Quantum Reprogramming: Ensembles and Single Systems: A Two-Tier Approach to Quantum Mechanics

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Many, perhaps most textbooks of quantum mechanics present a Copenhagen, single system angle, this text puts forward; fewer present the subject matter as an instrument for treating ensembles, but the two methods have been silently coexisting since the mid-1930s. This lingering dichotomy of purpose for a major physical discipline has greatly clouded further insights into the foundations of quantum theory. This work aims to resolve this long-standing dichotomy by examining the mutual relation between single systems and ensembles, assigning each its own tools for treating the subject at hand: i.e., Schrodinger-Dirac methods for ensembles versus period integrals for single systems. A unified treatment of integer and fractional quantum Hall effects and a finite description of the electron’s anomalies are mentioned as measures of justification for the chosen procedure of resolving an old-time dichotomy. The methods of presentation are, in part, elementary, with repetitive references needed to delineate differences with respect to standard methods. The parts on period integrals are developed with a perspective on elementary methods in physics, thus leading up to some standard results of de Rham theory and algebraic topology.