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.
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.
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.
Causal localizations describe the position of quantum systems moving not faster than light. They are constructed for the systems with finite spinor dimension. At the center of interest are the massive relativistic systems. For every positive mass, there is the sequence of Dirac tensor-localizations, which provides a complete set of inequivalent irreducible causal localizations. They obey the principle of special relativity and are fully Poincaré covariant. The boosters are determined by the causal position operator and the other Poincaré generators. The localization with minimal spinor dimension is the Dirac localization. Thus, the Dirac equation is derived here as a mere consequence of the principle of causality. Moreover, the higher tensor-localizations, not known so far, follow from Dirac’s localization by a simple construction. The probability of localization for positive energy states results to be described by causal positive operator valued (PO-) localizations, which are the traces of the causal localizations on the subspaces of positive energy. These causal Poincaré covariant PO-localizations for every irreducible massive relativistic system were, all the more, not known before. They are shown to be separated. Hence, the positive energy systems can be localized within every open region by a suitable preparation as accurately as desired. Finally, the attempt is made to provide an interpretation of the PO-localization operators within the frame of conventional quantum mechanics attributing an important role to the negative energy states.
On plain physical grounds localization of relativistic quantum particles is extended to the achronal regions of Minkowski spacetime. Achronal localization fulfills automatically the requirements of causality. It constitutes the frame which complies most completely with the principle of causality for quantum mechanical systems. Achronal localization is equivalent to the localization in the regions of the causal logic. Representations of the causal logic are constructed for the systems with mass spectrum of positive Lebesgue measure and every definite spin. Apparently no representation of the causal logic for a real mass system has been achieved in the past.
Abstract On plain physical grounds, localization of relativistic quantum particles is extended to the achronal regions of Minkowski spacetime. Achronal localization fulfills automatically the requirements of causality. It constitutes the frame which complies most completely with the principle of causality for quantum mechanical systems. Achronal localization is equivalent to the localization in the regions of the causal logic. Covariant representations of the causal logic are constructed for the systems with mass spectrum of positive Lebesgue measure and every definite spin. Apparently, no representation of the causal logic for a real mass system has been achieved in the past.
A physical principle that has consequences for the topology of a relativistic space-time is the principle of Einstein-Weyl causality. We show that these consequences also have quantum mechanical manifestations. Borchers and Sen have rigorously investigated the mathematical implications of Einstein-Weyl causality, regarded as a partial order, for the underlying spaces. This partial order was axiomatized by them and shown to admit Q^2 as an ordered space. They subsequently proved that, given certain topological assumptions, these spaces satisfying Einstein-Weyl causality are densely imbedded in spaces that have locally the structure of differentiable manifolds. They were left, however, with important philosophical paradoxes regarding the nature of the physical real line E, e.g., whether E = R, the real line of mathematics. We can view their original result, without further assumptions, as an insight into a denumerable space-time. This would suggest an investigation into a constructible foundation for relativistic physics. A set theoretical foundation is proposed including an axiom of countable constructibility. We first show that this constructible theory actually contains polynomial functions which are locally homeomorphic with a dense, denumerable metric space R*. Eigenfunctions governing fields can then be effectively obtained by iteration. Postulating a Lagrangian for fields in a compactified space-time, we get a general field description of which the Schrodinger equation is a special case. From these results we can then also show that this denumerable space-time is relational (in the sense that space is not infinitesimally small if and only if it contains a quantized field) and that, since Q^2 is imbedded in R*^2, it directly fulfills the strict topological requirements for Einstein-Weyl causality. Therefore, the theory predicts that E = R*.
This paper derives the Dirac equation as the logically forced relativistic completion of the de Broglie wave dynamics established in prior work of this series. Starting from the logical foundation that yields Lorentz/Poincaré invariance, we promote the non-relativistic framework (continuous spacetime, Fourier duality, local plane-wave structures, and probability conservation via statistical energy flux) to the relativistic regime. We show that the requirements of exact local relativistic plane-wave structures (with phase gradients satisfying the mass-shell condition pμpμ=m2c2), first-order time evolution (needed for clean probability conservation), and full Lorentz covariance together force the introduction of a specific non-commuting algebraic structure and a multi-component wave function. The Dirac equation emerges as the unique linear differential operator of order at most one in the spacetime derivatives that consistently glues these local structures across smooth backgrounds while preserving a conserved four-current with positive-definite time component. This multi-component structure arises naturally as the minimal algebraic consequence of the constraints, without any separate postulate for intrinsic angular momentum. In exact analogy with the emergence of the finite invariant causal speed in special relativity, the Dirac dynamics is revealed not as an ad-hoc linearization of the Klein--Gordon equation, but as the mathematically inevitable description of indivisible particles in continuous relativistic spacetime. The derivation thereby completes the single-particle sector of the series, demonstrating that core structures of relativistic quantum mechanics flow deductively from a small number of deep, nearly tautological commitments.