The Dual Dynamical Foundation of Orthodox Quantum Mechanics

This paper combines mathematical, philosophical, and historical analyses in a comprehensive investigation of the dynamical foundations of the formalism of orthodox quantum mechanics. The results obtained include: (i) A deduction of the canonical commutation relations (CCR) from the tenets of Matrix Mechanics; (ii) A discussion of the meaning of Schrödinger’s first derivation of the wave equation that not only improves on Joas and Lehner’s 2009 investigation on the subject, but also demonstrates that the CCR follow of necessity from Schrödinger’s first derivation of the wave equation, thus correcting the common misconception that the CCR were only posited by Schrödinger to pursue equivalence with Matrix Mechanics; (iii) A discussion of the mathematical facts and requirements involved in the equivalence of Matrix and Wave Mechanics that improves on F. A. Muller’s classical treatment of the subject; (iv) A proof that the equivalence of Matrix and Wave Mechanics is necessitated by the formal requirements of a dual action functional from which both the dynamical postulates of orthodox quantum mechanics, von Neumann’s process 1 and process 2, follow; (v) A critical assessment, based on (iii) and (iv), of von Neumann’s construction of unified quantum mechanics over Hilbert space. Point (iv) is our main result. It brings to the open the important, but hitherto ignored, fact that orthodox quantum mechanics is no exception to the golden rule of physics that the dynamics of a physical theory must follow from the action functional. If orthodox quantum mechanics, based as it is on the assumption of the equivalence of Matrix and Wave Mechanics, has this “peculiar dual dynamics,” as von Neumann called it, then this is so because by assuming the equivalence one has been assuming a peculiar dual action.