Journal of Operator Theory

Volume 60, Issue 1, Summer 2008 pp. 199-215.

Evolution integrals

Authors: Rocco Duvenhage
Author institution: Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa

Summary: A framework analogous to path integrals in quantum physics is set up for dynamical systems in a $W^{\ast}$-algebraic setting. We consider spaces of evolutions, defined in a specific way, of a $W^{\ast}% $-algebra $A$ as an analogue of spaces of classical paths, and show how integrals over such spaces, which we call ``evolution integrals'', lead to dynamics in a Hilbert space on a ``higher level'' which is viewed as an analogue of quantum dynamics obtained from path integrals. The measures with respect to which these integrals are performed are projection valued.

Keywords: Dynamical systems, $W^{\ast}$-algebras, spectral measures.

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