Discrete Physics and the Emergence of Quantum Mechanics

Math, UIC
University of Illinois at Chicago
Chicago, Illinois 60607-7045
USA

We show how taking a discrete stance from the very beginning for physical processes and physical measurements leads to the non-commutativity of position and momentum in physical observation. Thus discrete physics, viewed in this way, yields a framework for quantum mechanics. We then discuss the role of i where i^2 = -1, and its appearance in the commutator of position and momentum.
In order to bring i into this structure we analyze the nature of i in relation to discrete time and identify i as standing for a fundamental temporal process. We take the dictum inspired by special relativity and replace t by it, interpreting this substitution in terms of i as a discrete dynamical system. With this identification we can write pq - qp = i hbar in our discrete model and thus arrive at the elements of quantum mechanics.
We will elaborate on the foundations of discrete physics in terms of non-commutativity and the view of i as the recursion i -> -1/i.
The symbolic i is a solution to i = -1/i. The temporal i is the simplest alternating iterant +1,-1,+1,-1,... The purpose of this talk is to discuss the possibilities in new ways to view quantum echanics as emergent from discrete and non-commutative measurement.