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

• URL: http://arxiv.org/abs/2102.00861v1
• Date: Fri, 29 Jan 2021 18:57:29 GMT
• Title: A Review of The Algebraic Approaches to Quantum Mechanics. Appraisals on Their Theoretical Relevance
• Authors: Antonino Drago
• Abstract summary: The various foundations of quantum mechanics have been suggested since the birth of this theory till last year. They are: Heisenberg-Born-Jordan (1936), Weyl (1928), Dirac (1930), von Neumann (1947), Segal (1947), T.F. Jordan (1986), Morchio and Strocchi (2009) and Buchholz and Fregenhagen (2009)
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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