arXiv subjects

Don Ridgeway

Publications and source records attributed to Don Ridgeway.

3 recordsLinked to original sources

arXiv2007

An algebraic theory of infinite classical lattices III: Theory of single measurements

This is the third in a series of papers dealing with the algebraic theory of infinite classical lattices. This paper presents a theory of single measurements on a lattice which we represent as comprising a finite subvolume--the system of measurement--immersed in an infinite surround or ``heat bath'' which determines the system's state. We consider the class of all stationary distributions on the set of microcanonical states of the infinite lattice. The theory addresses the question, ``For a lattice initially in state A, say, what is the probability that measurement of a certain quantity will take a value in (a,b)?'' Discussion includes description of the source of randomness in a measurement as well as characterization of the given states A.

math-ph↗

arXiv2005

An algebraic theory of infinite classical lattices I: General theory

We present an algebraic theory of the states of the infinite classical lattices. The construction follows the Haag-Kastler axioms from quantum field theory. By comparison, the *-algebras of the quantum theory are replaced here with the Banach lattices (MI-spaces) to have real-valued measurements, and the Gelfand-Naimark-Segal construction with the structure theorem for MI-spaces to represent the Segal algebra as C(X). The theory represents any compact convex set of states as a decomposition problem of states on an abstract Segal algebra C(X), where X is isomorphic with the space of extremal states of the set. Three examples are treated, the study of groups of symmetries and symmetry breakdown, the Gibbs states, and the set of all stationary states on the lattice. For relating the theory to standard problems of statistical mechanics, it is shown that every thermodynamic-limit state is uniquely identified by expectation values with an algebraic state.

math-ph↗

arXiv2005

An algebraic theory of infinite classical lattices II: Axiomatic theory

We apply the algebraic theory of infinite classical lattices from Part I to write an axiomatic theory of measurements, based on Mackey's axioms for quantum mechanics. The axioms give a complete theory of measurements in the sense of Haag and Kastler, taking the traditional form of a logic of propositions provided with a classical spectral theorem. The results are expressed in terms of probability distributions of individual measurements. As applications, we give a separation theorem for states by the set of observables and discuss its relationship to the equivalence of ensembles in the thermodynamic-limit program. We also introduce a weak equivalence of states based on the theory.

math-ph↗