Quantum Mechanics and Number Theory: an

T. R. Robinson

The natural numbers play a key role in axiomatic set theory and hence in the foundations of mathematics. The natural numbers also have a major role in quantum mechanics. The idea is explored that von Neumann's construction (vNC) of the natural numbers, within the framework of Zermelo-Fraenkel (ZF) axiomatic set theory, can serve as a blueprint for deriving some key basic equations of quantum mechanics. This approach obviates any need for quantizing classical mechanics and provides further support for the view that quantum mechanics is perfectly applicable to fields beyond those in physics for which it was originally intended.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

Cécile de Witt-Morette

Alfredo Iorio

We discuss the q deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the q-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. The von Neumann theorem in Quantum Mechanics (QM) states that for systems with finite number of degrees of freedom the representations of the canonical commutation relations (ccr), also called the Weyl representations, are each other unitarily equivalent. In the limit of infinite number of degrees of freedom, namely in Quantum Field Theory

Donald Gary Swanson

Quantum mechanics is essentially a 20th century development that is based on a number of observations that defied classical explanations. While some of these experiments have semiclassical explanations, the triumph of quantum mechanics is that it gives precise verification for an overwhelming number of experimental observations. Its extensions into relativistic quantum mechanics through quantum electrodynamics (QED), electro-weak theory, and quantum chromodynamics (QCD) have led us to the Standard Model of today. While the number of unanswered questions remains approximately constant at each stage of development, the number of answered questions that relate theory with experiment continues to grow rapidly. In this chapter, some of the historical landmarks in the development of the theory are noted. The resolution of the dilemmas presented by classical theory will be dealt with in later chapters, but these are listed to motivate the break from classical mechanics. Because of the apparently unphysical nature of the postulates upon which quantum mechanics is founded, we supply motivation and justification for this break. In the first part of this chapter, we will review some of the experiments that confounded classical theory, and then introduce a formalism that provides some rationale for the postulates upon which quantum mechanics is based. In the end, these postulates will stand on their own.