Stochastic dynamics of a relativistic particle in a wave-packet field

Georgin Moiseevich Zaslavskiî

Eric Aboagye Amankwah

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.

Khavtgain Namsrai

Slobodan Danko Bosanac

Classical and quantum dynamics of wave packets for spin- 1/2 particles are analysed in the presence of an electromagnetic field. An almost exact agreement is found between these two approaches, both in the strong and the weak coupling limits. The question of negative energy states is discussed. and it is shown that they are not produced in such an interaction. The divergence problem of the perturbation expansion of the particles wavefunction is discussed, the source of which is explained.

Eduard Prugovečki

Eliade N. Stefanescu

We consider a quantum particle as a wave packet in the coordinate space. When the conjugate wave packet in the momentum space is considered, we find that the group velocities of these two wave packets, which describe the particle dynamics, are in agreement with the Hamilton equations only if in the time dependent phases one considers the Lagrangian instead of the Hamiltonian which leads to the conventional Schrodinger equation. We define a relativistic quantum principle asserting that a quantum particle has a finite frequency spectrum, with a cutoff propagation velocity c as a universal constant not depending on the coordinate system, and that any time dependent phase variation is the same in any system of coordinates. From the time dependent phase invariance, the relativistic kinematics is obtained. We consider two types of possible interactions: 1) An interaction with an external field, by a modification of the time dependent phase differential with the terms proportional to the differentials of the space-time coordinates multiplied with the components of this field four-potential, and 2) an interaction by a deformation of the space-time coordinates, due to a gravitational field. From the invariance of the time dependent phase with field components, we obtain a mechanical force of the form of Lorentz’s force, and three Maxwell equations: The Gauss-Maxwell equations for the electric and magnetic fluxes, and the Faraday-Maxwell equation for the electromagnetic induction. When the fourth equation, Ampere-Maxwell, is considered, the interaction field takes the form of the electromagnetic field. For a low propagation velocity of the particle waves, we get a packet of waves with the time dependent phases proportional to the relativistic Hamiltonian, as in Dirac’s famous theory of spin, and a slowly-varying amplitude with a phase proportional to the momentum and this velocity. In the framework of our theory, the spin is obtained as an all quantum effect, without any additional assumption to the quantum theory. When a space-time deformation is considered in the time dependent phase of a quantum particle, from the group velocity we get the particle dynamics according to the general theory of relativity. In this way, the relativistic dynamics, the electromagnetic field, and the spin of a quantum particle are obtained only from the invariance of the time dependent phases of the particle wave functions.