A Review of The Algebraic Approaches to Quantum Mechanics. Appraisals on Their Theoretical Relevance

TL;DR

This paper reviews the historical and theoretical development of algebraic foundations in quantum mechanics, highlighting key approaches, misinterpretations, and the lack of progress in foundational understanding.

Contribution

It provides a comprehensive historical analysis of algebraic approaches to quantum mechanics and discusses unresolved foundational issues using a problem-based, constructive mathematics perspective.

Findings

Historical algebraic approaches have evolved from matrices to C*-algebras.

No significant progress has been made in foundational understanding of QM.

The algebraic approach's formalism increased without clarifying physical foundations.

Abstract

I review the various algebraic foundations of quantum mechanics. They have been suggested since the birth of this theory till up to last year. They are the following ones: Heisenberg-Born-Jordan (1925), Weyl (1928), Dirac (1930), von Neumann (1936), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi (2009) and Buchholz and Fregenhagen (2019). Three cases are stressed: 1) the misinterpretation of Dirac foundation; 2) von Neumann conversion from the analytic approach of Hilbert space to the algebraic approach of the rings of operators; 3) the recent foundation of quantum mechanics upon the algebra of perturbation Lagrangians. Moreover, historical considerations on the go-and-stop path performed by the algebraic approach in the history of QM are offered. The level of formalism has increased from the mere introduction of matrices till up to group theory and C*-algebras. But there was no progress in approaching closer the foundations of physics; therefore the problem of discovering an algebraic formulation of QM organized as a problem-based theory and making use of no more than constructive mathematics is open.