On Segal's postulates for general quantum mechanics

S. Sherman

Stanley Gudder, S. Boyce

John P. Lowe

Maria Victoria Velasco, Enrique Aycart-Maldonado

Abstract In this paper, we review Segal’s postulates for general Quantum Mechanics, updating and clarifying their mathematical formulation to simplify them. Our approach provides additional insights as well as a pathway to easily contribute new results to the body of knowledge on systems of observables. Moreover, we introduce the notion of complex system of observables, which arises naturally from the analysis carried out. As we show, these last systems of observables are related to the classical (real) ones through the process of complexification. Taking advantage of this, algebraic systems of observables are completely determined in both the real and complex cases.

John P. Lowe, Kirk A. Peterson

David B. Lowdenslager

A collection of postulates for the observable quantities in a general quantum mechanical system has been given by Segal, [4]. In this system, the observables form a real Banach space S which is partially ordered by a closed convex positive cone P, and which has a unit element e interior to the positive cone, all satisfying Pf(e-P) =S1(e)CSi(0), where Sr(x) is the solid sphere of radius r, center x. Alternatively, S is a partially ordered linear space with an element e>O which is a Banach space under the norm llxi =min {X: -Xe ?x <Xe }, where the minimum is attained. Let us call such a space a partially ordered Banach space with unit. We shall show that any such Banach space S with order unit e can be given the algebraic structure postulated by Segal, in possibly different ways. In this proof it is very useful to have an alternative formulation of Segal's postulates stated in terms of the partial ordering alone. We are then able to give examples of such systems with various pathologies and to classify the three dimensional ones. Finally we shall give a necessary and sufficient condition that a system of operators satisfying these postulates be a C* special Jordan algebra: the system of hermitean elements of a C* algebra. DEFINITION. A spectral function L on a partially ordered Banach space S with order unit e is a function assigning to each point x ES a