Achronal Localization, Representations of the Causal Logic for massive systems

TL;DR

This work addresses the challenge of achieving causality-consistent localization in relativistic quantum mechanics by extending localization from spacelike regions to achronal regions in Minkowski space $\mathbb{R}^4$. It introduces and proves the equivalence of Achronal Localization (AL) and Representation of the Causal Logic (RCL), and constructs covariant representations for massive systems with a mass spectrum of positive Lebesgue measure across all spins using a Doplicher–Regge–Singer–style imprimitivity framework. The main technical contribution is a complete construction of projection-valued AL and RCL via a canonical cross section and a posited $W^{pos}$, followed by a covariant realization of the causal logic for massive systems through direct integrals over mass and spin, including a detailed decomposition of momentum and spinor spaces. The results generalize localization to achronal and causal-logical regions while preserving Poincaré covariance and causality, offering a principled framework for region-based localization beyond traditional Euclidean-space notions. Overall, the paper provides a mathematically rigorous pathway to causal, covariant localization for massive relativistic systems with arbitrary spin, highlighting both the power and the current limits of mass-smearing approaches.

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. 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.