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14050032

Consider discrete physics with a minimal time step taken to be

tau. A time series of positions q,q',q'', ... has two classical

observables: position (q) and velocity (q'-q)/tau. They do not commute,

for observing position does not force the clock to tick, but observing

velocity does force the clock to tick. Thus if VQ denotes first observe

position, then observe velocity and QV denotes first observe velocity,

then observe position, we have

VQ: (q'-q)q/tau

QV: q'(q'-q)/tau

(since after one tick the position has moved from q to q').

Thus [Q,V]= QV - VQ = (q'-q)^2/tau. If we consider the equation

[Q,V] = k (a constant), then k = (q'-q))^2/tau and this is recognizably

the diffusion constant that arises in a process of Brownian motion.

Thus, starting with the simplest assumptions for discrete physics, we are

lead to recognizable physics. We take this point of view and follow it

in both physical and mathematical directions. A first mathematical

direction is to see how i, the square root of negative unity, is related

to the simplest time series: ..., -1,+1,-1,+1,... and making the

above analysis of time series more algebraic leads to the following

interpetation for i. Let e=[-1,+1] and e'=[+1,-1] denote, as ordered

pairs, two phase-shifted versions of the alternating series above.

Define an operator b such that eb = be' and b^2 = 1. Regard b as a time

shifting operator. The operator b shifts the alternating series by one

half its period. Regard e' = -e and ee' = [-1.-1] = -1 (combining term by

term). Then let i = eb. We have ii = (eb)(eb) = ebeb = ee'bb = -1. Thus ii = -1

through the definition of i as eb, a temporally sensitive entity that

shifts it phase in the course of interacting with (a copy of) itself.

By going to i as a discrete dynamical system, we can come back to the

general features of discrete dynamical systems and look in a new way at

the role of i in quantum mechanics. Note that the i we have constructed is

already part of a simple Clifford algebra generated by e and b with

ee = bb = 1 and eb + be = 0. We will discuss other mathematical physical

structures such as the Schrodinger equation, the Dirac equation and the

relationship of a simple logical operator (generalizing negation) with

Majorana Fermions.

©2012 Institut Périmètre de Physique Théorique