Philippe Blanchard

Jürg Fröhlich

This collection of essays is above all intended to pay tribute to the fact that while QM today is a refined and incredibly successful instrument, many issues concerning the internal consistency and the interpretation of this theory are still not nearly as well understood as they ought to be. In addition, whenever possible these essays take the opportunity to link foundational issues to the many exciting developments that are often linked to major experimental and technological breakthroughs in exploiting the electromagnetic field and in particular, its quantum properties and its interactions with matter, as well as to advances in solid state physics (such as new quantum Hall liquids, topological insulators and graphene). The present volume also focuses on various areas, including new interference experiments with very large molecules passing through double-slits, which test the validity of the Kochen-Specker theorem; new tests of the violation of Bell’s inequalities and the consequences of entanglement; new non-demolition measurements and tests of “wave-function collapse” to name but a few. These experimental developments have raised many challenging questions for theorists, leading to a new surge of interest in the foundations of QM, which have puzzled physicists ever since this theory was pioneered almost ninety years ago. The outcome of a seminar program of the same name on foundational issues in quantum physics (QM), organized by the editors of this book and addressing newcomers to the field and more seasoned specialists alike, this volume provides a pedagogically inspired snapshot view of many of the unresolved issues in the field of foundational QM.

Attempts Towards a Synthesis

The Message of Quantum Science

Alan D. Sokal

Roberto Fernandez

Simple random walks - or equivalently, sums of independent random vari ables - have long been a standard topic of probability theory and mathemat ical physics. In the 1950s, non-Markovian random-walk models, such as the self-avoiding walk,were introduced into theoretical polymer physics, and gradu ally came to serve as a paradigm for the general theory of critical phenomena. In the past decade, random-walk expansions have evolved into an important tool for the rigorous analysis of critical phenomena in classical spin systems and of the continuum limit in quantum field theory. Among the results obtained by random-walk methods are the proof of triviality of the cp4 quantum field theo ryin space-time dimension d (::::) 4, and the proof of mean-field critical behavior for cp4 and Ising models in space dimension d (::::) 4. The principal goal of the present monograph is to present a detailed review of these developments. It is supplemented by a brief excursion to the theory of random surfaces and various applications thereof. This book has grown out of research carried out by the authors mainly from 1982 until the middle of 1985. Our original intention was to write a research paper. However, the writing of such a paper turned out to be a very slow process, partly because of our geographical separation, partly because each of us was involved in other projects that may have appeared more urgent.

Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory

Thomas Kerler

This book reviews recent results on low-dimensional quantum field theories and their connection with quantum group theory and the theory of braided, balanced tensor categories. It presents detailed, mathematically precise introductions to these subjects and then continues with new results. Among the main results are a detailed analysis of the representation theory of U (sl ), for q a primitive root of unity, and a semi-simple quotient thereof, a classfication of braided tensor categories generated by an object of q-dimension less than two, and an application of these results to the theory of sectors in algebraic quantum field theory. This clarifies the notion of "quantized symmetries" in quantum fieldtheory. The reader is expected to be familiar with basic notions and resultsin algebra. The book is intended for research mathematicians, mathematical physicists and graduate students.

Jürg Fröhlich

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