Dirac and Weyl Fermions -- the Only Causal Systems

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 compact localized Dirac and Weyl wave-functions 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 δ-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. -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 compact localized eigenstates of the projections to non-spacelike flat strips are latechange states.

Spectral Representations of the Wiener-Hopf Operator for the sinc Kernel and some Related Operators

“…As we will prove in (9), the spectral functions q s of K multiplied by the normalization constants n(s) := i π (2s) −1/2 gives rise to a kernel q for an integral operator, which determines an Hilbert space isomorphism V from L 2 (0, 1) onto L 2 (R + ). The following definition considers both cases q s = q + s | R+ and q − s | R− .…”

Incompatibility of Frequency Splitting and Spatial Localization: A Quantitative Analysis of Hegerfeldt's Theorem

“…In a series of later articles [11,10,12], Hegerfeldt discussed these results and their observational consequences in greater detail. Hegerfeldt's theorem has applications to quantum theory in the context of causal localizations (see for example [5,4] and the references therein for more recent developments). In [12] Hegerfeldt addresses the question why the Dirac equation is not a counter example: the original result is based on the assumption that the Hamiltonian of the system is positive definite, which obviously is not the case for the Dirac Hamiltonian.…”