arXiv subjects

Robert A. Van Wesep

Publications and source records attributed to Robert A. Van Wesep.

4 recordsLinked to original sources

arXiv2012

Satisfaction relations for proper classes: Applications in logic and set theory

We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension T of ZF (Zermelo-Fraenkel set theory) there is a finitely axiomatizable extension T' of GB (von Neumann-Bernays-Gödel class theory) that is a conservative extension of T. We also prove a conservative extension result that justifies the use of this predicate to characterize ground models for forcing constructions.

math.LO↗

arXiv2006

Compatibility of representations of quantum systems

There is a natural equivalence relation on representations of the states of a given quantum system in a Hilbert space, two representations being equivalent iff they are related by a unitary transformation. There are two equivalence classes, with members of opposite classes being related by a conjugate-unitary (anti-unitary) transformation. These two conjugacy classes are related in much the same way as are the two imaginary units of a complex field, and there is a priori no basis on which to prefer one over the other in any individual case. This is potentially problematic in that the choice of conjugacy class of a representation determines the sign of energy and other quantities defined as generators of continuous symmetries of the system in question, so that it would appear that principles like conservation of energy for a compound system may hold or fail depending on relative choices of conjugacy class of representations of its subsystems. We show that for any finite set of quantum systems there are exactly two ways of choosing conjugacy classes of representations consistent with the usual tensor-product construction for representing the compound system composed of these. Each is obtained from the other by reversing the conjugacy of all the representations at once. The relation of unitary equivalence for representations of a single system is therefore uniquely extendible to representations of all systems that can interact with it.

quant-ph↗

arXiv2005

Hidden variables in quantum mechanics: Generic models, set-theoretic forcing, and the emergence of probability

The hidden-variables premise is shown to be equivalent to the existence of generic filters for algebras of commuting propositions and for certain more general propositional systems. The significance of this equivalence is interpreted in light of the theory of generic filters and boolean-valued models in set theory (the method of forcing). The apparent stochastic nature of quantum observation is derived for these hidden-variables models.

quant-ph↗

arXiv2005

Many worlds and the emergence of probability in quantum mechanics

The interpretation of the squared norm as probability and the apparent stochastic nature of observation in quantum mechanics are derived from the strong law of large numbers and the algebraic properties of infinite sequences of simultaneous quantum observables. It is argued that this result validates the many-worlds view of quantum reality.

quant-ph↗