A Review of The Algebraic Approaches to Quantum Mechanics. Appraisals on Their Theoretical Relevance
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: HeisenbergBornJordan (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 goandstop 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 problembased theory and making use of no more than constructive mathematics is open.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2102.00861
 Bibcode:
 2021arXiv210200861D
 Keywords:

 Physics  History and Philosophy of Physics;
 Mathematical Physics;
 Quantum Physics;
 20G42;
 46L67;
 B.0
 EPrint:
 I show that in the past the scholars reflecting on the foundations of quantum mechanics missed to take into account the progressive relevance acquired by the algebraic approach. 13 pages