Algebras of Measurements: the logical structure of Quantum Mechanics

In Quantum Physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. This paper studies algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space, justifying their properties on epistemological grounds. A central topic of interest is the commutation of measurements. The work presents a new class of abstract structures, termed algebras of measurements (M-algebras), designed to capture the logic of physical measurements and the differences between classical and quantum mechanics, where classical physics is viewed as a limited case of quantum physics. It further provides a generalized view of logic, highlighting classical logic as a specific instance. This exploration is inspired by the foundational work of G. Birkhoff and J. von Neumann, who posited that experimental propositions are closed subspaces of a Hilbert space, a topic that has received limited attention since. The paper proposes an enriched framework for understanding the logical underpinnings of quantum measurements, aiming to bridge gaps in current philosophical and physical frameworks.