Dirac and Weyl Fermions -- the Only Causal Systems

TL;DR

This paper classifies causal relativistic quantum systems with real mass and finite spin, establishing Dirac and Weyl fermions as the only irreducible solutions, and explores their localization, Lorentz contraction, and causal logic representations.

Contribution

It demonstrates that Dirac and Weyl fermions are the unique irreducible causal systems under specified conditions and analyzes their localization properties and causal logic representations.

Findings

Dirac and Weyl fermions are the only irreducible causal systems with real mass and finite spin.

All Dirac and Weyl wave-functions are subject to Lorentz contraction.

Causal systems extend to non-timelike hyperplanes, satisfying specific projection-valued measure conditions.

Abstract

Causal systems describe the localizability of relativistic quantum systems complying with the principles of special relativity and elementary causality. At their classification we restrict ourselves to real mass and finite spinor systems. It follows that (up to certain not yet discarded unitarily related systems) the only irreducible causal systems are the Dirac and the Weyl fermions. Their wave-equations are established as a mere consequence of causal localization. - The bounded localized Dirac and Weyl wavefunctions are studied in detail. One finds that, at the speed of light, the carriers shrink in the past and expand in the future. For every direction in space there is a definite time at which the change from shrinking to expanding occurs. A late changing time characterizes those states, which shrink to a delta-strip if boosted in the opposite direction. Using a density result for these late-change states one shows that all Dirac and Weyl wave-functions are subjected to Lorentz contraction. The latter is discussed in some detail. - We tackle the question whether a causal system induces a representation of a causal logic and thus provides a localization in proper space-time regions rather than on spacelike hyperplanes. The causal logic generated by the spacelike relation is shown to do not admit representations at all. But the logic generated by the non-timelike relation in general does, and the necessary condition is derived that there is a projection valued measure on every non-timelike non-spacelike hyperplane being the high boost limit of the localization on the spacelike hyperplanes. Dirac and Weyl systems are shown to satisfy this condition and thus to extend to all non-timelike hyperplanes, which implies more profound properties of the causal systems. The bounded localized eigenstates of the projections to non-spacelike flat strips are late-change states.