Foundations of Quantum Mechanics : An Introduction to the Physical Background and Mathematical Structure

Barbara Piechocinska

Motivated by reductionism's current inability to encompass the quantum theory we explore an indivisible and dynamical wholeness as an underlying foundation for physics. After reviewing the role of wholeness in the quantum theory we set a philosophical background aiming at introducing an ontology, based on a dynamical wholeness. Equipped with the philosophical background we then propose a mathematical realization by representing the dynamics with a non-trivial elementary embedding from the mathematical universe to itself. By letting the embedding interact with itself through application we obtain a left-distributive universal algebra that is isomorphic to special braids. Via the connection between braids and quantum and statistical physics we show that a the mathematical structure obtained from wholeness yields known physics in a special case. In particular we point out the connections to algebras of observables, spin networks, and statistical mechanical models used in solid state physics, such as the Potts model. Furthermore we discuss the general case and there the possibility of interpreting the mathematical structure as a dynamics beyond unitary evolution, where entropy increase is involved.

Guozhen Wu

Franco Strocchi

James T. Cushing, Gary Bowman

Introduction It is a pleasure to be able to contribute to this volume in honour of Professor Michael Redhead. In the foundations of physics community, his influence has been felt both through his Incompleteness, Nonlocality and Realism (1987), which has largely denned the terms of discourse in this field for the last decade, and by the new ‘Cambridge School’ of the philosophy of physics that he established there during his Professorship in the History and Philosophy of Science Department. Michael Redhead's work illustrates well what has always seemed to be a central tenet of his: that philosophers of science should use the conceptual background and the mathematical formalism of modern, successful scientific theories – especially quantum mechanics and quantum field theory – as a guide to understanding the ontology of the world (Redhead 1983, 1990) and to reexamining traditional philosophical questions (French & Redhead 1988; Redhead & Teller 1991, 1992). In his Tarner Lectures (1995) he used this type of work to anchor a general position that can be characterized as structural realism , which holds that the structures – mathematical formalism, models, analogies – of a successful scientific theory eventually define the real, even as regards central, unobservable entities. While Professor Redhead has done an outstanding job of pressing forward with this general programme, it remains arguable that he and some of his co-workers, who have, in the main, stayed within the framework of standard quantum theory and relativity, may have effectively bracketed a different and potentially fruitful alternative theory.

Franco Strocchi