2001 Series on Knots and Everything  
Starting from the principles of finiteness, discreteness, finite computability and absolute nonuniqueness, we develop the ordering operator calculus, a strictly constructive mathematical system having the empirical properties required by quantum mechanical and special relativistic phenomena. We show how to construct discrete distance functions, and both rectangular and spherical coordinate systems (with a discrete version of 'Y). The richest discrete space constructible without a preferred axis
more » ... ut a preferred axis and preserving translational and rotational invariance is shown to be a discrete 3-space with the usual symmetries. We introduce a local ordering parameter with local (proper) time-like properties and universal ordering parameters with global (cosmological) time-like properties. Constructed "attribute velocities" connect ensembles with attributes that are invariant as the appropriate time-like parameter increases. For each such attribute, we show how to construct attribute velocities which must satisfy the "relativistic Doppler shift" and the "relativistic velocity composition law," as well as the Lorentz transformations. By construction, these velocities have finite maximum and minimum values. In the space of all attributes, the minimum of these maximum velocities will predominate in all multiple attribute computations, and hence can be identified as a fundamental limiting velocity. General commutation relations are constructed which under the physical interpretation are shown to reduce to the usual quantum mechanical commutation relations. * It is usually permissible that a finite number of values of n violate the inequality. We do not allow this. t This is not an ontological statement. /
doi:10.1142/9789812810090_0005 fatcat:apnfjsn46vac3oyawjxrrrpu3q