamental coherence scale governing the structure of matter fields. DLWPT thus proposes a unified conceptual picture in which particles are self-coherent spacetime wave packets, interactions arise from coherence structures and their orientation modes, and spacetime geometry reflects the collective behavior of coherence energy at the Planck scale. The present work develops the relativistic fermionic sector of the theory, deriving the modified Dirac equation, associated dispersion relations, and propagator structure. Potential observational signatures include Planck-suppressed deviations in high-energy particle propagation and coherence-induced modifications to scattering processes.
Abstract This work introduces a covariant reformulation of quantum dynamics in which the laboratory time ttt is replaced by a geometric parameter τ\tauτ, interpreted as a potential-time variable along a phase flow. The standard Schrödinger equation,iℏ ∂tψ=H^ψi\hbar\,\partial_t \psi = \hat{H}\psiiℏ∂tψ=H^ψ,is generalized toiℏ DψDτ=H^τψi\hbar\,\frac{D\psi}{D\tau} = \hat{H}_\tau \psiiℏDτDψ=H^τψ,withDDτ:=uμ(∇μ+iΩμ)\frac{D}{D\tau} := u^\mu(\nabla_\mu + i\Omega_\mu)DτD:=uμ(∇μ+iΩμ),where uμu^\muuμ is the unit phase flow and Ωμ\Omega_\muΩμ the time connection, whose curvatureKμν=∇μΩν−∇νΩμK_{\mu\nu} = \nabla_\mu \Omega_\nu - \nabla_\nu \Omega_\muKμν=∇μΩν−∇νΩμdefines a measurable time curvature. The resulting framework embeds Schrödinger evolution within a geometric covariant structure that remains unitary, allows a natural definition of a time operator or covariant time POVM, and recovers the standard form in the flat limit (Ωμ→0\Omega_\mu \to 0Ωμ→0, R(3)→0R^{(3)} \to 0R(3)→0). Probability conservation, phase transport, and curvature coupling are derived explicitly, showing that amplitude focusing obeysRμνuμuν≈2 D2ln⁡A/Dτ2+(2/ℏ) DΩτ/DτR_{\mu\nu}u^\mu u^\nu \approx 2\,D^2\ln A/D\tau^2 + (2/\hbar)\,D\Omega_\tau/D\tauRμνuμuν≈2D2lnA/Dτ2+(2/ℏ)DΩτ/Dτ.This approach unifies time, geometry, and quantum evolution in a single covariant dynamical law.