Equivalence Relations in Quantum Theory: An Objective Account of Bases and Factorizations

TL;DR

This paper offers an invariant, objective account of bases and factorizations in quantum theory, addressing the perspectival relativism that challenges rational understanding of quantum states and entanglement.

Contribution

It introduces an invariant framework based on the logos categorical approach, linking mathematical formalism with quantum phenomena to clarify equivalence relations.

Findings

Provides an invariant account of bases and factorizations.

Establishes a conceptual bridge between formalism and phenomena.

Addresses the relativism in quantum reference frames.

Abstract

In orthodox Standard Quantum Mechanics (SQM) bases and factorizations are considered to define quantum states and entanglement in relativistic terms. While the choice of a basis (interpreted as a measurement context) defines a state incompatible to that same state in a different basis, the choice of a factorization (interpreted as the separability of systems into sub-systems) determines wether the same state is entangled or non-entangled. Of course, this perspectival relativism with respect to reference frames and factorizations precludes not only the widespread reference to quantum particles but more generally the possibility of any rational objective account of a state of affairs in general. In turn, this impossibility ends up justifying the instrumentalist (anti-realist) approach that contemporary quantum physics has followed since the establishment of SQM during the 1930s. In contraposition, in this work, taking as a standpoint the logos categorical approach to QM -- basically, Heisenberg's matrix formulation without Dirac's projection postulate -- we provide an invariant account of bases and factorizations which allows us to to build a conceptual-operational bridge between the mathematical formalism and quantum phenomena. In this context we are able to address the set of equivalence relations which allows us to determine what is actually the same in different bases and factorizations.