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International Journal of Quantum Technologies

Emergent Relativistic Quantum Wave Equation, Dynamics, and Topological Structures of Bosons and Fermions in 1+1D from a Flip-Flop Dual-Component Model

Jau Tang

Published Feb 25, 2026• 0 citations
Publisher

Abstract

We present a physically intuitive and mathematically rigorous framework for deriving the relativistic quantum wave equations, dynamics, and topological structures of bosons and fermions in 1+1-dimensional spacetime. Starting from a simple flip-flop dual-component system that models internal oscillations, we show how first-order linear rate equations naturally give rise to the Klein-Gordon and Dirac equations. For bosons, the system leads to the familiar Klein-Gordon equation, while for spin-½ fermions—augmented by an internal clock degree of freedom—the Dirac equation emerges in 1+1D. The topological distinction between bosons and fermions is revealed through their rotational symmetry: bosons follow a 360° closed loop structure, while fermions are represented by a Möbius band, requiring a 720° rotation to return to their original state. We also introduce two distinct Lorentz transformation structures: hyperbolic (sinh–cosh) for bosons and trigonometric (sin–cos) for fermions. This approach provides a clear, unified, and pedagogical interpretation of relativistic quantum dynamics and internal particle structure.

DOI
10.66311/3068-9090.02.01.14

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The Dirac equation stands as the cornerstone of relativistic quantum mechanics, successfully predicting spin and antimatter. However, its derivation is typically presented as a formal algebraic requirement—the linearization of the Hamiltonian—without a clear physical ontology explaining why nature demands such a structure. In this work, building upon a previously established model of particles as topologically anchored wave-defects in a continuous medium, we demonstrate that the Dirac equation is not merely an algebraic curiosity but a structural necessity. We show that the requirement to describe the causal, first-order evolution of a relativistic standing wave compels the state variable to possess a multi-component spinor structure. We explicitly derive the Clifford algebra of the dynamical operators and identify the emergent 4-component state space with the physical degrees of freedom of the defect: a local vibrational duality (Spin) coupled to a global topological orientability (Charge). Thus, the ”quantum strangeness” of the fermion—its spin-1/2 nature and antiparticle partner—are demystified, emerging as the natural description of a relativistic object with non-trivial topology.
Abstract This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.
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