Representation of the causal logic for the Dirac system and the electron

• URL: http://arxiv.org/abs/2607.21266v1
• Date: Thu, 23 Jul 2026 12:38:41 GMT
• Title: Representation of the causal logic for the Dirac system and the electron
• Abstract summary: We construct a covariant representation of the causal logic for the Dirac system.<n>Full Dirac achronal localization is projection valued and represents achronal separateness.<n>We discuss the electron--positron decomposition and the state transformation induced by position measurements.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We construct a covariant representation of the causal logic for the Dirac system and the electron, based on the conserved Dirac probability current and a general current-to-localization procedure for achronal regions. A polynomial decay estimate, derived by a non-stationary phase argument, verifies the conditions required to define covariant achronal localization for the full Dirac system. Extending this localization to complete spacetime regions yields the corresponding causal-logic representation; restriction to the positive-energy invariant subspace gives the analogous construction for the electron. We establish several structural properties. On Euclidean space, Dirac localization agrees with canonical projection-valued localization, while its position operator differs from the Newton--Wigner operator by an explicitly determined bounded self-adjoint correction. Full Dirac achronal localization is projection valued and represents achronal separateness by orthogonality, thus satisfying microscopic causality. After compression to the electron subspace, it becomes positive-operator valued while preserving causality. We further extend known results from spatial to general achronal regions, prove separation and norm-one criteria, and analyze the high-boost limit, obtaining Lorentz contraction in a precise probabilistic sense. Finally, we discuss the electron--positron decomposition and the state transformation induced by position measurements, showing that measurement-induced positron production is universal and independent of the specific measuring device.

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