Assuming that “quantum spacetime” is fundamentally discrete, how might this discreteness show itself? Some of its potential effects are more evident, others less so. The atomic and molecular structure of ordinary matter influences the propagation of both waves and particles in a material medium. Classically, particles can be deflected by collisions and also retarded in their motion, giving rise in particular to viscosity and Brownian motion. In the case of spatio-temporal discreteness, viscosity is excluded by Lorentz symmetry, but fluctuating deviations from rectilinear motion are still possible. Such “swerves” have been described in and. They depend (for a massive particle) on a single phenomenological parameter, essentially a diffusion constant in velocity space. As far as I know, the corresponding analysis for a quantal particle with mass has not been carried out yet, but for massless quanta such as photons the diffusion equation of can be adapted to say something, and it then describes fluctuations of both energy and polarization (but not of direction), as well as a secular “reddening” (or its opposite). A more complete quantal story, however, would require that particles be treated as wave packets, raising the general question of how spatiotemporal discreteness affects the propagation of waves . Here, the analogy with a material medium suggests effects such as scattering and extinction, as well as possible nonlinear effects. Further generalization to a “second-quantized field” might have more dramatic, if less obvious, consequences.
Assuming that “quantum spacetime” is fundamentally discrete, how might this discreteness show itself? Some of its potential effects are more evident, others less so. The atomic and molecular structure of ordinary matter influences the propagation of both waves and particles in a material medium. Classically, particles can be deflected by collisions and also retarded in their motion, giving rise in particular to viscosity and Brownian motion. In the case of spatio-temporal discreteness, viscosity is excluded by Lorentz symmetry, but fluctuating deviations from rectilinear motion are still possible. Such “swerves” have been described in and. They depend (for a massive particle) on a single phenomenological parameter, essentially a diffusion constant in velocity space. As far as I know, the corresponding analysis for a quantal particle with mass has not been carried out yet, but for massless quanta such as photons the diffusion equation of can be adapted to say something, and it then describes fluctuations of both energy and polarization (but not of direction), as well as a secular “reddening” (or its opposite). A more complete quantal story, however, would require that particles be treated as wave packets, raising the general question of how spatiotemporal discreteness affects the propagation of waves . Here, the analogy with a material medium suggests effects such as scattering and extinction, as well as possible nonlinear effects. Further generalization to a “second-quantized field” might have more dramatic, if less obvious, consequences.
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.
DESCRIPTION This work presents a complete, deterministic framework for quantum mechanics derived entirely from a single, testable postulate: the density of quantum space. The Core Problem For over a century, quantum mechanics has been interpreted as a theory of fundamental randomness, wave-particle duality, instantaneous jumps, and irreducible uncertainty. The standard Copenhagen interpretation treats the wave function as a probability amplitude, measurement as a "collapse" of that probability, and the Heisenberg uncertainty principle as a fundamental limit of nature. This work demonstrates that all these "mysteries" are not properties of nature—they are artifacts of our limited perception. We observe quantum phenomena through slow, macroscopic instruments that cannot resolve the actual continuous dynamics occurring in a space that is 6.54 trillion times denser than what we perceive as "empty space." The Fundamental Postulate The foundation of this work is a single, geometrically derived constant: ρ = (number of Planck lengths in 1 mm) / (number of millimeters in 1 light-year) = 6.54 × 10¹² Physical Meaning: One millimeter of physical space contains 6.54 trillion times more quantum spatial units than one light-year contains millimeters. What we perceive as "empty space" is actually a dense, discrete, information-rich medium. The entire observable universe (93 billion light-years in diameter) maps to only 0.014 light-years in quantum space—well within our Solar System. Consequence: The density inversion explains why quantum phenomena appear strange: we are observing a dense, fast, discrete reality through slow, coarse, macroscopic instruments. The apparent "weirdness" is not in nature—it is in our perception. The Comprehensive Law From the density postulate and classical physics (Newton, Coulomb, Maxwell, Einstein), a single comprehensive law is derived that contains all known physics as limit cases: E(t,v,n) = (m₀c²/ρ) · ‖n‖ · √(1 + (γv/c)²) · sin(2πt/T) This formula contains: Einstein's E = mc² as the limit ρ → 1, ‖n‖ → 1, sin → 1, v ≪ c Newtonian mechanics as the limit v ≪ c Special relativity as the limit ρ → 1, ‖n‖ → 1, sin → 1 Quantum mechanics as the discrete geometry n ∈ ℤ³ and resonance sin(2πt/T) Each term has a precise physical meaning: the rest mass energy (m₀c²) is scaled by the quantum space density (1/ρ), modified by the position relative to the absolute zero point Z₀ (‖n‖), adjusted for relativistic motion (√(1 + (γv/c)²)), and modulated by temporal resonance (sin(2πt/T)). Resolution of All Quantum Paradoxes Wave-Particle Duality = Optical Illusion The electron is always a localized particle with a definite trajectory. The "wave" is the charge density distribution—a standing wave in the dense quantum medium. From the standing wave condition 2πr = nλ, the Poynting vector in the far field is exactly zero (S = 0), meaning the electron does not radiate. The apparent "wave behavior" is an optical illusion caused by the density ρ smearing the point-like trajectory in our low-resolution projection. Wave Function = Energy Density, Not Probability The squared modulus of the wave function |Ψ|² is the energy density fraction, not a probability density. The particle itself follows a deterministic trajectory y(t) = v₀t - ½g₀t². Probability appears only because we cannot resolve the trajectory (due to t_total = 118 as ≪ Δt_instr) and we do not know the initial phase φ of the oscillation. Schrödinger Equation = Effective Approximation The Schrödinger equation is an effective, averaged description of how the energy density evolves. The fundamental law is classical motion in the dense quantum medium. The Schrödinger equation emerges when we average over the fast oscillation sin(2πt/T), ignore the density force F_ρ, and treat trajectories statistically. Quantization = Resonance, Not Arbitrary Jumps Energy levels are the resonant harmonics of the quantum space medium. The standing wave condition 2πr = nλ gives E_n = -13.6 eV/n²—exactly the same physics as a vibrating string. Only frequencies that satisfy the standing wave condition are stable. The transition between levels is not an instantaneous "jump" but a continuous trajectory y(t) = v₀t - ½g₀t² with total duration t_total = 118 as. The apparent "instantaneous jump" is an artifact of our slow instruments. Heisenberg Uncertainty = Collision, Not Fundamental Limit The uncertainty product depends on the measurement geometry: Δx·Δp = h·tan(θ/2), where θ is the scattering angle of the measurement photon. As θ → 0 (forward scattering with perfect detection), the product approaches zero. There is no fundamental lower bound. The uncertainty is a practical limitation of invasive measurement—we disturb the particle's trajectory when we hit it with a photon. Quantum Randomness = Ignorance Residual "quantum randomness" is simply unmodeled environmental force noise. The apparent randomness arises from three sources: temporal resolution (t_total = 118 as ≪ Δt_instr—we cannot resolve the trajectory), initial phase ignorance (we do not know the phase φ of sin(2πt/T)), and measurement perturbation (we do not know the exact collision parameters). If we could measure the outgoing photon completely and know the initial phase, the outcome would be fully deterministic. Randomness is in our ignorance, not in nature. Five Testable Predictions The theory makes five clear, quantitative predictions that distinguish it from standard quantum mechanics: Prediction 1 - Electron Transit Time: The electron transit time during the n=1 → n=2 transition in hydrogen is 118 ± 5 attoseconds. Standard quantum mechanics predicts zero duration (instantaneous jump). Testable with attosecond pump-probe spectroscopy. Prediction 2 - Heisenberg Product Can Approach Zero: With better detectors, the uncertainty product Δx·Δp can approach zero, not ħ/2. Standard quantum mechanics predicts Δx·Δp ≥ ħ/2 always. Testable with weak measurement experiments. Prediction 3 - Energy Oscillation at 116 kHz: The energy of gold-197 oscillates at 116 kHz with amplitude 1.252 × 10⁻²⁶ J. Standard quantum mechanics predicts constant energy for a given state. Testable with precision spectroscopy. Prediction 4 - Bell Inequality Violations Decrease with Measurement Strength: As measurement impulse strength decreases, Bell inequality violations decrease. Standard quantum mechanics predicts violations are independent of measurement strength. Testable with weak measurement entanglement experiments. Prediction 5 - Quantum Randomness is Environmental Noise: Residual "quantum randomness" equals unmodeled environmental force noise. Full force accounting restores classical determinism. Standard quantum mechanics predicts randomness is fundamental and irreducible. Testable with complete force reconstruction in closed quantum systems. The Absolute Zero Point Z₀ This work introduces Z₀ = (0,0,0) as a universal, absolute reference point in quantum space. Every point in space has a discrete address: n = (n_r, n_θ, n_φ), ‖n‖ = √(n_r² + n_θ² + n_φ²) Z₀ is physically grounded in the cosmic microwave background (CMB) rest frame, vacuum energy (the lowest possible energy state), and the geometric center of the quantum space lattice. This eliminates the relativity of reference frames—providing an absolute coordinate system that standard physics lacks. Mathematical Consistency The work provides rigorous derivations of: The density ρ from first principles (Planck length, electron mass, fine structure constant) The quantum coordinate norm ‖n‖ from the discrete metric of quantum space The oscillation period T from the de Broglie relation The coupling constant ε from fundamental constants The regularized metric eliminating all singularities The thermodynamic properties of quantum space (entropy, temperature, heat capacity) The covariant Einstein equation in quantum space The connection to quantum anomalies and topological invariants The Lamb shift from quantum space density oscillation Dark energy from quantum space density variation Comparison with Other Approaches String theory requires approximately 10⁵⁰⁰ free parameters and offers no testable predictions with current technology, while introducing new physics such as strings, extra dimensions, and supersymmetry. Loop quantum gravity requires approximately 10 parameters and also offers no testable predictions, introducing spin networks and loop quantization. Both approaches have difficulty connecting to the Standard Model. In contrast, this theory has only one parameter (ρ, which is derived from fundamental constants rather than being free), offers five testable predictions with current or near-future technology, requires no new physics beyond standard physics applied in a dense medium, and connects directly to the Standard Model through the energy-momentum tensor. The Core Message Quantum mechanics is not magic. It is not a new kind of physics. It is the same old physics—Newton, Maxwell, Coulomb—operating in a space that is 6.54 trillion times denser than we perceive. Three things to remember: Space is dense, not empty. What we call "empty space" is actually a dense, discrete medium. We are the ones who are sparse—not the universe. The particle has a trajectory. Every particle follows a precise, continuous path. We don't see it because our instruments are too slow. The path exists—we just can't resolve it. Energy breathes. Energy is not a static number. It oscillates in time, like a heartbeat. The universe is alive with resonance. Conclusion This work demonstrates that quantum mechanics is not a theory of fundamental randomness and irreducible uncertainty. It is the low-resolution, static-space approximation of a deeper, deterministic, dynamic reality. The deeper reality is quantum space—dense, discrete, resonant, and absolute. The final statement of this
Newtonian mechanics focuses on dx and dt units which tend to zero and so presents, it seems, a smoothed over or averaged type of motion. This motion in principle extends to dx and dt values which are almost zero. There is, however, a priori no reason not to expect that such a smooth averaging is at some length scale based on another kind of motion. The question is: What should that “other” kind of motion be? If it is deterministic in x and t, one may reapply the idea of this motion breaking down at yet some smaller scale. Thus it seems one should consider decoupling of x and t because one cannot follow a particle from one x point to another in time within some small length scale as this would be deterministic. The opposite approach seems to be a probabilistic one with t and x decoupled. The external ruler and clock which exist in the Newtonian picture, however, do not disappear simply because a length scale is small. Thus having x positions based on probabilities at some length scale still uses the notion of the probability itself being smoothed over because it contains x which may take on any value. This is not necessarily a problem because the probabilistic behaviour must still appear to be Newtonian on average and so x and t based on the external clock/ruler with infinite resolution need to “appear” again. (For example, an electron has a wavelength and exhibits interference from two slits, but while moving towards the slits, distance = velocity* time with distance and time measured by an external clock. It is a mathematical generalization, however, to expect that d=vt actually fully describes motion (i.e. other than on average) within a wavelength length. If it did there would be no interference pattern. The idea of a smoothed over (or averaged motion) based on probabilistic motion also appears in the classical world, but for many particles. For example, a Maxwell-Boltzmann gas with no potential has uniform density. If one considers a hypothetical box containing many molecules, it is like a Newtonian stationary object with a balance of pressure. In the case of a potential (1-D) one may consider a dx unit (with many molecules) and balance Force(x) density(x) with the difference of pressure to find the density distribution proportional to exp(- V(x)/T). These ideas are completely Newtonian, but they are smoothed over average notions. At a microscopic level molecules undergo probabilistic collisions linked to T (temperature). Thus the microscopic picture is not static - it is only the equilibrium average or big picture which is static. Even in the case of no potential, focusing on a tiny region of space smaller than the interparticle average separation and arguing that density is constant seems to be an average argument which bypasses much of the reality. If x and t may then become probabilistic and -Et+px is a Lorentz invariant, then why should there not be stochasticity linked with E and p? In fact we argue they are == as shown by Maxwell’s solution of a photon from his electromagnetic equations. E and p (energy and momentum) are average values. Their density is given by: .5C1( Electric field dot Electric field + C2 Magnetic field dot Magnetic field) and the Poynting vector for momentum. Both the Electric field and Magnetic field fluctuate as cos(-Et+px) with E and p being average values. At first this was ascribed to wave motion in a medium, but in 1905 Einstein’s photoelectric arguments against energy being spread out seem to suggest a breakdown of deterministic physics describing a wave in a medium. Instead a photon is probabilistic and has a wavefunction of Electric + i Magnetic field. (2)
Understand the intrinsic physical meaning of quantum formalism leads ontologically to admit that one fundamental side of reality is purely contradictory, irreducibly random, objectively indeterminate and intrinsically independent of relativistic space-time. It means also that physical reality has another fundamental side, i.e. space-time, and that both are irreducible to each other. Quantum reality, discretized by the Planck constant, assume coherent states and partially ordered variables (that are commutative or not), while on the other side, relativistic space-time, structured by the speed of light, assume coherent variables and partially ordered states (that are causally related or not). From this point of view, fields would be only a derived mixture of these two sides. But quantum reality and relativistic space-time seem to be mixed in two kinds of fields themselves irreducible to each other. If, on one hand, quantum field theory has already formalized the way that quantum reality locates, evolves and interacts on a relativistic background, on the other hand, quantum gravity tends to formalize the way that smooth and causally ordered space-time emerges on a quantum background. But no unification of the two fundamental sides of reality must be waited at this derived level. Unification could perhaps intervene only upstream, in a pure variation preceding and underlying its double distinction in variables and states of variation, i.e. its quantization or its localization and their derived mixtures.
\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 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) 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 (□g+ℏ2m2c2)Φ=0describes the dynamics of a scalar quantum field Φ propagating in a curved spacetime background. Here, □g denotes the covariant d'Alembert operator associated with the metric tensor gμν, defined as □gΦ=−g1∂μ(−ggμν∂νΦ), where g=det(gμν) is the determinant of the spacetime metric. The factor −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 gμν 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, gμν=ημν+hμν,∣hμν∣≪1,the geometry deviates slightly from Minkowski space 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 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 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 ϕ(x,t)=ϕˉ(x,t)+δϕ(x,t)separates spacetime structure into a smooth background component ϕˉ and a small fluctuating component δϕ. The background term represents the classical gravitational potential arising from large-scale geometry, while δϕ 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 asH=H0+δH,we distinguish between ordinary quantum evolution in a smooth geometry (H0) and additional corrections induced by spacetime fluctuations (δH). The termδH≈mδϕ−mc2ℏ2δϕ∇2shows 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, while the small ripples correspond to δϕ. 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 iℏ∂tψ=[−2mℏ2∇2+mϕˉ+mδϕ−mc2ℏ2δϕ∇2]ψ 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) 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⟨δϕ(x,t)⟩=0states that there is no preferred bias in the fluctuations: on average, spacetime remains centered around the smooth background ϕˉ. The two-point correlation function⟨δϕ(x,t)δϕ(x′,t′)⟩=C(x−x′)δ(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 HamiltonianHint(t)=m∫d3xδϕ(x,t)∣x⟩⟨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. The resulting term∂tρˉ(x,x′,t)=−Γ(x−x′)ρˉ(x,x′,t)describes decoherence: of
One of the most intriguing quantum phenomena of light is the wave–particle duality which is related to the inability of the classical wave and particle concepts to fully describe the behavior of quantum objects. Theoretically, the wave features of the electromagnetic field, such as interference, follow from the solution of Maxwell's equations while particle features are a consequence of the field quantization, in which the electromagnetic field is described to consist of discrete energy packets, photons. Due to the wave–particle duality, for example, simultaneous description of optical absorption, emission, and interference of propagating fields has conventionally not been considered feasible using local optical models but has required using Maxwell's equations with stochastic source terms accounting for the wave–particle duality. Also the description of the momentum of light in a medium has been problematic.
In the research described in this thesis, the quantized fluctuational electrodynamics (QFED) model is derived based on combining the electromagnetic Green's functions to the field quantization to describe the position dependence of the photon number and local thermal balance in general non-equilibrium conditions, and to separate the electromagnetic field into parts propagating in different directions in resonant structures, which has conventionally been problematic due to interference effects. The QFED method is shown to resolve the previously found anomalies in the canonical commutation relations of photon ladder operators. The QFED method is also used to derive quantum optical field–matter interaction parameters fully capturing the interference, emission, damping, and scattering of propagating photons in stratified media leading to an interference-exact radiative transfer equation (RTE) model. Therefore, the derived interaction parameters solve the problem of simultaneously describing interference and losses in the widely used RTE framework.
The particle aspects of light are also studied by directly applying the conservation laws of nature and the special theory of relativity to show that light propagating in a medium must be described as mass-polariton (MP) quasiparticles, covariant coupled states of the field and matter. These quasiparticles are shown to have momentum of the Minkowski form and a nonzero mass that is carried by a mass density wave associated with light. The field–matter interaction related to the MP quasiparticles also leads to the dissipation of photon energy. These particle model results are also derived using a wave-based optoelastic continuum dynamics (OCD) approach following from the electrodynamics of continuous media and the continuum mechanics. The obtained results verify the full agreement between the wave and particle models when they are correctly applied. The key finding that a light pulse propagating in a medium is inevitably associated with an experimentally measurable mass implies a fundamental change in the prevailing perceptions of light.; Eras kiehtovimmista valon…