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Physica Scripta

Preparation-resolved spectral decomposition of localized free Dirac wave packets

Jaseer Ahmed, S B Faruque

Published Oct 21, 2026• 0 citations

Abstract

Abstract We investigate the effect of spinor preparation of a localized one-dimensional Gaussian Dirac packet on its positive and negative energy spectral content and subsequent free evolution. The normalized constant spinor is varied and projected onto the eigenspaces of the free Dirac Hamiltonian keeping the scalar Gaussian envelope fixed. This procedure leads to a compact geometric expression for the integrated negative energy fraction in terms of a Gaussian averaged Dirac direction. The result identifies the constant spinor with the smallest negative energy content. Yet, for a packet with finite momentum width, it remains distinct from a completely positive energy packet, which requires a momentum dependent spinor. The spectral decomposition is then related to branch separated density evolution, coherent current interference, overlap decay, and leading dispersive broadening. These analytical results are then tested against the direct numerical quadrature of the exact spectral representation, retaining the full momentum dependence of the dispersion relation and the eigenspinors. The study, in general, reveals two qualitatively distinct effects. First, the density separates because the two energy branches move in opposite directions. Secondly, the zitterbewegung-like current arises from coherent interference between the branches through the mixed current matrix element. Overall, the formulation provides a concise preparation-resolved reference for more general Dirac wave-packet problems.

DOI: 10.1088/1402-4896/aeb759

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Abstract We investigate the effect of spinor preparation of a localized one-dimensional Gaussian Dirac packet on its positive and negative energy spectral content and subsequent free evolution. The normalized constant spinor is varied and projected onto the eigenspaces of the free Dirac Hamiltonian keeping the scalar Gaussian envelope fixed. This procedure leads to a compact geometric expression for the integrated negative energy fraction in terms of a Gaussian averaged Dirac direction. The result identifies the constant spinor with the smallest negative energy content. Yet, for a packet with finite momentum width, it remains distinct from a completely positive energy packet, which requires a momentum dependent spinor. The spectral decomposition is then related to branch separated density evolution, coherent current interference, overlap decay, and leading dispersive broadening. These analytical results are then tested against the direct numerical quadrature of the exact spectral representation, retaining the full momentum dependence of the dispersion relation and the eigenspinors. The study, in general, reveals two qualitatively distinct effects. First, the density separates because the two energy branches move in opposite directions. Secondly, the zitterbewegung-like current arises from coherent interference between the branches through the mixed current matrix element. Overall, the formulation provides a concise preparation-resolved reference for more general Dirac wave-packet problems.

Zitterbewegung (ZBW), arising from the interference between positive- and negative-energy solutions of the Dirac equation, is a characteristic feature of relativistic single-particle dynamics, but its explicit manifestation depends on the choice of position operator. We formulate relativistic vortex electron wave packets as coherent superpositions of Dirac plane-wave states and derive their exact spacetime evolution. For the canonical Dirac position, we show that orbital angular momentum substantially enhances and spatially structures the relative-coordinate contribution conventionally associated with ZBW. We then analyze the operator dependence of this result by separating the canonical position into a Foldy–Wouthuysen/Newton–Wigner (FW/NW) mean position and an oscillatory relative coordinate. For the vortex packets considered here, the FW/NW mean position exhibits only the ordinary centroid drift and no transverse trembling. However, the same vortex-enhanced radial scale survives in the positive-energy sector as a Berry-type geometric dressing of the projected position operator. The present work therefore identifies not a representation-independent trembling of the one-particle centroid, but a vortex-amplified remnant of Dirac interband mixing in an operationally meaningful position observable.

Abstract The Pryce (e) spin and position operators of the quantum theory of Dirac's free field were re-defined and studied recently with the help of a new spin symmetry and suitable spectral representations [Eur. Phys. J. C 82, 1073 (2022)]. This approach is generalized here, associating a pair of integral operators acting directly on particle and antiparticle wave spinors in momentum representation to any integral operator in configuration representation, acting on mode spinors. This framework allows an effective quantization procedure, giving a large set of one-particle operators with physical meaning as the spin and orbital parts of the isometry generators, the Pauli-Lubanski and position operators, or other spin-type operators proposed to date. Special attention is paid to the operators that mix the particle and antiparticle sectors whose off-diagonal associated operators have oscillating terms producing Zitterbevegung. The principal operators of this type, including the usual coordinate operator, are derived here for the first time. As an application, it is shown that an apparatus measuring these new observables may prepare and detect one-particle wave packets moving uniformly without Zitterbewegung or spin dynamics, spreading in time normally as any other relativistic or even non-relativistic wave packet.

Spectral constraints can alter the observable manifestation of quantum geometry without changing the local eigenspace. Relativistic electron vortices provide a sharp example: the spin orbit interaction redistributes spin and orbital angular momentum, yet its magnetic contribution differs between localized three-dimensional wave packets and stationary monochromatic beams. We resolve this discrepancy by showing that fixing the energy replaces ambient momentum derivatives in position-weighted current moments by shell-tangent derivatives and derive a Hermitian selection rule for first-order observables in momentum-dependent internal subspaces. Its extension-independent form applies beyond Dirac systems, including a broad class of antisymmetric first moments of scalar-charge velocity currents. For the free positive-energy Dirac doublet, the magnetic correction is an exact non-Abelian Berry-curvature contraction with a pointwise saturable quantum-metric bound. Changing the relative phase between the two

σ_z

spinor components switches this correction from zero in a conserved transverse-helicity sector to a finite value in a radial coherent superposition. At fixed finite Lorentz factor, the radial response scales as

O(θ/γ)

, compared with

O(θ^2)

for longitudinal polarization. Spectral reduction and internal coherence thus provide distinct controls of the observable imprint of quantum geometry.

The free particle solutions of the relativistic Dirac equation are characterized by plane waves with infinite uncertainty in position. However, many practical implementations of the solutions require a wave packet configuration, which can be utilized to represent a localized Dirac particle. Unlike the traditional wave packet generation method by superposing multiple plane waves, this study of ours presents an alternative approach towards obtaining a wave packet solution of a free particle relativistic Dirac equation. In this paper, we present Dirac’s free particle equation with a modification in the generalized momentum. The modification is achieved by coupling the momentum with a spatially varying logarithmic function, and this alteration does not affect the relativistic dispersion relation of the particle. Moreover, a solution of this modified Dirac equation is provided as well, which has been calculated using a trial wave function. The wave function solution is carried out in one dimension, where it behaves as a wave packet for a given ratio of the envelope parameter to the reduced Planck's constant greater than unity, where the envelope parameter regulates the width of the wave packet. The solution, being subject to this constraint, represents a bound particle with spin and a continuous energy spectrum.

Abstract Localized spinor solutions of the zero-mass free-space Dirac equation, which are eigenstates of the angular momentum component along the direction of propagation, satisfy the three-dimensional wave equation. General expressions for and special compact forms of space-time localized spinors are given. An exact evaluation of the norm, energy, momentum, and angular momentum of such wavepackets is made. These are all finite. The integrals of the densities over all space, at fixed time, are shown to be equivalent to integrals over the wavenumber weight function. The localized wavepackets converge onto or diverge from a focal region, and their net momentum is smaller than their energy divided by the speed of light. They can be transformed to a (net) zero-momentum frame.