Constructibility in Quantum Mechanics

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BenDaniel, David Jacob
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CONSTRUCTIBILITY IN QUANTUM MECHANICS D.J. BENDANIEL Cornell University ( Ithaca NY, 14853, USA We pursue an approach in which space-time proves to be relational and its differential properties fulfill the strict requirements of Einstein-Weyl causality. Space-time emerges here from a set theoretical foundation for a constructible mathematics. In this theory the Schrödinger equation can be obtained by adjoining a physical postulate of action symmetry in generalized wave phenomena. This result now allows quantum mechanics to be considered conceptually cumulative with prior physics. A set theoretical foundation is proposed in which the axioms are the theory of Zermelo-Frankel (ZF) but without the power set axiom, with the axiom schema of subsets removed from the axioms of regularity and replacement and with an axiom of countable constructibility added. Four arithmetic axioms are also adjoined; these formulae are contained in ZF but must be added here as axioms. All sets of finite natural numbers in this theory are finite and hence definable. The real numbers are countable. We first show that this constructible theory gives polynomial functions of a real variable, which are of bounded variation and locally homeomorphic. Eigenfunctions governing physical fields can then be effectively obtained. By using an integral of the Lagrange density of a field over a compactified space, we produce a nonlinear sigma model. Finally, the Schrödinger equation follows directly from the postulate and a sui generis proof in this theory of the discreteness of the space-like and time-like terms of the model.