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arXiv:2605.08248

Generalized Catability of Relativistic Quantum States Measurement in a Unified Lie-Algebraic Foldy-Wouthuysen (FW) Framework

Abdelmalek Bouzenada, Rami Ahmad El-Nabulsi

quant-phhep-thSubmitted May 7, 2026• Announced May 8, 2026• 0 citations• Revised Jun 6, 2026 (v2)
v1 · May 7, 2026v2 · Jun 6, 2026
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Abstract

In this work, a unified Lie-algebraic formulation of catability is constructed for relativistic quantum systems with arbitrary spin within this framework. In this case, the analysis starts with constructing catability as a quantitative measure for superposed coherent states, where coherence structure and quantum interference properties are studied using algebraic representations in this framework. Also, a generalized Foldy-Wouthuysen transformation is formulated within a Lie algebraic framework, delivering a systematic procedure for block-diagonalization of relativistic Hamiltonians and separation of positive- and negative-energy components in this framework. Within this formalism, a phase-sensitive catability operator is introduced to study phase correlations and coherence effects in the relativistic quantum dynamics framework. The approach is applied to Dirac spin-$1/2$ particles, where relativistic fermionic catability is analyzed in relation to spinorial structures and symmetry generators framework. The formalism is extended through a unified geometric and Lie-algebraic treatment, establishing a consistent description of catability in a relativistic quantum mechanics framework. In this context, the generalized framework is constructed for arbitrary spin-$s$ fields, enabling investigation of higher-spin relativistic quantum states within the same algebraic structure framework. In this context, the obtained results show a generalized theoretical platform for investigating relativistic quantum coherence, superposition effects, and algebraic symmetries in the framework of fermionic and bosonic systems.

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Testing Catability and Coherent Superposition of 2D2\mathcal{D}2D Graphene Quantum system
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The problem of the position and spin in relativistic quantum mechanics is analyzed in detail. It is definitively shown that the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one) are quantum-mechanical counterparts of the classical position and spin variables. The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions. The relativistic spin operators are discussed. The spin-orbit interaction does not exist for a free particle if the conventional operators of the orbital angular momentum and the rest-frame spin are used. Alternative definitions of the orbital angular momentum and the spin are based on noncommutative geometry, do not satisfy standard commutation relations, and can allow the spin-orbit interaction.