(C) 1998-2007 by William D. Eshleman, All Rights Reserved

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April, 2007

Abstract.The conventional method for specification of a formalism for reality is to assume and determine for the chosen space, the following collection of mathematical objects: 1) the state as a vector in a vector space, 2) the observable as an operator that acts on the vector space, and 3) an algebra determined by the set of operators and certain constraints. Usually, this requires a detailed analysis of how vectors add in the chosen space. That is, the evolution of the state of a vector is a process whereby differential vectors are added to a vector. The original many-worlds interpretation took this approach and assumed the Hilbert space.Detailed analysis of how vectors could magnify one another in the chosen space is usually avoided due to a change of the type of a vector upon multiplication; e.g., the product of two position tensors is not a tensor, but sums of tensors remain tensors. The report presented here attempts to investigate hypothetical mathematical products that predict equivalent results, as compared to conventional results, for relatively low-energy trajectories, and then to compare the differences for high-energy motion.

The assumed collection of mathematical objects for this report is: 1) state as a product of factors (resembling vectors) in a world of an observer, 2) observable as the logarithm of a particular factor that acts on the space of the observer, and 3) the algebra of convergent products, the notion of the equivalence of products, and the notion of the entropy of products. That is, the evolution of the state of a vector is a process whereby vectors magnify to produce change; the Product Paradigm.

This report claims the mathematical form for a Relativistic Hamiltonian consistent with the sub-factorization of Lorentz factors.

In addition to his short thesis Everett wrote a much larger exposition of his ideas, which was never published. The present volume contains both of these works, together with a handful of papers by others on the same theme. Looked at in one way, Everett's interpretation calls for return to naive realism and the old fashioned idea that there can be direct correspondence between formalism and reality. Because physicists have become more sophisticated than this, and above all because the implications of his approach appear to them so bizarre, few have taken Everett seriously. Nevertheless his basic premise provides such a stimulating framework for discussions of the quantum theory of measurement that this volume should be on every quantum theoretician's shelf.

"... a picture, incomplete yet not false, of the universe as Ts'ui Pen conceived it to be. Differing from Newton and Schopenhauer, ... [he] did not think of time as absolute and uniform. He believed in an infinite series of times, in a dizzily growing, ever spreading network of diverging, converging and parallel times. This web of time -- the strands of which approach one another, bifurcate, intersect or ignore each other through the centuries -- embraces every possibility. We do not exist in most of them. In some you exist and not I, while in others I do, and you do not, and in yet others both of us exist. In this one, in which chance has favored me, you have come to my gate. In another, you, crossing the garden, have found me dead. In yet another, I say these very same words, but am an error, a phantom."

"Actualities seem to float in a wider sea of possibilities from out of
which they were chosen; and somewhere, indeterminism says, such possibilities
exist, and form a part of the truth."

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