A Relativistic Pseudo-Unitary Version of Schwinger's Quantum Mechanical Symbolism of Atomic Measurements and a Prospect for a New Relativistic Quantum Information Theory

TL;DR

This work extends Schwinger's non-relativistic measurement symbolism into a relativistic regime by embedding atomic measurement operators in Cartan's space, leveraging the $SU(2,2)$ structure to describe covariant measurement correlations across spacetime frames via the orthochronous Poincaré subgroup. It develops a formalism of states, observables, and measurement operations within an indefinite inner product framework, employing pseudo-Hermitian and pseudo-unitary operators to realize covariant measurement processes. The results suggest a foundational path toward a relativistic quantum information theory and covariant quantum gates, while provoking reassessment of nonlocality and invariance of computation under relativistic transformations. The framework also points to applications in relativistic particle-antiparticle systems and covariant measurement-based quantum computation, with further work anticipated to flesh out these ideas.

Abstract

The measurement processes that are traditionally described within the realm of non-relativistic quantum mechanics are transcribed into the covariant framework of Cartan's space, the four-valued representation space of the restricted conformal group for special relativity. It is assumed at the outset that the non-relativistic quantum measurement mechanisms of state reductions as well as the definition of Born probabilities should remain unaltered when the passage to the covariant framework is worked out. The correlations between observations registered in different spacetime frames, concerning intermediate steps and outcomes of microscopic measurements, are attained through the implementation of the orthochronous proper Poincaré subgroup of an appropriate realization of SU(2,2). It appears that the overall work may supply an elementary theoretical background to the construction of relativistic quantum computational gates whilst suggesting that the non-locality feature of the old quantum mechanics should be reconsidered within a relativistic formulation. The important question as to whether quantum computational processes should bear an invariant character is then raised.