Uma abordagem operacional conectando transformações e dinâmicas no espaço físico

Abstract

This work investigates the operational connection between the laboratory preparation of quantum systems and their representation as states in Hilbert space, as well as the behavior of transformations in light of this correspondence. By employing the Hopf fibration, we present an explicit operational description of the relationship between “laboratory variables” and quantum states, illustrated through the Stern–Gerlach experiment and a simple optical setup. Within this framework, we also show how spatial rotations in three dimensions naturally correspond to unitary transformations acting on quantum states, under constraints such as purity preservation and reversibility, leading to an operational reconstruction of the two-to-one homomorphism between the groups SU(2) and SO(3). Moreover, by extending this approach to higher-dimensional spaces, we contribute to the characterization of emergent dynamics in coarse-graining systems. Beyond its theoretical relevance, this approach provides a concrete and pedagogically accessible perspective, rendering experimentally tangible both the abstract relationship between symmetry groups and the geometry of the Bloch sphere and the physical mechanisms underlying the emergence of effective dynamics on coarse-graining problems.