Let the Mathematics of Quantum Speak: Allowed and Unallowed Logic

TL;DR

This paper explores the mathematical structure of quantum physics, emphasizing how non-commutative algebras restrict the types of logic applicable, especially contrasting quantum and classical systems.

Contribution

It clarifies the role of non-commutative algebra in quantum logic and argues that classical logic cannot be directly derived from quantum formalism without additional assumptions.

Findings

Quantum formalism is characterized by non-commutative algebras.

Classical logic emerges only in approximate, macroscopic, commutative regimes.

Quantum non-commutativity prevents a straightforward 'yes-no' logic in the quantum realm.

Abstract

Some notes about quantum physics, an interpretation if one wishes, are put forward, insisting on `closely following the mathematics/formalism, the `nuts and bolts of what quantum physics says'. These, basically well-known, issues seem to highlight some rather bold points about the `logic' aspect in quantum physics, necessarily restricting when and which logic may be admissible. And one may understand why that path is hardly followed in the literature. The mathematics/formalism of quantum, compared with classical, physics, may be fairly basically characterized by non-commutative algebras replacing commutative. These classically appearing, in fact, in dealing with systems of possibilities (say, all possible planetary motions under gravity of which one is the actual one). In particular, contrary to too common usage, the quantum non-commutativity should make it impossible to simply `transcend' the `system of possibilities' aspect into a `yes-no' logic essential for an `actual world'. One may have the latter only in a `haven' of approximately commutative algebras of `quasi-classical macroscopic observables', and moreover that `yes-no actual world' would plainly be an `extra ingredient' to the base quantum theory itself.