The Ultimate Constituents of the Material World: In Search of an Ontology for Fundamental Physics 9783110326123, 9783110325270

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Meinard Kuhlmann The Ultimate Constituents of the Material World In Search of an Ontology for Fundamental Physics

Philosophische Analyse Philosophical Analysis Herausgegeben von / Edited by Herbert Hochberg • Rafael Hüntelmann • Christian Kanzian Richard Schantz • Erwin Tegtmeier Band 37 / Volume 37

Meinard Kuhlmann

The Ultimate Constituents of the Material World In Search of an Ontology for Fundamental Physics

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.

This book won the “Award for Furthering Research in Ontology” conferred by the German Society for Analytical Philosophy (GAP) in September 2009 at the GAP7 conference in Bremen. North and South America by Transaction Books Rutgers University Piscataway, NJ 08854-8042 [email protected] United Kingdom, Ireland, Iceland, Turkey, Malta, Portugal by Gazelle Books Services Limited White Cross Mills Hightown LANCASTER, LA1 4XS [email protected]

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2010 ontos verlag P.O. Box 15 41, D-63133 Heusenstamm www.ontosverlag.com ISBN 978-3-86838-072-9 2010 No part of this book may be reproduced, stored in retrieval systems or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use of the purchaser of the work Printed on acid-free paper FSC-certified (Forest Stewardship Council) This hardcover binding meets the International Library standard Printed in Germany by buch bücher.de

Contents

I

Ontology and Quantum Field Theory

xi

1 Introduction

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2 Philosophical Background 2.1 Atomism in the History of Philosophy . . . . . . . . . . . . 2.2 Philosophical Versus Scientific Atomism . . . . . . . . . . 2.3 Atomism and Reductionism . . . . . . . . . . . . . . . . .

7 8 13 14

3 Ontology and Physics 3.1 Some Main Themes in Ontology . . . . . . . . . . . . . . 3.2 A Brief History of Ontology . . . . . . . . . . . . . . . . 3.3 The Analytical Tradition of Ontology . . . . . . . . . . . 3.4 No-Go Theorems as Tools for the Ontological Practician 3.5 Symmetries, Heuristics and Objectivity . . . . . . . . . .

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27 27 31 32 35 37

5 Alternative Approaches 5.1 Deficiencies of the Standard Formulation of QFT . . . . . 5.2 The Algebraic Point of View . . . . . . . . . . . . . . . . . 5.3 Basic Ideas of AQFT . . . . . . . . . . . . . . . . . . . . .

41 41 42 44

4 History and Basic Structure of QFT 4.1 The Early Development . . . . . . . 4.2 The Emergence of Infinities . . . . . 4.3 The Taming of Infinities . . . . . . . 4.4 The Lagrangian Formulation of QFT 4.5 Interaction . . . . . . . . . . . . . . .

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CONTENTS

iv 6 The 6.1 6.2 6.3 6.4

II

Ontological Significance of QFT and AQFT QM Versus QFT . . . . . . . . . . . . . . . . . . AQFT and the Ideal Language Philosophy . . . . QFT Versus AQFT . . . . . . . . . . . . . . . . . The Philosophical Interest in (A)QFT . . . . . .

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Classical Ontologies

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7 Classical vs. Revisionary Ontologies 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 7.2 Aristotle’s Theory of Substances . . . . . . . . . . . 7.3 Substance Ontologies . . . . . . . . . . . . . . . . . 7.4 Substances Under Attack . . . . . . . . . . . . . . . 7.5 Substance Ontology and Quantum Physics . . . . . 7.5.1 Incompatible Observables . . . . . . . . . . 7.5.2 Non-Vanishing Vacuum Expectation Values

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8 Particle Interpretation of QFT 8.1 The Particle Concept . . . . . . . . . . . . . . . . . . . . 8.1.1 General Features . . . . . . . . . . . . . . . . . . 8.1.2 Wigner’s Analysis of the Poincar´e Group . . . . . 8.2 Theory and Experiment in Elementary Particle Physics: Is a Particle Track a Track of a Particle? . . . . . . . . . 8.3 Localization Problems . . . . . . . . . . . . . . . . . . . 8.3.1 The Clash of Causality and Localizability . . . . 8.3.2 Locating the Origin of Non-Localizability: A Comparative Study . . . . . . . . . . . . . . . 8.4 Further Problems for a Particle Interpretation of QFT . 8.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Field Interpretations of QFT 9.1 The Field Concept . . . . . . . . . . . . . . 9.2 Fields as Basic Entities of QFT . . . . . . . 9.2.1 The Role of Field Operators in QFT 9.2.2 Indirect Evidence for Fields . . . . . 9.3 Fields Versus Algebras . . . . . . . . . . . .

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63 63 66 72 77 79 79 81

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113 113 115 115 117 118

CONTENTS III

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Revisionary Ontologies

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10 Process Ontology 10.1 The Strands of Process Ontology . . . . 10.2 Why Process Ontology in QM and QFT? 10.3 A Case Study: Feynman Diagrams . . . 10.4 Evaluation of the Case Study . . . . . . 10.5 Remaining Problems . . . . . . . . . . . 11 Trope Ontology I: The Ontological Status of Properties 11.1 The Problem of Universals . . . . . 11.2 The Traditional Responses . . . . . 11.3 A New Solution: Trope Ontology . 11.4 An Evaluation of the Debate . . . . 11.5 Conclusion and Outlook . . . . . .

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12 Trope Ontology II: Properties and Things

IV

The Trope Bundle Interpretation

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13 Dispositional Trope Ontology 157 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 157 13.2 Trope Bundles and Many-Particle Systems . . . . . . . . . 159 13.2.1 ‘Elementary Particles’ . . . . . . . . . . . . . . . . 159 13.2.2 Individuality of Quantum Objects . . . . . . . . . . 160 13.2.3 Dispositions and Tropes . . . . . . . . . . . . . . . 164 13.2.4 An Example . . . . . . . . . . . . . . . . . . . . . . 167 13.3 The Trope Bundle Interpretation of AQFT . . . . . . . . . 169 13.3.1 AQFT as a Model of Trope Ontology . . . . . . . . 170 13.3.2 An Algebraic Argument for the Bundle Conception 171 13.3.3 Representations and Properties/Tropes . . . . . . . 174 13.3.4 Outlook on Potential Problems and Further Work . 178 13.3.5 The Explanatory Power of the Trope Bundle Interpretation . . . . . . . . . . . . . . . . . . . . . 179

CONTENTS

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13.4 Summing Up . . . . . . . . . . . . . . . . . . . . . . . . .

V

Concluding Remarks

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14 Physics and Philosophy

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15 Summing Up 15.1 General Remarks . . . . . . . . . . . . . 15.2 Comparison of Ontologies for QFT . . . 15.2.1 Particles Versus Fields . . . . . . 15.2.2 Processes Versus Tropes . . . . . 15.2.3 The Merits of Dispositional Trope

187 187 189 189 192 194

VI

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Appendices

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Abbreviations and Notation

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A Special Relativity Theory: Some Notation and Required Results

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B Ontologically Oriented Survey of Quantum Mechanics B.1 The Hilbert Space Formalism . . . . . B.1.1 States and Observables . . . . . B.1.2 Probability Interpretation . . . B.1.3 Dynamics . . . . . . . . . . . . B.2 Problems for an Ontology of QM . . . B.2.1 The Problem of Individuation . B.2.2 The Problem of Reidentifiability B.2.3 The Measurement Problem . . .

203 205 205 207 212 213 214 216 218

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C Advanced Foundational Topics in QFT 221 C.1 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . 221 C.2 Effective Field Theories and Renormalization . . . . . . . . 223 C.3 String Theory . . . . . . . . . . . . . . . . . . . . . . . . . 225

CONTENTS

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D Assumptions and Results of AQFT 227 D.1 Assumptions of AQFT . . . . . . . . . . . . . . . . . . . . 227 D.2 Representations and States . . . . . . . . . . . . . . . . . . 231 D.3 Superselection Sectors . . . . . . . . . . . . . . . . . . . . 234 References

240

Physics Glossary

257

Philosophy Glossary

273

Preface Many people have contributed to the making of this book. I am very grateful to various colleagues whose feedback was instrumental in advancing my thinking about (A)QFT and ontology: Guido Bacchiagaluppi, Jeff Barrett, Harvey Brown, Detlev Buchholz, Jeremy Butterfield, Dennis Dieks, Klaus Fredenhagen, Holger Lyre, David Malament, Bryan Roberts, Simon Saunders, Johanna Seibt, David Wallace, and Andrew Wayne. I also wish to express thanks to the many attentive and critical audiences in Bremen, Chicago, Cologne, Dortmund, Florence, G¨ottingen, Hamburg (Desy), Montreal, Oxford and Utrecht. This book was chosen for the 2009 ontos-Award for scientific research in the field of analytical ontology and metaphysics by the German Society for Analytical Philosophy (Gesellschaft f¨ ur Analytische Philosophie e.V., GAP). To the ontos verlag I convey my best thanks for publishing my book in conjunction with this prize. I would like to thank Rafael H¨ untelmann of the ontos verlag for his commitment and cooperation which were of particular help during the final stages of the publication process. Most of all I owe my sincerest thanks to Manfred St¨ockler for his exceptionally committed and competent support of my work. His enthusiasm and intellectual companionship are what made this book possible. And finally, I would like to thank my friends, especially Christina Thiel, and my family, for the support and patience they have shown me throughout.

Part I

Ontology and Quantum Field Theory

Chapter 1 Introduction Since the very beginning of western philosophy reflections about the material world which go beyond the directly observable play a central role in philosophy. Starting with the presocratics it has always been a point of debate what the fundamental characteristics of the material world are. Is everything constantly changing or are there certain permanent features? What is basic and what is merely a matter of perspective and appearance? In the course of time various answers have been given and conflicting views have often been alternating in their predominance. Quantum Field Theory (QFT)—the mathematical and conceptual framework for contemporary elementary particle physics—is presently the best starting point for analysing the fundamental features of matter and interactions. During the last two decades QFT became a more and more vividly discussed topic in the philosophy of physics. Since mathematical reasoning dominated the heuristics of QFT, its interpretation is still open in most areas which go beyond immediate empirical predictions. Philosophical analysis can help to clarify its semantics. QFT taken seriously in its metaphysical implications seems to give a picture of the world which is at variance with central classical conceptions like particles and fields and even with some features of (non-relativistic) quantum mechanics. An ontological analysis of QFT can yield crucial insights about the fundamental properties of the material world. Which questions will be explored? The philosophical topic of this book is ontology, the investigation of the most general structures of what there is in the world. I will ask which kinds of things and modes of being there are in a very general sense. However, I will explore these issues 1

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CHAPTER 1. INTRODUCTION

mostly in relation to modern physics. The main question is which ways to conceive of the world are compatible with the mathematical formalism and the physical assumptions of quantum field theory (QFT). The truth of QFT, its relation to reality as well as being as such will only be issues as far as they have a direct relevance for the following ontological investigation of QFT which is taken as the starting point. An analysis of the conceptual consistency and maturity of QFT and the status of QFT within physics is therefore an important preparatory step. The immediate question why QFT of all scientific theories has been chosen for this enterprise has a straightforward answer. If any scientific theory about nature can lay claim on being the most fundamental one it is QFT. This is not to say that everything can or should be reduced to QFT. However, when particular sciences come under ontological consideration at all, QFT is of outstanding importance. Accordingly, it is of particular interest which picture of the world this theory paints. The term ‘ontology’ is often used in a two-fold way, at least in the tradition of analytical philosophy which is the philosophical background of the present work. Besides the search for, or the theory of, the most general structures of being, ‘ontology’ denotes the domain itself to which a language or theory refers. Following this tradition I will freely make use of both senses of ‘ontology’ as well. It is not presupposed that there is one definite set of basic entities to which QFT refers and which one could simply, or after some closer investigation, read off QFT. The only two things I will presuppose with respect to ontological questions is, first, that these questions make sense at all and, second, that particular sciences like physics deal with objects whose conceptual analysis yields a valuable contribution to the general ontological questions. What kind of answers should be expected? I will present some ways one can and some ways one cannot imagine the world to be in line with QFT. However, it is not yet the time to supply the right or appropriate ontology of QFT. This is a field of ongoing research where new options are still presented and various ontologically relevant aspects need further analysis. Moreover, there might never be an uncontroversial unified answer. Although a final answer to the posed questions should not be expected it will become clear in the course of this investigation that not all problems

3 are equally important and not all options are equally viable. Eventually, I will introduce a new option for the ontology of QFT and justify my preference for this option. Historical and systematical background. The ontological analysis of QFT is a relatively new area of philosophical concern. Nevertheless, historically as well as systematically, it is situated on a background of closely related issues. In a sense it is a follow-up to the famous discussion about the wave-particle duality which arose with the formation of quantum mechanics (QM). Quantum objects seem to defy a representation in classical terms. With the quantum measurement problem the very idea of an objective ascription of properties to things came under suspicion. Moreover, Heisenberg’s uncertainty relations together with the non-classical behaviour of systems containing so-called identical particles endanger the individuality of quantum objects. All these questions still linger on in the present investigation. There is, however, a much longer tradition of kindred questions in the history of atomism beginning with ancient Greek philosophy. Here we have similar considerations about the building blocks of the world and their nature. Nevertheless, there is a pivotal difference between the old atomist’s reasoning and the way one proceeds nowadays. Except for its very last period, the history of atomism consists exclusively of conceptual considerations. In contrast to that the ontological study of QFT starts with a theory which has been exceedingly well corroborated by a plethora of experiments. Main results. Besides an account of some parts of the current state of research and of some systematical and historical foundations this work contains four new contributions to the ongoing research. The first point consists in the embedding of the main topic of my investigation into the philosophy of physics on the one side and general philosophy on the other side, in particular analytical ontology. The other three contributions concern different options for the ontology of QFT. The context of the first contribution is an argumentation against one option for an ontology of QFT, viz. the particle interpretation. It deals with localization problems of relativistic N-particle states. The relation of two conceptually important no-go theorems will be explored in section 8.3.2 and it will be shown that

4

CHAPTER 1. INTRODUCTION

the analysis leads to a surprising result. Other than the first contribution the second one as such is neutral with respect to the choice between different ontologies. Its purpose is to carve out interpretative consequences of the most radical proposal for an ontology of QFT, namely process ontology. The new study carried out in this book has to do with the interpretation of Feynman diagrams. It will show in sections 10.3 and 10.4 that and which interpretative differences follow from the choice of different ontological approaches. The last and most important contribution is the proposal of a new ontology of QFT, which I call ‘dispositional trope ontology’. I will show in part IV why I consider a trope-ontological approach to be a very palatable new alternative for the ontology of QFT and I will argue that dispositional properties are pivotal in such an interpretation of QFT. By these contributions I try to push a subject further ahead which is vividly discussed in these days. I hope that my results will help to further stimulate the debate. Since there is still much more to explore about the ontological aspects of QFT I have no doubt that this will remain an area of lively research for many years to come. Accordingly, the exposition of the state of research as well as my own results represent only a small part of what can, and hopefully will, be done. The choice of topics. Two choices to be made regard the ontological alternatives that will be examined on the one side and the aspects to be considered for each ontology on the other side. I do not aspire to completeness with respect to either side. Rather, one intention is to describe and to embed new results I found out during my own research. One of these results is about an already existing ontological interpretation of QFT, namely particle ontology. The other two results are attempts to establish and evaluate new alternatives for an ontology of QFT. These are process and trope ontology where trope ontology is the alternative which I will finally give my own preference. The reason why I included field ontology in my exposition is not that I had much new to say about it, leaving evaluative aspects aside. By including this ontological alternative I wish to sharpen the understanding for those alternatives which are my main concern. Moreover, an account about the ontology of QFT would be misleading if it did not comprise those conceptions which are held by many researchers after all. These remarks should make it clear that the length of chapters does not indi-

5 cate the importance or popularity of the respective ontological alternatives considered. It could be objected that a complete list of alternatives as well as of all the arguments for and against the single options would be helpful. One might further hope that by doing this one could sort out all but one option which would then be the right one. I agree. This would be a nice thing to have. However, currently we are by no means in a position to do that. We are just at the beginning of an ontological investigation of QFT. Moreover, even if we should finally reach a state of research when we know all the options and have all arguments and counterarguments on the table who tells us that there is one single alternative which would emerge out of this cost-benefit analysis? Structure of Presentation. The first part on ‘Ontology and QFT’ (chapters 2 - 6) is the longest and prepares the ground for the ensuing investigation. The second part (chapters 7 - 9) and the third and fourth part (chapters 10 - 13) are the two halfs of the main study, where I present my own proposal in chapter 13. Although the investigation of ontological aspects of QFT is a rather new area of philosophical research the issue cannot even be nearly understood properly without a recognition and analysis of its historical as well as systematical background. A thorough preparation is indispensible for any further investigations. Accordingly, the introductory chapters in part I are more significant than a merely didactically motivated lead-in. Historically, the discussion concerning atomism is the precursor of today’s ontological examination of quantum physics. The ways of atomism and the role of its ideas in the research process of modern physics will be traced back and explored in chapter 2. The next stride in this line is the embedding of QFT into twentieth century physics. Chapter 4 introduces some basis structures of QFT with an eye on their ontological relevance. I have tried to make it accessible for philosophers with some familiarity with the basic ideas of standard QM and special relativity theory. Chapter 5 deals with some issues that have led to dissatisfaction with standard QFT and which have motivated the search for alternative formulations. In particular, an axiomatic reformulation of QFT called algebraic quantum field theory (AQFT) will be introduced which is of special im-

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CHAPTER 1. INTRODUCTION

portance for considerations in chapters 8 and 13. Section 6.2 reflects on the status of AQFT in relation to standard QFT. The more generally philosophical chapters and sections appear at different places in the introductory part as well as in the main study. The concept of ontology is a very intricate topic inside general philosophy already. Discussions concerning ontology are always apt to provoke much controversy. Some strands of tradition and current stances concerning ontology will be laid out in chapters 3 and 7 as well as in the introductory passages to specific ontological approaches. The first sections in chapter 3 are tailored for readers with no or little background in ontology, in particular physicists. The main investigation of different ontological conceptions for QFT falls into two parts. Chapters 7 - 9 are concerned with classical ontologies, chapters 10 - 13 with revisionary ontologies. The classical-revisionary-split is derived from the division into descriptive and revisionary metaphysics which stems from the philosophical tradition of Analytical Ontology. Although the idea behind this division will be intuitively clear to everyone the matter is not as easy as it looks. The origin and the meaning of these concepts will be given further thought in chapter 7. The concluding part V serves two purposes. In chapter 14 some thought will be given to the question whether and how the physicist and the philosopher can benefit from each other when ontological considerations are concerned. Chapter 15 is the final summary which collects and evaluates the main points of my investigation.

Chapter 2 Philosophical Background The purpose of this chapter is to delineate the tradition of conceptual investigations about theories of nature from ancient until modern times. The emphasis lies on those strands which constitute the context of the present study. In fact, there is a coherent tradition of philosophical thinking about nature from early ancient philosophy to our days. One indication for this coherence is the fact that some of the most outstanding twentieth century physicists like Schr¨odinger and Heisenberg1 put considerable emphasis on the linkage between quantum physics and ancient Greek philosophy. However, something has changed in modern times which makes us less aware of this tradition. Before modern times there was, besides astronomy, no separate discipline corresponding to theoretical physics. The only equivalent in ancient and medieval times can be found in the work of thinkers which we classify as philosophers and theologians today. Although the distinction between philosophy and physics as subjects is relatively sharp today this is not always the case with respect to the people involved. This is particularly so in regard to theoretical quantum physics where conceptual research takes place in a continuum from the physics to the philosophy community. 1

The monographs Schr¨odinger (1954) and Heisenberg (1959) are just two of their explicitly philosophical works.

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2.1

CHAPTER 2. PHILOSOPHICAL BACKGROUND

Atomism in the History of Philosophy

The focus of human thinking about nature has shifted decisively from ancient to modern times. While originally it was mainly speculative, one of the key concepts of modern science is experimental scrutiny. The development of and the debate about atomism is a favourable example in case. The turning point from speculative to experimental emphasis occurred in the seventeenth century. It is very instructive to realize that the elegance and purity of certain philosophical positions was now left behind in favour of eclectic conceptions which were mainly assessed by their explanatory power in view of observable phenomena. From that time onward speculative thinking about nature which did not take scientific results as its starting point came more and more under suspicion. The history of atomism has some aspects which render it particularly interesting in the context of the ontological analysis of quantum field theory (QFT) in this book. It is of exceptional methodological interest since it is one of the few cases of an interplay of philosophy and physics. However, not only this formal aspect of the historical development of atomism is closely related to the issues treated in this work but also the content. First, it is a famous example for an ontological debate and, second, it can in some respects be seen as the forerunner of the questions which will be pursued regarding QFT. After all, the immediate candidates for the most basic entities to which QFT refers are the modern counterparts of atoms, namely electrons, quarks etc. Most expositions of the history of atomism simply presuppose that the atomistic account of matter has turned out to be the correct view so that historical studies can use this yardstick in order to evaluate how well former philosophers have come off in making the right guess even without empirical evidence. This procedure is flawed in at least two respects which mirror the more general arguments in the opening paragraphs of this chapter. First, the ‘atomistic account of matter’ consists of a large spectrum of views and one could defend the position that there are non-atomistic accounts which are closer to the results of modern physics than some of the atomistic views. Second, if one takes atomism to be the view that the material world consists of unchangeable atoms (whatever they are) on the one hand and the void on the other hand then one has to conclude that modern physics actually arrived at a non-atomistic point of view. One can add as a third

2.1. ATOMISM IN THE HISTORY OF PHILOSOPHY

9

point that there is no uniform and fixed point of view of modern physics anyway. By these considerations I wish to point out that an examination of various older atomistic views as well as the arguments for and against atomism might have more than just historical value. It may well be that looking back at ancient thinking about atomism reveals conceptual results which are helpful for an understanding of today’s physics as well. The lack (and neglect) of empirical evidence in premodern times might even turn out to have some virtue in it. The variety of different conceptions is larger and the argumentation richer since there is nothing else one could appeal to. Atomism has emerged as a proposal to solve the conflict between two opposite views in the sixth and early fifth century BC.2 On the one side Heraclitus was famous for maintaining that everything is in flux. Whether or not Heraclitus as a historical figure is correctly described by this statement is not very important in our context. What matters more is that this view was ascribed to him by his contemporaries, possibly by Parmenides, certainly by various later philosophers. Parmenides on the other side believed that the impression of change is just an illusion. Both philosophers had more or less convincing arguments for their views but obviously they could not both be right. In the fifth century BC Leucippus had an elegant compromise to offer which Democritus worked out to a consistent philosophy. They agreed with Parmenides that change is inconceivable on the fundamental level of the material world. Instead of then denying the possibility of any change, however, they assumed basic unchangeable building blocks out of which everything else is composed. Since these building blocks were thought of to be the smallest parts of matter they were called ,
atoms (greek α τ oµoς, uncuttable, undivisible). With the assumption of atoms it was possible for Leucippus and Democritus to give an account of observable change in the world without admitting any change on the basic level of atoms. The recombination of atoms is responsible for the change on higher levels. Things change but the atoms out of which these things are composed stay unchanged. Atomism thus allows to maintain the view 2

Dijksterhuis (1956) and Sambursky (1962) are two classical monographs. van Melsen (1967) is a good short account of atomism. Pabst (1994) although having atomism in the Middle Ages as its main issue contains a helpful and modern summary of ancient atomism as well as an interesting evaulation of references.

CHAPTER 2. PHILOSOPHICAL BACKGROUND

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that matter is basically or intrinsically unchangeable while at the same time accounting for change in the world of experience without marking the impression of change as a mere illusion.3 Let us have a closer look at the origin and some details of the first atomistic theories. For Democritus atoms can only differ in shape and size but not in any other properties. Since ancient times it is a common practice to classify this opinion by saying that Democritus’ atoms differ quantitatively but not qualitatively.4 The multiplicity of observable appearances stems from the infinite number of possibilities for their combination. The elegance of this atomistic theory of matter becomes clear on the background of some conceptual problems to which it provides a solution. The Eleatic philosopher Parmenides and his favourite student Zeno considered paradoxes which still today confuse most people when they first learn about them. Zeno’s most famous paradoxes have to do with an infinite division of a given space or time interval and the sum of these infinitely many ever smaller quantities. Zeno’s arguments hinge on the implicit presupposition that an infinite sum of non-vanishing positive numbers, say 1 1 1 1 2 + 4 + 8 + 16 + ..., must give an infinite result. Zeno has a twofold r´esum´e. First, movement is an illusion since any given process in time can be divided and summed up as above showing that it would have to take infinitely long. Second, divisibility is not conceivable since it would lead to the conclusion that things must have either infinite or zero size. These so-called problems of the continuum are a vital part of the background on which atomism appeared. Leucippus and Democritus had the idea to solve Zeno’s paradoxes by saying that things are in fact divisible but only at certain places, viz. in the void between atoms.5 3

Further ancient theories of atomism can be found for instance in the work of Diodorus and Epicurus in the early third century BC. Conceptually their points of view do not differ very much from the one of Democritus with the exception of the introduction of the new distinction between physical and mathematical atomism. 4 See, e. g., van Melsen (1967), p. 194f. Although I can see the point I doubt that it is a good characterisation especially with respect to the property of shape. At most I would mark this difference as between extrinsic and intrinsic properties. 5 In modern times it turned out that there is an easier solution to Zeno’s paradoxes. Zeno was simply mistaken in assuming that an infinite series of non-vanishing positive numbers must always give an infinite sum. Whether or not there is a finite sum depends on how fast the terms in this series are decreasing. It is not the case that Achill never reaches the tortoise. In a way Zeno’s thought experiment corresponds to slow motion

2.1. ATOMISM IN THE HISTORY OF PHILOSOPHY

11

The next important step in the history of atomism is Plato’s theory. In his amply discussed dialogue Timaios Plato seems to lay out an atomistic theory of matter which is reminiscent of highly mathematical theories in twentieth century physics. Plato takes up Empedocles’ famous doctrine (fifth century B. C.) which assumes four basic elements, viz. earth, water, air and fire, out of which everything is composed. While it is an open question whether Empedocles had a corpuscular theory of matter in mind Plato linked each element to a kind of atom which is characterized by a certain geometrical figure, for instance the tetrahedron in the case of fire. Using the somewhat problematic quantitive/qualitative-distinction again one can say that Plato proposed a hybrid between theories based on the permanence of qualitative properties - as Empedocles did - and theories where quantitive properties are taken to be basic - as in Democrit’s atomism. Evaluating Plato’s theory, it has to be pointed out that the similarity with the role of mathematical elegance and beauty in the development of theories in modern physics is superficial in at least one important respect. In contrast to these theories Plato had no empirical basis for his thesis. Plato’s theory was highly speculative and—in contrast to modern physical theories—it was not meant to be checked by later experiments.6 There are various opponents to atomism in antiquity. Most influential among them is Aristotle who, like the Stoics, defended the continuum thesis. The writings of the Sceptic Sextus Empiricus include a valuable collection of arguments against atomism. Most expositions of the historical development of atomistic theories simply make a jump from late antiquity to early modern times without further commentary. The classical standard view about the history of atomism in the Middle Ages is that there is no such history.7 However, the modern research on the significance and dissemination of atomistic ideas in the which is getting ever slower. But this does not make the observed process itself infinitely long. 6 More recent studies show that it is not clear whether Plato’s atomism aimed at physics at all. It is possible that the objective is an ethical model rather than a physical theory. Lothar Sch¨afer, emeritus professor of Hamburg University, propounded this thesis in his talk “Naturordnung und Herrschaft in Platons Timaios” at the University of Bremen, 2nd of February 2000. 7 For a long time the most notable exception was Lasswitz’ famous history of atomism from the Middle Ages until Newton Lasswitz (1890).

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12

Middle Ages is surprisingly interesting, and a difficult issue in itself. The prejudices about the supposedly dark Middle Ages with little independent and productive intellectual life is largely responsible for this inappropriate evaluation. In the case of atomism the facts were almost turned upside down. In his study on the history of atomistic theories in the Latin Middle Ages Bernhard Pabst (1994) could prove that there were at least 18 medieval advocates of atomistic theories. According to Pabst one can claim that in a certain period of the Middle Ages atomism was more widespread and readily accepted than at any time in antiquity. And not only that. Contrary to the common belief that medieval thought was based on an uncritical trust in authorities neither did any of the medieval authorities held an atomistic view nor did any of the above-mentioned 18 medieval adherents of atomistic theories copy an ancient theory of atomism. Only the general idea of atomism was transmitted from antiquity to the Middle Ages. The heyday of medieval atomism was between 1100 and 1150. None of the atomistic theories was based on a materialistic view of the world which was a central trait of ancient atomism. This difference is clearly due to the predominance of religion and theology in medieval thought. The atomists of the Middle Ages aimed at pragmatic explanations for natural phenomena. It is not only this neutral attitude which renders medieval atomism as closer to modern thinking about nature than ancient theories. Various developments over ancient conceptions were generated and vividly discussed. From the modern point of view the major improvement was the qualitative charaterisation of atoms. In the second half of the twelfth century the atomistic movement declined. Aristotle’s works became accessible through translations of Arabic copies and with these translations Aristotle’s arguments against atomism spread.8 Since the medieval versions of atomism in the 1100-1150-period were particularly vulnerable for Aristotle’s arguments atomism quickly lost its popularity. 8

Plato’s Timaios had no influence on this period of medieval atomism since the only available translation by Calcidius contains just the first half of the Timaios which does not comprise Plato’s corpuscular theory.

2.2. PHILOSOPHICAL VERSUS SCIENTIFIC ATOMISM

2.2

13

Philosophical Versus Scientific Atomism

Atomism has a philosophical and a scientific dimension. Until the 17th century it was primarily a philosophical issue. From that time onward the focus shifted to the scientific interest. Central in this shift was the identification of kinds of atoms with chemical elements by Dalton around the turning point from the 18th to the 19th century. The shift from ancient, medieval and early modern philosophical theories of atomism to scientific theories of atomism can be characterized by an essential change of emphasis and aims. While philosophical theories of atomism tried to explain the very possibility of change on the basis of something stable, scientific theories of atomism aim at the explanation and prediction of quantitative details of observable phenomena. This fundamental difference of intentions brings about a difference in the character of reasoning as well. Philosophical theories of atomism often had a tendency to be dogmatic about a number of aspects. It is simply postulated that at a certain level matter is not divisible any more. Moreover, it is assumed that there is a certain number of kinds of atoms with certain spatial structures. After all, no more was intended than to find arguments for the general conceivability of changing qualities without the need to assume that everything is in flux which would render the world utterly inexplicable and barred to cognition. Scientific theories of atomism have a very different standard for the evaluation of atomistic theories. The detailed properties of atoms do matter a lot since they have consequences for the prediction of observable phenomena. Atomistic models are as good as they give accurate quantitative predictions for experiments. Philosophical inconsistencies are less acutely felt than a lack of numerical precision. Since the experimental checking yields a very fine method of control for scientific theories of atomism there is less dogmatism to be found here than in the philosophical period of atomism. Due to experimental results the assumptions about atoms in scientific theories can undergo fundamental revisions. As we see it today atoms are actually divisible and what takes their place as the smallest building blocks (maybe quarks and leptons, maybe superstrings) are not eternal entities. They can undergo change themselves and can even begin and cease to exist. Modern scientific theories of atomism are so different from philosophical theories because it has turned out that only radically modified theories are successful in predicting observable phenomena.

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There is another characteristic change from ancient and medieval philosophical theories of atomism to scientific theories of atomism. While philosophical atomism is very figurative and visualizable, scientific atomism has a different emphasis. When looking at scientific atomistic models it is rated much higher how mathematically elegant and simple its description is and how numerically manageable and precise the predictions are. One consequence of this scientific attitude is the fact that the resulting atomistic theories loose their connection to the way we conceive of the natural world. Instead, they are primarily predictive tools with little impact on our picture of the world. One of the aims of this book is the attempt to fill a part of this gap by pointing out which ways to imagine the natural world are compatible with current theories of modern physics.

2.3

Atomism and Reductionism

I conclude this chapter with some brief remarks about a general attitude behind the search for atomistic explanations and my own stance towards it. Atomism can be seen as one form of reductionism with its assertion that everything can be reduced to some basic building blocks. In order to clarify possible ways to understand this assertion and some of its consequences it is helpful to introduce the distinction between methodological and ontological reductionism. A methodological reductionist holds that all scientific theories can and should be reduced to a fundamental theory which is generally taken to be found in physics. One can express this claim the other way round as well. Whenever something is to be explained one has to start with the most basic theory and derive the explanation for the phenomenon in question by specifying a sufficient number of constraints and boundary conditions for the general fundamental laws. An example for this procedure is the explanation of an atomic spectrum by calculating the excited energy states of a many-body system containing the relevant constituents. An ontological reductionist is more modest. He agrees with the methodological reductionist that the reduction of higher-level theories to lowerlevel theories is possible in principle. However, the ontological reductionist thinks that a reduction to the lowest possible level is often neither practically feasible nor even desirable. In order to explain the shift of power in the

2.3. ATOMISM AND REDUCTIONISM

15

last election nobody is interested in getting information involving quarks, gluons and electroweak interaction. And even in the case of the spectrum of a uranium atom this is not the appropriate point to start. Note that methodological reductionism presupposes ontological reductionism but not vice versa. The attitude of the ontological reductionist, to which I agree, is not without consequences for the practice of science. Although an actual reduction might often not be completely feasible, aiming at it can sometimes have a high methodological fruitfulness. The Laser Theory as established by H. Haken and others is a famous example. In this theory it was attempted to reduce the already known phenomenon of laser light to the most basic theory, i. e. QFT. Although certain less profound explanations for laser light were available before the establishment of Laser Theory researchers felt that more could be done. My point now is that it is almost of secondary importance whether the aim of a complete reduction to the most fundamental laws was actually achieved since while attempting to get there numerous technically highly significant effects and modulation possibilities were discovered. This would never have been achieved if one had been satisfied with less fundamental explanations. Closely connected with the issue of reductionism is the one of realism, both of which are almost two sides of the same medal.9 Again I agree with the attitude of the realist and again out of the same main reason as above. I think that the realist’s stance has a high heuristic value, no matter whether he is actually “right” with his attitude in the end. I think that the realist’s belief in the existence and recognizability of an external world is methodologically much more fruitful than the point of view of an antirealist. The antirealist cuts off a lot of questions whose investigation can yield interesting insights and discoveries. 9

According to Bartelborth (1997) the core claims of (scientific) realism are, first, that most entities that are postulated in our accepted scientific theories exist, second, in a way that is independent of our opinions and cognitive capacities where, and this is the third claim, the postulated entities have more or less the properties that are specified by our theories. The phrase “entities that are postulated in our accepted scientific theories” is equivalent to the widespread expression “theoretical entities’. While strong realists hold on to all three claims, advocates of a weaker realism accept only some claims. Ian Hacking, e. g., accepts only the first and the second claim. For more detailed expositions see Bartelborth (1997) and Glymour (1992).

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I think that the same is true for the philosophical investigation of scientific theories as well. Although the analysis undertaken in this book would loose some of its relevance without a realist’s attitude I think it is still of interest for those who do not share this standpoint. It seems to me that the truth of realism or antirealism is of minor importance in the end. What really matters is how fruitful each of these attitudes is for the actual research whether it be in natural sciences or in philosophy.

Chapter 3 Ontology and Physics Through its history the philosophical subdiscipline ontology has undergone various fundamental changes and gave rise to severe criticism. More than in regard to most other disciplines it has often been questioned whether ontology constitutes a genuine field of research at all. In some quarters, at least, it lingers the impression that the ontologist can do nothing more than the scientist or that there is no yardstick to measure possible answers given to the questions of ontology. The notion and the plan of an ontological analysis as such need clarification and justification. Viewing the issue in its historical perspective can both illuminate the opposing points of view and sharpen the understanding of the approach taken in this work.

3.1

Some Main Themes in Ontology

In short, ontology as a philosophical discipline is concerned with existence, or being, in generality. It “is a general theory of everything”.1 The range of philosophical investigations which go by the name ‘ontology’ is large, in particular because of a wide divergence about what to consider as interesting and legitimate questions. The emphasis can be either on the meaning of the concept ‘being’ or on the more specific question of what there is in the world in the most general sense.2 In the first case, the ontologist can be interested in the famous question “why is there something rather than nothing?” and he can further search for the principles or reasons of “being 1

Simons (1998a), 251. “On What There is” is the title of Quine’s famous paper Quine (1948) which is reprinted in Quine (1961). 2

17

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as being”. In the second case, the ontologist asks which types of objects and modes of being there are and which general properties they have, can have or must have. He considers which entities are the fundamental ones and which entities are only derived in whatever sense. The present work is in the vein of this second emphasis in ontology since the investigations start on the basis of a given theory, viz. QFT, and ask to which basic entities it refers. For now I leave it open whether this specific question is just one among various further equally important enterprises which consider other sciences and fields of reality. A number of examples of classical debates with at least strong ontological components should help to make the general orientation of this second approach to ontology clearer. The first exemplary debate is concerned with the ontological status of properties. While Plato maintained that properties like redness exist as universals outside of our mind others argued that there are only particulars like one red flower. These two traditional points of view are called realism about universals on the one side and nominalism on the other side. The vivid medieval debate about this issue is particularly famous. The current debate about the ontological status of properties will be the topic of chapter 11 and plays an important role for the ensuing chapters. Another discussion concerns the existence and relation of mind and matter. While the materialist claims that everything, including mental states, can be reduced to matter, the classical Cartesian dualist contends that there is a dichotomy of res cogitans and res extensa as the two fundamental and thus irreducible parts of reality. On the very opposite side of materialism lies idealism, propounded by Berkeley, with the view that there are only mental objects, called ‘ideas’, which are in the mind of God. A third and last example for a debate with largely ontological significance is the one about space and time. Is existence only possible in space and time? In connection to this question follows another one. Which kind of reality do numbers and laws of nature have? What is the ontological status of space and time themselves? Do they have an existence independent of material objects (substantialism) or are they only a means to describe the relation of these objects (relationism)? Kant argued that all appearances are in space and time but these are only the form in which the things-inthemselves are inevitably given to us.

3.2. A BRIEF HISTORY OF ONTOLOGY

19

As one can see there often is a broad spectrum of possible answers to ontological questions. When pursuing one specific ontological question it is helpful to place the questions as well as the proposed solutions in their historical and systematical context. Hard questions usually reoccur in disguise so that a knowledge of other debates can help to evaluate the debate in which one is primarily interested. The following short historical sketch of the development of ontology as a philosophical discipline is meant to further explicate the understanding of ontology that underlies the present study on QFT by placing it in its historical as well as systematical context.

3.2

A Brief History of Ontology

The history of philosophical investigations which are explicitly labeled ‘ontology’ begins with scholastic philosophers in the seventeenth century.3 Nevertheless the enterprise itself was important in ancient times already. Parmenides considered ‘being’ as the central philosophical concept. And notably Aristotle put much thought into ‘being as being’.4 Although the expressions ‘metaphysics’ and ‘ontology’ were both unknown to Aristotle the issues which go by these names can be found at length in Aristotle’s works, notably in his most important work which was later given the name Metaphysics. The importance of Aristotle’s contributions for ontology does not only consist in the answers given by him but particularly in his explicit reflections about the nature of ontology as a philosophical discipline, his illuminating formulations of those questions ontology tries to tackle and, last not least, in the way how Aristotle addresses these questions. Instead of ‘metaphysics’ Aristotle himself speaks of ‘first philosophy’. The relation of metaphysics and ontology has long been and still is a point of disagreement. While in the seventeenth century both terms were mostly taken to be synonymous, in the traditional view (e. g. Christian Wolff) ontology is classified as a part of metaphysics besides cosmology and psychology. In the twentieth century there are strands which consider 3

The term itself was probably coined by Rudolf Goclenius and first appeared in his “Lexicon philosophicum” published in 1613. 4 Metaphysics Γ (i. e. book IV),1, 1003 a 21.

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20

ontology to be metaphysically neutral5 or they are even explicitly critical of metaphysics in contrast to ontology (Heidegger). In the eighteenth century Christian Wolff canonized and completed ontology as a discipline. Since Wolff’s time ontology is separated from natural theology in contrast to the Aristotelian tradition in which first philosophy and metaphysics in the sense of natural theology were seen as a unity. In the Leibniz and Wolff tradition ontology was a matter of necessary truths which are derivable from unquestionable first principles like the principle of contradiction and the principle of sufficient reason. Thus in order to pursue ontology one does not have to look at the actual order of the world. According to this line of thought particular sciences6 have no relevance for questions of ontology. Kant considered this traditional understanding of ontology to be presumptuous and refuted it by showing that a priori inquiries can only find out something about the general form of possible experience. In Kant’s view one cannot say anything substantial (i. e. synthetic) about the contents of experience without first making experiences. A vital step in Kant’s argumentation against the derivability of ontological matters from first principles was his ‘second antinomy of pure reason’ which demonstrates internal contradictions in the traditional conception of ontology. The intention of Kant was to replace traditional ontology by his - in comparison - moderate transcendental philosophy. Another famous part of Kant’s philosophy is his argument against the ontological proof of the existence of God which will be discussed in a bit more detail in chapter 6.2. Kant argued that existence must not be considered as a property which something can have in addition to other properties.

3.3

The Analytical Tradition of Ontology

Analytical ontology can be characterized as the rehabilitation of the old ontological questions while maintaining the new and powerful instruments 5

For instance Simons (1998b). Although I would like to use the expression ‘special sciences’ I have chosen to talk of ‘particular sciences,’ including physics, in order to avoid confusion with Fodor’s wellknown usage (1974) of ‘special sciences’, excluding physics. 6

3.3. THE ANALYTICAL TRADITION OF ONTOLOGY

21

of the philosophical analysis of language. It is mostly within this framework that the investigation in this work will be carried through. One important reason is the fact that the analysis of languages is structurally similar to the investigation of scientific theories. The three most important steps of the rehabilitation of metaphysics within the analytical tradition are marked by Carnap, Quine and Strawson, with an increasingly positive attitude. Carnap’s philosophy rests on a radical criticism of the very question of traditional ontology.7 Carnap makes a distinction between internal and external questions about existence. Internal questions ask for the existence of an entity within a given “linguistic framework”. An example for Carnap’s broad notion of a linguistic framework is the order of space and time and the entities one can ask for could be material everyday things in space and time. External questions concern the existence of the very linguistic framework itself. In our example this would be the question whether (the points of) space and time exist. Carnap considers external questions concerning the reality of the framework itself as pseudo questions without cognitive content. In Carnap’s view the choice of the linguistic framework is a matter of pure convention. The only sensible measure is how practical the choice is. According to Carnap, any further questions lead astray. Quine rejected Carnap’s view by denying the tenability of his basic distinction between internal and external questions. Quine pointed out that there are no clear criteria to make this distinction. Instead, Quine prefers to speak of the “ontological commitments” of a theory. Suppose that a given theory T is a true theory, Quine asks which are the ontological commitments. One famous definition of ontological commitments of a theory runs as follows: a theory is commited to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true. Quine (1948) Somewhat more succinct is Quine’s well-known slogan “To be is to be a value of a variable.”8 Another way to express Quine’s attitude is to say 7

“Empiricism, Semantics, and Ontology” Carnap (1950), which is reprinted in Carnap (1956), is a classical source for his attitude towards ontology. 8 Quine in a talk in 1939.

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that ontology asks for the truth makers of a given theory (or language). What is looked for are those entities which one has to assume in order to make a particular theory (or language) true. According to Quine the truth of the theory itself is not the question in philosophy. Quine argues that the most appropriate way to ontology is to look at the best science available at a time. The attitude which underlies my following investigation about ontological aspects of QFT is in its rough outline akin to the Quinean approach. I agree with Quine that the “best science available” is of pivotal importance for general ontological questions. And I agree with Quine that the attempt to discover and unfold the ontological commitments of the relevant scientific theories is a fruitful approach. Nevertheless, there are some points where I disagree with Quine. I do not believe that the situation is as easy as Quine describes it. It is not as if one could simply take the “best science available”, determine its ontological commitments and there you have the best ontology available. It seems to me that there is no straightforward way to figure out to which entities a given theory is committed. I think that probably all one can achieve is to find out which ontological conceptions are compatible with a given scientific theory. In the case of quantum physics I believe the most one can hope for is that one finds an ontological conception which explains puzzling features most naturally. But I am afraid that such arguments will never have the power of a commitment. Moreover, neither is it always clear what the “best science available” is. I will address this problem specifically with respect to quantum field theory in chapters 5 and 6. In the case of quantum physics (be it non-relativistic or relativistic) it turned out that one powerful tool for actually carrying out ontological studies is indirect since it is sometimes easier to exclude some options than to positively say what the ontological commitments are. Since these so-called ‘no-go theorems’ are pivotal for some steps in the ensuing investigation I will briefly describe their role in quantum physics and in particular their significance for ontological studies.

3.4. NO-GO THEOREMS AS TOOLS FOR THE ONTOLOGICAL 23

3.4

No-Go Theorems as Tools for the Ontological Practician

There are different ways to find out something about the ontology of a scientific theory. For QM and QFT a very precise and successful method is to look for no-go theorems like the famous one by John Bell on hidden variable theories or a more recent one found by David Malament on the impossibility of a certain particle interpretation for relativistic QM.9 The advantage of such no-go theorems is a very high degree of precision. However, most no-go theorems suffer from a very limited scope. Malament’s no-go theorem for instance only shows that non-relativistic QM of a fixed number of localizable particles cannot be reconciled with relativity theory. It thus does not rule out a particle interpretation for QFT because here the precondition of a fixed number of particles is not met. The relevance of this no-go theorem for the interpretation of QFT is not immediately clear therefore. Further thought is thus necessary for an understanding of its ontological significance with regard to QFT. I will depict and discuss Malament’s no-go theorem in the second part of section 8.3.1. The importance of no-go theorems in quantum physics rests partly on the fact that there is no undebated correct way to understand various entities appearing in the formalism of QM, which I will introduce in chapter B. So far, the most successful way to handle this situation is the construction or discovery of proofs which demonstrate that certain sets of assumptions (e. g. locality, separability, determinism, value definiteness of all possible physical quantities etc.) lead to contradictions. Assuming that an interpretation of a (piece of) formalism can sometimes be condensed into a set of assumptions one can thus at least exclude some interpretations. Since one can by this exclusion procedure show that one or the other interpretative option is not an admissible way to go these results are called ‘no-go theorems’. There are three particularly famous examples for no-go theorems. The first one is John von Neumann’s alleged proof of the impossiblity of hidden variable theories, i. e. theories that assume further not directly accessible quantities in addition to the ones that occur in standard QM. Bohmian 9

See “In defense of dogma: Why there cannot be a relativistic quantum mechanics of (localizable) particles” by Malament (1996).

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quantum mechanics is the best-known example of a hidden variable theory. It is not hit by von Neumann’s proof, however, since it later turned out that this proof rests on implicit assumptions which narrow the applicability of his result considerably.10 Von Neumann’s proof is not a legitimate argument against Bohm’s version of quantum mechanics since this is an explicitly holistic, or non-local, theory, contrary to the assumptions of von Neumann’s proof. The second famous example of a no-go theorem is Bell’s theorem.11 Here the so-called ‘Bell inequalities’ are derived under certain conditions and it can be shown that they are violated by QM as well as directly by experiments.12 The third example are non-objectification theorems against the possibility of an ignorance interpretation of QM, a keyword here are nonvanishing ‘interference terms’.13 I will not go into any further details of these three examples since the main purpose of this section was to highlight only the general significance of no-go theorems for ontological investigations with regard to quantum physics. Moreover, complete no-go theorems will be given in the course of this work. One further issue that is particularly important for ontological analyses of physical theories are symmetries.

3.5

Symmetries, Heuristics and Objectivity

Symmetries play a central role in QFT. In order to characterize a special symmetry one has to specify transformations T and features, which remain unchanged during these transformations (invariants I), symmetries are thus pairs {T, I}. The basic idea is that the transformation change elements of the mathematical description (the Lagrangians for instance) 10

Jammer (1974) is the authoritative account of the historical background as well as of the change in the evaluation of Neumann’s alleged proof. 11 See “On the problem of hidden variables in quantum mechanics” Bell (1966) and “On the Einstein-Podolsky-Rosen paradox” Bell (1964). Interestingly, Bell himself seems to have been annoyed by this business and in “On the impossible pilot wave” Bell (1982), he wondered “[...] why did people go on producing ‘impossibility’ proofs [...]”, p. 160 in Bell (1987), mentioning famous names like J. M. Jauch, C. Piron, B. Misra, S. Kochen, E. P. Specker, S. P. Gudder and, last but not least, himself!? Probably the reason for this stance are almost ideological feelings concerning Bohmian Mechanics. 12 See, e. g., Redhead (1987) for details. 13 See chapter 4 in Mittelstaedt (1998).

3.5. SYMMETRIES, HEURISTICS AND OBJECTIVITY

25

whereas the empirical content of the theory is unchanged. There are spacetime transformations and so-called internal transformations. Whereas spacetime symmetries are universal, i. e., they are valid for all interactions, internal symmetries characterize special sorts of interaction (strong, electromagnetic or weak interaction). Symmetry transformations define properties of particles/quantum fields which are conserved if the symmetry is not broken. The invariance of a system defines a conservation law, e. g., if a system is invariant under translations the linear momentum is conserved, if it is invariant under rotation the angular momentum is conserved. Inner transformations such as so-called gauge transformations are connected with more abstract properties. Symmetries are not only defined for Lagrangians but they can also be found in empirical data and phenomenological descriptions. Symmetries can thus bridge the gap between descriptions which are close to empirical results (‘phenomenology’) and the more abstract general theory which is a most important reason for their heuristic force. If a conservation law is found one has some knowledge about the system even if details of the dynamics are unknown. The analysis of many high energy collision experiments led to the assumption of special conservation laws for abstract properties like baryon number or strangeness. Evaluating experiments in this way allowed for a classification of particles. This phenomenological classification was good enough to predict new particles which could be found in the experiments. Free places in the classification could be filled even if the dynamics of the theory (for example the Lagrangian of strong interaction) was yet unknown. As the history of QFT for strong interaction shows considerable constraints on the construction of dynamics follow from symmetries found in the phenomenological description: The Lagrangian should not exhibit less symmetries than the phenomenology. Arguments from group theory played a decisive role in the unification of fundamental interactions. In addition, symmetries bring about substantial technical advantages. For example, by using gauge transformations one can bring the Lagrangian into a form which makes it easy to prove the renormalizability of the theory. Symmetries are not only instruments of provisional physics used in not yet fully developed theories. Symmetries also supply some sort of ‘justification’, they are often used in the beginning of a chain of explanation.

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To a remarkable degree the present theories of elementary particle interactions can be understood by deduction from general principles. Under these principles symmetry requirements play a crucial role in order to determine the Lagrangian. For example, the only Lorentz invariant and gauge invariant renormalizable Lagrangian for photons and electrons is precisely the original Dirac Lagrangian. In this way symmetry arguments acquire an explanatory power and help to minimize the unexplained basic assumptions of a theory. Heisenberg concludes that in order “to find the way to a real understanding of the spectrum of particles it will therefore be necessary to look for the fundamental symmetries and not for the fundamental particles.” (Blum et al. (1985), p. 507). Since symmetry operations change the perspective of an observer but not the physics an analysis of the relevant symmetry group can yield very general information about those entities which are unchanged by transformations. Such an invariance under a symmetry group is a necessary (but not sufficient) requirement for something to belong to the ontology of the considered physical theory. Hermann Weyl propagated the idea that objectivity is associated with invariance14 Symmetries help to separate objective facts from the conventions of descriptions. (See Kosso in Brading and Castellani (2003).) Symmetries are typical examples for such abstract mathematical structures that show much more continuity in scientific change than assumptions about the entities of a theory (light as particles, as waves, as quantum fields). For that reason structural realists consider abstract structures as “the best candidate for what is ‘true’ about a physical theory” (Redhead (2002b), p. 34). Physical entities (like electrons) are similar to fictions and, in the end, should not be taken seriously. In the epistemic variant of structural realism structure is all we know about nature whereas the objects which are related by structures might exist but they are not accessible to us. In the ontic variant of structural realism nature seems to be reduced to mathematical objects. In the world, there is nothing but structure.15 14

See, e.g., his authoritative work Weyl (1952), p. 132. For a detailed account of structural realism see Ladyman (1998). Symmetry considerations are also of central importance in Auyang (1995) where the connection between properties of physically relevant symmetry groups and ontological questions is stressed. A recent anthology with various philosophical studies about symmetries in physics is Brading and Castellani (2003). 15

Chapter 4 History and Basic Structure of QFT 4.1

The Early Development

The historical development of QFT is very instructive until the present day.1 Its first achievement, namely the quantization of the electromagnetic field is “still the paradigmatic example of a successful quantum field theory” Weinberg (1995). Ordinary QM cannot give an account of photons which constitute the paradigmatic case of relativistic ‘particles’. Since photons have the rest mass zero, and correspondingly travel in the vacuum at the velocity, naturally, of light c it is ruled out that a non-relativistic theory such as ordinary QM could give even an approximate description. Photons are implicitly contained in the emission and absorption processes which have to be postulated, for instance, when one of an atom’s electrons makes a transition from a higher to a lower energy level or vice versa. However, only the formalism of QFT contains an explicit description of photons. Looking back one would say that most topics in the early development of quantum theory (1900-1927) were related with the interaction of radiation and matter and should be treated by quantum field theoretical methods. However, the way to quantum mechanics formulated by Dirac, Heisenberg and Schr¨odinger (1926/27) started from atomic spectra and did not rely very much on problems of radiation. As soon as the conceptual framework of quantum mechanics was developed, a small group of theoreticians im1

The first chapter in Weinberg (1995) is a very good short description of the earlier history of QFT. Detailed accounts of the historical development of QFT can be found, e. g., in Darrigol (1986), Schweber (1994) and Cao (1997). Various historical and conceptual studies of the standard model are gathered in Hoddeson et al. (1997) and of renormalization theory in Brown (1993).

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CHAPTER 4. HISTORY AND BASIC STRUCTURE OF QFT

mediately tried to extend the methods to electromagnetic fields. A good example is the famous three-man paper von M. Born, W. Heisenberg, and P. Jordan (1926). Especially P. Jordan was acquainted with the literature on light quanta and made important contributions to QFT. The basic analogy was that in QFT field quantities, i. e. the electric and magnetic field, should be represented by matrices in the same way as in QM position and momentum are represented by matrices. The ideas of QM were extended to systems having infinite degrees of freedom. The inception of QFT is usually dated 1927 with Dirac’s famous paper Dirac (1927) on “The quantum theory of the emission and absorption of radiation.” Here Dirac coined the name quantum electrodynamics (QED) which is the part of QFT that has been developed first. Dirac supplied a systematic procedure for transferring the characteristic quantum phenomenon of discreteness of physical quantities from the quantum mechanical treatment of particles to a corresponding treatment of fields. Employing the quantum mechanical theory of the harmonic oscillator, Dirac gave a theoretical description of how photons appear in the quantization of the electromagnetic radiation field. Later, Dirac’s procedure became a model for the quantization of other fields as well. During the following three years the first approaches to QFT were further developed. P. Jordan introduced creation operators for fields obeying Fermi statistics. So the methods of QFT could be applied to equations resulting from the quantum mechanical (field like) treatment of particles like the electron (e. g. Dirac equation). Schweber points out (Schweber (1994), p. 28) that the idea and procedure of that “second quantization” goes back to Jordan (1927) while the expression itself was coined by Dirac. Some difficult problems concerning commutation relations, statistics and Lorentz invariance could be solved. The first comprehensive account of a general theory of quantum fields, in particular the method of canonical quantization, was presented in Heisenberg and Pauli (1929). Whereas the actual objects of Jordan’s second quantization procedure are the coefficients of the normal modes of the field, Heisenberg and Pauli (1929) started with the fields themselves and subjected them to the canonical procedure. Heisenberg and Pauli thus established the basic structure of QFT which can be found in any introduction to QFT up to the present day. Fermi and Dirac, Fock and Podolski presented different formulations which played a heuristic role

4.1. THE EARLY DEVELOPMENT

29

in the following years. Quantum electrodynamics, the historical as well as systematical entr´ee to QFT, rests on two pillars.2 The first pillar results from the quantization of the electromagnetic field, i. e. it is about photons as the quanta or quantized excitations of the electromagnetic field. As Weinberg points out the “photon is the only particle that was known as a field before it was detected as a particle” so that it is natural that QED began with the analysis of the radiation field (see Weinberg (1995), p. 15). The second pillar of QED consists in the relativistic theory of the electron, with the Dirac equation in its centre. The easiest way to quantize the electromagnetic (or: radiation) field consists of two steps. First, one Fourier analyses the vector potential of the classical field into normal modes (using periodic boundary conditions) corresponding to an infinite but denumerable number of degrees of freedom. Second, since each mode is described independently by a harmonic oscillator equation, one can apply the harmonic oscillator treatment from non-relativistic quantum mechanics to each single mode. The result for the Hamiltonian of the radiation field is Hrad =



k

r




ωk a†r (k)ar (k)

1 + , 2

(4.1)

where a†r (k) and ar (k) are operators which satisfy the following commutation relations [ar (k), a†s (k
)] = δrs δkk
[ar (k), as (k
)] = [a†r (k), a†s (k
)] = 0.

(4.2)

with the index r labeling the polarisation. These commutation relations imply that one is dealing with a bosonic field. The operators a†r (k) and ar (k) as well as their product a†r (k)ar (k) have interesting physical interpretations as so-called particle creation and annihilation operators. In order to see this one has to examine the eigenvalues of the operators (4.3) Nr (k) = a†r (k)ar (k) 2

See, for instance, the short and lucid “Historical Introduction” of Scharf’s original book Scharf (1995).

30

CHAPTER 4. HISTORY AND BASIC STRUCTURE OF QFT

which are the essential parts in Hrad . Due to the commutation relations (4.2) one finds that the eigenvalues of Nr (k) are the integers nr (k) = 0, 1, 2... and the corresponding eigenfunctions (up to a normalisation factor) are (4.4) | nr (k) = [a†r (k)]nr (k) | 0  where the right hand side means that a†r (k) operates nr (k) times on | 0, the state vector of the vacuum with no photons present. The interpretation of these results is parallel to the one of the harmonic oscillator. a†r (k) is interpreted as the creation operator of a photon with momentum
k and energy
ωk (and a polarisation which depends on r and k). That is, equation (4.4) can be understood in the following way. One gets a state with nr (k) photons of momentum
(k) and energy
ωk when the creation operator a†r (k) operates nr (k) times on the vacuum state | 0 . Accordingly, Nr (k) is called the ‘number operator’ and nr (k) the ‘occupation number’ of the mode that is specified by k and r, i. e. this mode is occupied by nr (k) photons. Note that Pauli’s exclusion principle is not violated since it only applies to fermions and not to bosons like photons. The corresponding interpretation for the annihilation operator ar (k) is parallel, when it operates on a state with a given number of photons this number is lowered by one. It is a widespread view (see e. g. Ryder (1996), p. 131) that these results complete “the justification for interpreting N (k) as the number operator, and hence for the particle interpretation of the quantized theory.” This is a rash judgment, however. For instance, the question of localizability or at least approximate localizability is not even touched while it is certain that this is a pivotal criterion for something to be a particle. All that is established so far is a certain discreteness of physical quantities which is one feature of particles. However, this is not yet conclusive evidence for a particle interpretation of QFT. Recalling how various entities of forerunner theories, e. g. the single particle wave function, gained a very different meaning in QFT one should be extremely cautious to jump to one’s conclusions when only certain aspects are rediscovered in QFT. It is not clear at this stage whether we are in fact talking about particles or about fundamentally different objects which only have this one feature of discreteness in common with particles.

4.2. THE EMERGENCE OF INFINITIES

4.2

31

The Emergence of Infinities

Quantum field theory started with a theoretical framework that was built in analogy to quantum mechanics. Although there was no unique and fully developed theory, quantum field theoretical tools could be applied to concrete processes. Examples are the scattering of radiation by free electrons (“Compton scattering”), the collision between relativistic electrons or the production of electron-positron pairs by photons. Calculations to the first order of approximation were quite successful, but most people working in the field thought that QFT still had to undergo a major change. On the one side some calculations of effects for cosmic rays clearly differed from measurements. On the other side and, from a theoretical point of view more threatening, calculations of higher orders of the perturbation series led to infinite results. The self-energy of the electron as well as vacuum fluctuations of the electromagnetic field seemed to be infinite. The perturbation expansions did not converge to a finite sum and even most individual terms were divergent. The various forms of infinities suggested that the divergences were more than failures of specific calculations. Many people tried to avoid the divergences by formal tricks (truncating the integrals at some value of momentum, or even ignoring infinite terms) but such rules were not reliable, violated the requirements of relativity and were not considered as satisfactory. Some people came up with first ideas of coping with infinities by a redefinition of the parameters of the theory and using a measured finite value (for example of the charge of the electron) instead of the infinite ‘bare’ value (“renormalization”). From the point of view of philosophy of science it is remarkable that these divergences did not give enough reason to discard the theory. The years from 1930 to the beginning of World War II were characterized by a variety of attitudes towards QFT. Some physicists tried to circumvent the infinities by more-or-less arbitrary prescriptions, others worked on transformations and improvements of the theoretical framework. Most of the theoreticians believed that QED would break down at high energies. There was also a considerable number of proposals in favour of alternative approaches. These proposals included changes in the basic concepts (e. g. negative probabilities), interactions at a distance instead of a field theoretical approach, and a methodological change to phenomenological methods

32

CHAPTER 4. HISTORY AND BASIC STRUCTURE OF QFT

that focuses on relations between observable quantities without an analysis of the microphysical details of the interaction (the so-called S-matrix theory where the basic elements are amplitudes for various scattering processes). Despite the feeling that QFT was imperfect and lacking rigour, its methods were extended to new areas of applications. In 1933 Fermi’s theory of the beta decay started with conceptions describing the emission and absorption of photons, transferred them to beta radiation and analyzed the creation and annihilation of electrons and neutrinos (weak interaction). Further applications of QFT outside of quantum electrodynamics succeeded in nuclear physics (strong interaction). In 1934 a new type of fields (scalar fields) described by the Klein-Gordon equation could be quantized (another example of “second quantization”). This new theory for matter fields could be applied a decade later when new particles, pions, were detected.

4.3

The Taming of Infinities

After the end of World War II reliable and effective methods for dealing with infinities in QFT were developed, namely coherent and systematic rules for performing relativistic field theoretical calculations, and a general renormalization theory. On three famous conferences between 1947 and 1949 developments in theoretical physics were confronted with relevant new experimental results. In the late forties there were two different ways to address the problem of divergences. One of these was discovered by Feynman, the other one (based on an operator formalism) by Schwinger and independently by Tomonaga. In 1949 Dyson showed that the two approaches are in fact equivalent. Thus, Freeman Dyson, Richard P. Feynman, Julian Schwinger and Sin-itiro Tomonaga became the inventors of renormalization theory. The most spectacular experimental successes of renormalization theory were the calculations of the anomalous magnetic moment of electron and the Lamb shift in the spectrum of hydrogen. These successes were so outstanding because the theoretical results were in better agreement with high precision experiments than anything in physics before. The basic idea of renormalization is to avoid that divergences appear in physical predictions by shifting them into a part of the theory where

4.3. THE TAMING OF INFINITIES

33

they do not influence empirical propositions. Dyson could show that a rescaling of charge and mass (‘renormalization’) is sufficient to remove all divergences in QED to all orders of perturbation theory. In general, a QFT is called renormalizable, if all infinities can be absorbed into a redefinition of a finite number of coupling constants and masses. A consequence is that the physical charge and mass of the electron must be measured and cannot be computed from first principles. Perturbation theory gives well defined predictions only in renormalizable quantum field theories, and luckily QED, the first fully developed QFT belonged to the class of renormalizable theories. There are various technical procedures to renormalize a theory.3 One way is to cut off the integrals in the calculations at a certain value Λ of the momentum which is large but finite. This cut-off procedure is successful if, after taking the limit Λ → ∞, the resulting quantities are independent of Λ. Feynman’s formulation of QED is of special interest from a philosophical point of view. His so-called space-time approach is visualized by the famous Feynman diagrams that look like depicting paths of particles. Feynman’s method of calculating scattering amplitudes is based on the functional integral formulation of field theory.4 A set of graphical rules can be derived so that the probability of a specific scattering process can be calculated by drawing a diagram of that process and then using the diagram to write down the mathematical expressions for calculating its amplitude. The diagrams provide an effective way to organize and visualize the various terms in the perturbation series, and they seem to display the flow of electrons and photons during the scattering process. External lines in the diagrams represent incoming and outgoing particles, internal lines are connected with ‘virtual particles’ and vertices with interactions. Each of these graphical elements is associated with mathematical expressions that contribute to the amplitude of the respective process. The diagrams are part of Feynman’s very efficient and elegant algorithm for computing the probability of scattering processes. The idea of particles traveling from one point to another was heuristically useful in constructing the theory, and moreover, this in3

Part II of Peskin and Schroeder (1995) gives an extensive description of renormalization. 4 For an introduction to the theory and practice of Feynman diagrams see, e. g., chapter 4 in Peskin and Schroeder (1995).

CHAPTER 4. HISTORY AND BASIC STRUCTURE OF QFT

34

tuition is useful for concrete calculations. Nevertheless, an analysis of the theoretical justification of the space-time approach shows that its success does not imply that particle paths have to be taken seriously. General arguments against a particle interpretation of QFT clearly exclude that the diagrams represent paths of particles in the interaction area. Feynman himself was not particularly interested in ontological questions. In the beginning of the 1950s QED became a reliable theory which had left behind the preliminary status. It took two decades from writing down the first equations until QFT could be applied to interesting physical problems in a systematic way. The new developments made it possible to apply QFT to new particles and new interactions. In the following decades QFT was extended to describe not only the electromagnetic force but also weak and strong interaction so that new Lagrangians had to be found which contain new classes of ‘particles’ or quantum fields. The research aimed at a more comprehensive theory of matter and in the end at a unified theory of all interactions. New theoretical concepts had to be introduced, mainly connected with non-Abelian gauge theories and spontaneous symmetry breaking. 5 Today there are trustworthy theories of the strong, weak, and electromagnetic interactions of elementary particles which have a similar structure as QED. A combined theory associated with the gauge group SU(3) ⊗ SU(2) ⊗ U(1) is considered as ‘the standard model’ of elementary particle physics which was achieved by Glashow, Weinberg and Salam in 1962. According to the standard model there are three families of quarks and leptons, each of them containing 15 particles/fields with spin 1/2 (for example various quarks, the electron and its neutrino, or the muon and its neutrino). In addition it contains terms for the photon and other spin 1 particles/fields describing the forces between quarks and leptons. Altogether there is good agreement to experimental data, for example the masses of W + and W − bosons (detected in 1983) confirmed the theoretical prediction within one per cent deviation. 5

See Brading and Castellani (2008).

4.4. THE LAGRANGIAN FORMULATION OF QFT

4.4

35

The Lagrangian Formulation of QFT

The crucial step towards quantum field theory is in some respects analogous to the corresponding quantization in quantum mechanics by imposing the commutation relations. Its starting point is the classical Lagrangian formulation of mechanics, which is a so-called analytical formulation as opposed to the standard version of Newtonian mechanics. A generalized notion of momentum (the conjugate or canonical momentum) is defined by setting p = ∂L ∂ q˙ where L is the Langrange function L = T − V (T is the kinetic energy and V the potential) and q˙ ≡ dtd q . This definition can be motivated by looking at the special case of a Langrange function with a potential V which depends  only on the position so that (using Cartesian ∂L ∂ 1 2 coordinates) ∂ x˙ = ∂ x˙ 2 mx˙ = mx˙ = px . Under these conditions the generalized momentum coincides with the usual mechanical momentum. In classical Lagrangian field theory one associates with the given field φ a second field, namely the conjugate field π=

∂L ∂ φ˙

(4.5)

where L is a Lagrangian density. The field φ and its conjugate field π are the direct analogues of the canonical coordinate q and the generalized (canonical or conjugate) momentum p in classical mechanics of point particles. In both cases, QM and QFT, requiring that the canonical variables satisfy certain commutation relations implies that the basic quantities become operator valued. From a physical point of view this shift implies a restriction of possible measurement values for physical quantities some (but not all) of which can have their values only in discrete steps now. In QFT the canonical commutation relations for a field φ and the corresponding conjugate field π are [φ(x, t), π(y, t)] = iδ 3 (x − y) [φ(x, t), φ(y, t)] = [π(x, t), π(y, t)] = 0

(4.6)

which are equal-time commutation relations, i. e. the commutators always refer to fields at the same time. It is not obvious that the equal-time commutation relations are Lorentz invariant but one can formulate a manifestly covariant form of the canonical commutation relations. If the field

36

CHAPTER 4. HISTORY AND BASIC STRUCTURE OF QFT

to be quantized is not a bosonic field, like the Klein-Gordon field or the electromagnetic field, but a fermionic field, like the Dirac field for electrons one has to use anticommutation relations. In very loose terms, the operator valuedness of quantum fields means that to each space-time point (x, t) a field value φ(x, t) is assigned which is an operator. This is the fundamental difference to classical fields because an operator valued quantum field φ(x, t) does not by itself correspond to definite values of a physical quantity like the strength of the electromagnetic field. On this background, Teller has argued in Teller (1995) that the field interpretation of QFT is inappropriate since the alleged fields in QFT are not to be interpreted as physical fields with definite values of some sort which are assigned to space-time points, like in the case of the classical electromagnetic field. Rather, quantum fields are what Teller calls ‘determinables’ (p. 95), as it becomes manifest by the fact that quantum fields are described by mappings from space-time points to operators. Operators are mathematical entities which are defined by how they act on something. They do not represent definite values of quantities but they specify what can be measured, therefore Teller’s expression ‘determinables’. Further below I will discuss why this talk in terms of a field at a point has to be refined using the notion of a smeared field φ(f ). While there are close analogies between quantization in QM and in QFT there are also important differences. Whereas the commutation relations in QM refer to a quantum object with three degrees of freedom, so that one has a set of 15 equations, the commutation relations in QFT do in fact comprise an infinite number of equations, namely for each 4-tuple (x, t) there is a new set of commutation relations and there is, of course, a continuous set of space-time points (x, t). This infinite number of degrees of freedom embodies the field character of quantum field theory. Regarding differences between QM and QFT it is important to realize that the operator valued field φ(x, t) in QFT is not analogous to the wavefunction ψ(x, t) in QM, i.e. the quantum mechanical state in its position representation. Although in the development of QFT there is a continuity from the wave function, i. e. the quantum mechanical state in its position representation, to the field in QFT, it would be a misconception to understand these two quantities as analogues. Here, the ontologically relevant formal setting has changed in the transition from QM to QFT. While the

4.5. INTERACTION

37

wavefunction in QM is acted upon by observables, i. e. by operators, it is the (operator valued) field in QFT which itself acts on the space of states, i. e. on the states which are associated with the quantum field. In a certain sense one can say that the single particle wave functions have been transformed, via their reinterpretation as operator valued quantum fields, into observables. This step is sometimes called ‘second quantization’ because the single particle wave equations in relativistic QM already came about by a quantization procedure, e. g. in the case of the Klein-Gordon equation by replacing position and momentum by the corresponding quantum mechanical operators. Afterwards the solutions to these single particle wave equations, which are states in relativistic QM, are considered as classical fields which can be subjected to the canonical quantization procedure of QFT. The term ‘second quantization’ has often been criticized partly because it blurs the important fact that the single particle wave function φ in relativistic QM and the operator valued quantum field φ are fundamentally different kinds of entities despite their connection in the context of discovery. Summing up one can say that although in both, QM and QFT, the two fundamental irreducible kinds of entities are states on the one side and observables on the other side this fact is overshadowed by two confusing aspects in a comparison of QM and QFT. First, quantum fields which one expects to be somehow physically concrete like classical fields are on the side of observables although, as far as the development of theories is concerned, they are the successors of states (in their position representation, namely wave functions), e. g. in the Klein-Gordon equation of relativistic QM as described above. The second confusing aspect is that states are constantly mentioned in QM, whereas most of QFT is about quantum fields. States, which are comparatively abstract entities in QFT with no immediate spatio-temporal meaning, seem to be an appendage. Nevertheless, both states and observables are equally important in QM and QFT as the two fundamental kinds of entities.

4.5

Interaction

Up to this point, the aim was to develop a free field theory. Doing so does not only neglect interaction with other particles (fields), it is even

38

CHAPTER 4. HISTORY AND BASIC STRUCTURE OF QFT

unrealistic for one free particle because it interacts with the field that it generates itself. For the description of interactions—such as scattering in particle colliders—we need certain extensions and modifications of the formalism as so far exposed. The immediate contact between scattering experiments and QFT is given by the scattering or S-matrix which contains all the relevant predictive information about, e. g., scattering cross sections. In order to calculate the S-matrix the interaction Hamiltonian is needed. The Hamiltonian can in turn be derived from the Lagrangian density by means of the so-called Legendre transformation. In order to discuss interactions one introduces a new representation, the so-called interaction picture which is an alternative to the Schr¨odinger and the Heisenberg picture. For the interaction picture one splits up the Hamiltonian, which is the generator of time-translations, into two parts H = H0 + Hint where H0 describes the free system, i. e. without interaction, and gets absorbed in the definition of the fields and Hint is the interaction part of the Hamiltonian, or short the ‘interaction Hamiltonian’. Using the interaction picture is advantageous because the equations of motion as well as, under certain conditions, the commutations relations are the same for interacting fields as for free fields. Therefore, various results that were established for free fields can still be used in the case of interacting fields. The central instrument for the description of interaction is again the S-matrix, which expresses the connection between in and out states by specifying the transition amplitudes. In QED, for instance, a state |in describes one particular configuration of electrons, positrons and photons, i. e. it describes how many of these particles there are and which momenta, spins and polarizations they have before the interaction. The S-matrix supplies the probability that this state goes over to a particular |out state, e.g. that a particular counter responds after the interaction. Such probabilities can be checked in experiments. The canonical formalism of QFT as introduced in the previous section is only applicable in the case of free fields since the inclusion of interaction leads to infinities (see the historical part). For this reason perturbation theory makes up a large part of most publications on QFT. The importance of perturbative methods is understandable realizing that they establish the immediate contact between theory and experiment. Although the techniques of perturbation theory have become ever more sophisticated it

4.5. INTERACTION

39

is somewhat disturbing that perturbative methods could not be avoided even in principle. One reason for this unease is that perturbation theory is felt to be rather a matter of (highly sophisticated) craftsmanship than of understanding nature. Accordingly, the corpus of perturbative methods plays a small role in the philosophical investigations of QFT. What does matter, however, is in which sense the consideration of interaction effects the general framework of QFT.6

6

An overview is given in sec 4.1 (“Perturbation Theory—Philosophy and Examples”) of Peskin and Schroeder (1995).

Chapter 5 Alternative Approaches 5.1

Deficiencies of the Standard Formulation of QFT

From the 1930s onwards the problem of infinities as well as the potentially heuristic status of the Lagrangian formulation of QFT stimulated the search for reformulations in a concise and eventually axiomatic manner. A number of further aspects intensified the unease about the standard formulation of QFT. The first one is that quantities like total charge, total energy or total momentum of a field are unobservable since their measurement would have to take place in the whole universe. Accordingly, quantities which refer to infinitely extended regions of space-time should not appear among the observables of the theory as they do in the standard formulation of QFT. Another problematic feature of standard QFT is the idea that QFT is about field values at points of space-time. The mathematical aspect of the problem is that a field at a point, φ(x), is not an operator in a Hilbert space. The physical counterpart of the problem is that it would require an infinite amount of energy to measure a field at a point of spacetime. One way to handle this situation—and one of the starting points for axiomatic reformulations of QFT—is not to consider fields at a point but instead fields which are smeared out in the vicinity of that point using certain  functions, so-called test functions. The result is a smeared field φ(f ) = φ(x)f (x)dx with supp f ⊂ O (the support of the test function f) where O is the set of bounded open regions in Minkowski space-time. The third important problem for standard QFT which prompted reformulations is the existence of inequivalent representations. In the context of quantum mechanics, Schr¨odinger, Dirac, Jordan and von Neumann real41

CHAPTER 5. ALTERNATIVE APPROACHES

42

ized that Heisenberg’s matrix mechanics and Schr¨odinger’s wave mechanics are just two (unitarily) equivalent representations of the same underlying abstract structure, i. e. an abstract Hilbert space H and linear operators acting on this space. In 1931 von Neumann gave a detailed proof (of a conjecture by Stone) that the canonical commutation relations (CCRs) for position coordinates and their conjugate momentum coordinates in configuration space fix the representation of these two sets of operators in Hilbert space up to unitary equivalence (von Neumann’s uniqueness theorem). This means that the specification of the purely algebraic CCRs suffices to describe a particular physical system. In quantum field theory, however, von Neumann’s uniqueness theorem looses its validity since here one is dealing with an infinite number of degrees of freedom. Now one is confronted with a multitude of inequivalent irreducible representations of the CCRs and it is not obvious what this means physically and how one should cope with it.

5.2

The Algebraic Point of View

The described situation is the background for the establishment of algebraic reformulations of QFT. According to the algebraic point of view algebras of observables rather than observables themselves should be taken as the basic entities in the mathematical description of quantum physics. In the forties the mathematician I. E. Segal postulated that the C ∗ -algebra generated by all bounded operators should be the basic entity in the mathematical description of physics.1 It turned out that the adoption of an algebraic point of view could be the appropriate framework in order to handle all the above-mentioned problems of standard QFT, the representation problem for instance. In standard QM the algebraic point of view in terms of C ∗algebras makes no notable difference to the usual Hilbert space formulation since the Hilbert space representation and the C ∗ -algebra formalism are equivalent. However, in QFT this is no longer the case as described above. Since in QFT one is dealing with an infinite number of degrees of freedom there are unitarily inequivalent irreducible representations of a C ∗ -algebra. Sticking to the usual Hilbert space formulation thus means making an im1

See Segal (1947a) and Segal (1947b).

5.2. THE ALGEBRAIC POINT OF VIEW

43

plicit choice of one particular representation that is not equivalent to other available representations. Segal proposed to take one single C ∗ -algebra as the basic element of the mathematical description of a quantum physical system and he dismissed the availability of inequivalent representations as irrelevant to physics. Against this approach Haag, the most outstanding advocate of algebraic quantum field theory (AQFT), argued that inequivalent representations can be understood physically by a revision of Segal’s approach. The decisive idea is not to take individual C ∗ -algebras as the basic ingredients of the description but rather a so-called net of algebras (see below) since the relation between various algebras itself represents important physical information. Another point where algebraic formulations are advantageous derives from the fact that two quantum fields are physically equivalent when they generate the same algebras of local observables. Such equivalent quantum field theories belong to the same so-called Borchers class which entails that they lead to the same S-matrix. As Haag stresses Haag (1996), fields are only an instrument in order to “coordinatize” observables, more precisely, in order to coordinatize the sets of observables with respect to different finite space-time regions (the so-called net of local algebras in AQFT). The choice of a field system is to a certain degree conventional, namely as long as it belongs to the same Borchers class. From this point of view it is more appropriate to consider these algebras as the fundamental entities in QFT rather than quantum fields. In the fifties there was a strong tendency, in physics as well as in other fields, to reformulate grown theories in an axiomatic manner, partly in order to remove ad hoc features. A very prominent early attempt to axiomatise QFT is Arthur Wightman’s field axiomatics. Wightman imposed axioms on polynomial algebras P(O) of smeared fields, i. e. sums of products of smeared fields. A crucial point of Wightman’s approach is the replacement of the mapping x → φ(x), which supposedly expresses what is meant by a field, by the mapping O → P(O) from finite space-time regions O to P(O). Wightman’s smeared out field operators are unbounded which makes the approach cumbersome from a mathematical point of view and this is one of the differences to the approach I will introduce next where only bounded operators are considered. Algebraic Quantum Field Theory (AQFT) is arguably the most success-

CHAPTER 5. ALTERNATIVE APPROACHES

44

ful attempt to reformulate QFT in an axiomatic manner.2 AQFT originated in the late fifties by the work of Rudolf Haag and quickly advanced in collaboration with Huzihiro Araki and Daniel Kastler. AQFT itself exists in two versions, concrete AQFT (Haag-Araki) and abstract AQFT (Haag-Kastler). The concrete approach uses von Neumann algebras (or W  -algebras), the abstract one C  -algebras. The adjective ‘abstract’ refers to the fact that in this approach the algebras are characterized in an abstract fashion and not by explicitly using operators on a Hilbert space. In standard QFT, the CCRs together with the field equations can be used for the same purpose, i. e. an abstract characterization. One common aim of these axiomatisations of QFT is to avoid the usual approximations in standard QFT. Trying to do this in a strictly axiomatic way, however, they only get ‘reformulations’ which are not as rich as standard QFT from a physical point of view. The “algebraic approach [...] has given us a frame and a language not a theory” as Haag concedes in Haag (1996) with respect to AQFT.

5.3

Basic Ideas of AQFT

One of the main traits and possibly the most unusual one of AQFT is that so-called nets of algebras are seen as the primary objects of study. The idea is that the physical information in quantum field theories is not contained in individual algebras but in the mapping O → A(O) from spacetime regions O to algebras A(O) of local observables where the O’s are open and bounded regions in Minkowski spacetime. Since only finite regions are considered, the algebras are called local algebras. Physically, the elements of an algebra A(O) are seen as representing operations that can be performed in the region O that is associated with the algebra. The crucial point is that it is not necessary to specify observables explicitly in order to fix physically meaningful quantities. The very way how algebras of local observables are 2

Comprehensive introductions to AQFT are provided by the monographs Haag (1996) and Horuzhy (1990) as well as the overview articles Haag and Kastler (1964), Roberts (1990) and Buchholz (2000). Early pioneering monographs on axiomatic QFT were Streater and Wightman (1964) and Bogolubov et al. (1975). Mathematical aspects are emphasized in Bratteli and Robinson (1979).

5.3. BASIC IDEAS OF AQFT

45

linked to spacetime regions is sufficient to supply observables with physical significance. It is the partition of the so-called algebra Aloc of all local observables into subalgebras which contains physical information about the observables, i. e. it is the net structure of algebras which matters. The claim is that the allocation itself of observable algebras to finite space-time regions suffices to account for the physical meaning of observables. It is not necessary to start with any such information explicitly. The physical justification for this approach consists in the recognition that the experimental data for QFT are exclusively space-time localization properties of microobjects from which other properties are inferred. The Stern-Gerlach experiment is an illuminating example. All one gets in this experiment are certain space-time distributions of dots of detected particles which originated from a particle source and hit a photographic plate. Only in a second step one recognizes particles with certain spin directions after having passed an inhomogeneous magnetic field. This example might help to imagine that space-time localisation can specify or encode all other physical properties. Physically the most important notion of AQFT is the principle of locality which has an external as well as an internal aspect. The external aspect is the fact that AQFT considers only observables connected with finite regions of space-time and not global observables like the total charge or the total energy momentum vector which refer to infinite space-time regions. This approach was motivated by the operationalistic view that QFT is a statistical theory about local measurement outcomes with all the experimental information coming from measurements in finite space-time regions. Accordingly everything is expressed in terms of local algebras of observables. The internal aspect of locality is that there is a constraint on the observables of such local algebras: All observables of a local algebra connected with a space-time region O are required to commute with all observables of another algebra which is associated with a space-time region O
that is space-like separated from O. This principle of (Einstein) causality is the main relativistic ingredient of AQFT. The basic structure upon which the assumptions or conditions of AQFT are imposed are local observables, i.e. self-adjoint elements in local (noncommutative) von Neumann-algebras, and physical states, which are identified as positive, linear, normalized functionals which map elements of

CHAPTER 5. ALTERNATIVE APPROACHES

46

local algebras to real numbers. States can thus be understood as assignments of expectation values to observables. One can group the assumptions of AQFT into relativistic axioms, such as locality and covariance, general physical assumptions, like isotony and spectrum condition, and finally technical assumptions which are closely related to the mathematical formulation. As a reformulation of QFT, AQFT is expected to reproduce the main phenomena of QFT, in particular properties which are characteristic of it being a field theory, like the existence of antiparticles, internal quantum numbers, the relation of spin and statistics, etc. That this aim could not be achieved within AQFT on a purely axiomatic basis is partly due to the fact that the connection between the respective key concepts of AQFT and QFT, i.e. observables and quantum fields, is not sufficiently clear. It turned out that the main link between the theory of local observables and the quantum fields of standard QFT is the notion of superselection. Superselection rules are certain restrictions on the set of all observables and allow for classification schemes in terms of permanent or essential properties. It is only since the second half of the eighties that AQFT came into the focus of the philosophy of physics community. Some of the most fruitful discussions were stimulated by reexaminations of physical theorems from the sixties and seventies, in particular the Reeh-Schlieder theorem. Further results of interest are Haag’s theorem and a lemma by Borchers. The properties of the relativistic vacuum often play a central role in these discussions. Confer, e. g., Redhead (1995b), Redhead (1995a) and the monograph on The Philosophy of Vacuum Saunders and Brown (1991). A mayor reason for this interest is that the vacuum displays features which seem to be in conflict with standard ontological conceptions about what things are and how things and properties are related to each other. Some central issues in the philosophical debate about AQFT are questions about locality, localization and causality.3

3

For instance, see Saunders (1988), Redhead and Wagner (1998), Ruetsche (2003), R´edei and Summers (2002) as well as numerous fine papers by Rob Clifton, collected in Butterfield and Halvorson (2004)).

Chapter 6 The Ontological Significance of QFT and AQFT Before beginning with my main study I will address two questions which might occupy the thoughts of some readers by now. Why do I look at QFT while the much less complicated QM already displays the same problems in a far clearer way? Why does algebraic quantum field theory (AQFT) play such a prominent role in my investigation? These are the questions I will address in this chapter.1

6.1

QM Versus QFT

Beginning with the formation of QM in the twenties there has been a broad discussion about its conceptual foundations. One reason for these discussions is the fundamental difference between QM and classical mechanics in various respects. Heisenberg’s uncertainty relations (see section B.1.2) and the EPR correlations, for instance, are two well-known issues where peculiar quantum features are evident. Moreover, QM has severe internal problems connected with the measurement problem (see section B.2.3). And finally, there is the question of the compatibility of QM and relativity theory which has been only partly resolved with the development of QFT (see chapter 4). On the one hand the extensive discussions and analyses led to some clarification about the location and interconnection of various problems. 1

For a self-contained discussion of these and related philosophical questions together with a compact exposition of (A)QFT see Kuhlmann (2009).

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The establishment of no-go-theorems was particularly valuable and these results are arguably the highest achievements in this area of research (see section 3.4). Outstanding examples are Gleason’s theorem and the KochenSpecker theorem2 , Bell’s inequalities in the context of EPR correlations as well as non-objectification and probability theorems.3 On the other hand none of the proposed interpretations of or alternative approaches to QM is satisfactory in all respects. The most prominent proposals, most of which were introduced as solutions to the measurement problem, are Bohmian mechanics, nonlinear alternatives of the Schr¨odinger equation, Everettian approaches (different versions of many worlds interpretations and also the many minds interpretation), modal interpretations, consistent histories and, eventually, the decoherence approach (strictly speaking, not an independent solution to the measurement problem). Again see section B.2.3 for some more details. It seems that each proposal solves certain problems at the cost of having new problems at a different place or, figuratively speaking, it can only flatten the bulge in the carpet to the effect that the bulge appears somewhere else again. The impression is getting stronger that there will no sweeping new results on the foundations of QM in the foreseeable future. On first sight it might be surprising that the discussion on the conceptual foundations of the quantum domain has always been primarily concerned with QM and not with QFT. After all QFT is, in a certain sense, more comprehensive than QM and in particular relativistically invariant in contrast to QM. There have been at least two reasons for neglecting QFT in favour of QM regarding conceptual reflections. First, for a long time the attitude was dominating that the decisive philosophical problems show up in QM already so that a conceptual analysis of QFT appeared not to be necessary. It even seemed that looking at QFT would only blur the view on the central features since QFT is much more complex and mathematically advanced than standard QM. A second reason for neglecting QFT was the fact that QFT has not yet reached the status of a consistent and complete theory. For instance the lack of a quantum field theory of gravitation is felt as a pressing need. Since it cannot be excluded that the incorporation of 2

Section 1.5 and chapter 5 in Redhead (1987) in contain a systematic exposition. A rich and helpful historical survey can be found in Jammer (1974). 3 Mittelstaedt (1998), chapters 3 and 4, is an authoritative source for details.

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49

the fourth force might lead to deep changes of QFT as a whole, the current version of QFT must be seen as a preliminary theory. There have been various studies on the historical development of QFT and in particular Quantum Electrodynamics (QED).4 This is partly due to some charismatic figures involved, especially Richard Feynman, and some spectacular successes of methods like renormalization theory, Feynman diagrams and the extensive use of symmetry groups. For the preference of historical studies on QFT over philosophical ones it might have been more important, however, that history does not change afterwards like theories do. QFT as an object of philosophical reflection only began to receive wider attention around 1990. Apart from a certain saturation in the research on the conceptual foundations of QM the two above-mentioned arguments against QFT as a philosophical topic were weakened for the following reasons. First, a careful analysis of the specifically relativistic traits of QFT led to results which at least aggravate the conceptual problems of QM severely. Possibly they even surmount those problems qualitatively. Second, due to the development of QFT and of the theory of super strings in the last two decades the initial hope is fading away that QFT is near to its final completion. This hope flourished in the aftermath of the electroweak unification which seduced some people to euphorically anticipate the achievement of the final theory of everything. In my view the fact that the completion of QFT does not seem to lie in the foreseeable future speaks in favour of philosophical analyses of QFT today because otherwise one would have to postpone ontological thoughts about the very scientific theory, i. e. QFT, that is most important for ontology until never-never day. There are not only indirect arguments in favour of QFT as an object of philosophical research. Some further arguments support the hope that a conceptual analysis of QFT will deliver results which finally enable us to tackle problems which seemed insoluble when looking at QM. The fundamental difficulty to find and to understand the nature of the basic entities of the quantum regime might, looking at QFT, lead to a solution which only makes it necessary to explain why we have the impression of ‘elementary particles’. In that case there would be no need to take ‘elementary 4

Darrigol (1986), Schweber (1994) and Cao (1997) are arguably the most famous ones.

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particles’ ontologically serious. If that should be true, I want to argue, QM would no longer be sufficient for ontological questions. And that fits well with the fact that some newer results seem to make it almost impossible to maintain the wide-spread view that quantum field theory is just as well a particle as a field theory, despite of its supposedly misleading name. In particular the Reeh-Schlieder theorem and the Unruh effect seem to display features which show that QFT is essentially a field theory. Further results like a no-go theorem by David Malament and, e.g., Robert Wald’s research on QFT in curved space-time strongly support such a view, too. One problem for investigating the conceptual aspects of QFT consists in the fact that many results with conceptually important consequences can only be stated within a formalism which is mathematically involved, namely Algebraic Quantum Field Theory (AQFT), even if it is physically clearer than the standard formalism of QFT in at least some respects. Philosophical aspects cannot always be seen immediately, however. It is instructive to realize that, e.g., the Reeh-Schlieder theorem is already almost 50 years old without having received any notice from the philosophy of physics for more than three decades. I will deal with the ontological significance of AQFT more intensely in the following sections. At least until recently one (if not the longest and most intensive track) of discussions on the philosophy of QFT was an investigation by Paul Teller and Michael Redhead on different formal descriptions of many particle systems containing identical particles.5 This discussion which was initiated in the late eighties finally became a central part in Teller’s An Interpretive Introduction to Quantum Field Theory (1995) which is the first systematic monograph on the philosophy of QFT.6 Teller’s book displays two deficiencies, however. First, a major part of his studies can already be performed with respect to non-relativistic standard QM. Second, the formalism upon which Teller reflects is, on the one hand, somewhat out-of-date (about from the fifties) and, on the other hand, too restricted in its scope of application since only free field theory is considered. It can be seen as partly Teller’s 5

Redhead (1975), Redhead (1980), Redhead (1983), Teller (1983), Redhead (1988), Redhead and Teller (1991) and finally Teller (1995) is a selection of publications dealing at least partly with this question. 6 The anthology Philosophical Foundations of Quantum Field Theory Brown and Harr´e (1988) was the first book to appear about this field of research.

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51

merit, however, that a broad discussion on the conceptual foundations of QFT has begun in the recent years.7

6.2

AQFT and the Ideal Language Philosophy

Today, many investigations about the interpretation of QFT are carried out within AQFT, i.e. the algebraic formulation of QFT (see chapter 5). AQFT is not the most convenient formalism for working physicists who are interested in calculating scattering cross sections and other empirically relevant quantities in high energy physics.8 Nevertheless, despite its limited practical usefullness AQFT is a very fruitful and effective formalism in order to address conceptual questions. The program of AQFT can be compared to ideal language projects in the analytic tradition of philosophy. The starting point is a grown theory, i.e. standard QFT—comparable to an ordinary language like English—whose formulation is generically historical, a conglomeration of various old and new techniques and theoretical approaches which is unified by its remarkable success in predicting empirical quantities. These are parallel features to ordinary languages which obviously work relatively well in practice. Leaving the regime of practical purposes, however, it becomes evident that various ambiguities, disparities and the openness for unintended extensions are very inviting for conceptual conclusions which lead astray. The so-called ontological proof of the existence of God is an interesting example. Taking God as the epitome of the most positive attributes and assuming that being is an attribute which is better than not-being one concludes that God must necessarily exist. It can be argued that the accidental structure of historically grown western languages is the basis for such a heavy ontological conclusion. 7

Sunny S. Auyang’s book How is Quantum Field Theory Possible?, which appeared in 1995 as well, had fewer effects than Teller’s had at first. This might be partly due to the fact that it has relatively few connections with any recent contexts of discussions. Later on, Auyang’s book has received more attention because of a new interest, first, in event ontology and, second, in the role of symmetries in connection with the discussion on gauge theories. 8 Only recently theoretical physicists working on AQFT made some successful attempts to calculate standard “every-day quantities” using the formalism of AQFT with an expenditure comparable to that in conventional QFT. (Private communication with K. Fredenhagen at DESY, Hamburg.)

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An example from quantum physics illustrates the same point with respect to a physical formalism. The Schr¨odinger many-particle formalism for systems of identical particles contains an important index which clearly stems from labeling different particles in such a ‘many-particle-system’ (see subsection B.2.1). However, due to the so-called ‘indistinguishability of identical particles’ it turned out that these ‘particle labels’ cannot refer to different individual particles. In the sense of labeling any individual entities in the world they clearly lost their original meaning. Nevertheless, ‘particle labels’ linger on without a definite interpretation or, to put things more strongly, they remain as a piece of primarily historically justified, albeit still partly useful formalism.9 Another example from QFT is the confusion created by the procedure of so-called ‘second quantisation’.10 In this case—using the standard gloss—after having realized that the KleinGordon equation, which resulted from ‘quantisation’ of the classical oneparticle equation (see chapter 4), unfortunately cannot be the looked for relativistic quantum-mechanical one-particle equation, it turned out that treating the Klein-Gordon equation as describing a classical field which can again be ‘quantised’, one does eventually arrive at an equation that does have empirical significance, namely for certain many-particle systems. Since this procedure, i.e. ‘second quantisation’, proved to lead to a successful result it became an established part of QFT, albeit often accompanied by critical remarks. Such pieces of formalism as second ‘quantisation’ are inviting for ontological misconceptions since it seems, e.g., as if there was a classical Klein-Gordon field, as if there were two levels between a classical world of point particles and QFT, etc. These examples from ordinary languages as well as from physics show that some ontological misconceptions about a certain part or aspect or level of reality can be traced back to the very way how the leading theory or language in use is formulated. In general, of course, the formulation of theories and languages will, to a certain extent, represent our ontological ideas so that ontological misconceptions which are encorporated or implied by a particular formulation are simply those ontological misconceptions one 9

Redhead introduced the notion of surplus structure in this context in Redhead (1975). Cao (1997), section 7.3, gives a detailed historical discussion of the misunderstandings and ambiguities. A critical examination of the notion of ‘second quantisation’ can be found in Haag (1996), p. 46 ff. A comprehensive monograph on the matter is Berezin (1966). 10

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has started with. This general case is harmless and not the issue I am aiming at. The ontologically interesting cases in my comparison of ideal language philosophy and reformulation projects regarding physical theories are those where ontological implications of the initial formulation are not intended by and based on the ontological conception of that part (or aspect or level) of reality which the theory or language is meant to represent. In these cases the formulation is ontologically misleading although the theory or language may still be used with reasonable success. It seems possible, however, that the lack of a full success has the same roots as the ontological misconceptions. Therefore, even on an instrumentalist background it suggests itself to think about a reformulation of the theory or language without these deficiencies. With respect to language, reformulation projects were pursued in the analytical tradition of philosophy in the twenties and thirties. The early Wittgenstein is probably the best-known initiator of ideal language philosophy with his groundbreaking treatise Tractatus Logico-Philosophicus (1922). The idea of the so-called ideal language philosophy was to construct a new language which makes it impossible to even formulate sentences that are ambiguous and whose parts have no clear reference and/or function. Rudolf Carnap is among the most influential advocates of ideal language philosophy with his often polemical writings on the connection between the vagueness of ordinary languages and pseudo problems in philosophy and his own attempts to build a better formal language. His particularly provocative paper on “The elimination of metaphysics through logical analysis of language” contains some striking examples—e.g. about Heidegger’s use of nothing as a noun—which, despite Carnap’s harsh evaluation, display the main point in a clear way.11 Ordinary languages contain nontrivial ambiguities that leave room for preposterous ontological conclusions. In modern physics there are reformulation programs which were initiated by similar motives as the ideal language project. The first attempts to reformulate quantum mechanics in a unified and rigorous fashion originated in the late twenties so that there is even an interesting temporal coincidence between the beginning of the philosophical and the physical program. The first steps towards a concise and abstract reformulation of 11

Carnap (1931).

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QM were quickly achieved and found their way into most textbooks. The higher degree of abstractness is due to the generalization from different historically grown equivalent approaches. Similar reformulation programs with respect to QFT began only in the late forties and have been carried on up to now. In the beginning the aim was a thorough axiomatization of QFT.12 Arguably the most successful of these attempts is algebraic quantum field theory (AQFT) which will play a mayor role in my own proposal for an ontological understanding of QFT. The founders of AQFT proceeded in a fashion which is comparable to the ideal language program at least in its general outline and some of its original motives. AQFT is an attempt to reformulate QFT in a way which is (mathematically) rigorous, as economical as possible with respect to the basic entities and concepts and which displays a clear structure. These maxims were carried out in an axiomatic fashion by imposing fundamental physical conditions on the set of observable quantities. One hope was to get rid of the notorious infinities for quantities like mass and charge. The opinion was that such infinities are mathematical artefacts which should disappear in an ideally constructed formalism rather than being improperly wiped out from a dirty formalism by the method of renormalisation. Starting with axioms like (Einstein) locality and relativistic covariance the hope was that everything could be derived in a systematic way without ad-hoc moves and approximations. Given the original idea of reformulating QFT in an axiomatic manner and thus overcoming various problems, AQFT only had a partial success. First, a purely axiomatic version of QFT could not be established since none of the attempts led to a theory that was rich enough from a physical point of view. Therefore, the approach had to be enriched by non-axiomatic elements in order to get into contact with ‘real physics’. This is probably one reason why Haag, the main figure in the establishment of AQFT, prefers to talk about ‘local quantum physics’ thereby avoiding to put too much emphasis on axiomatization.13 A second for the limited success of AQFT is that, on the one hand, problems, e.g., with infinities appeared in 12

The most prominent figures of this era of search for an axiomatic reformulation of QFT are I. E. Segal, A. Wightman, R. Haag, H. Araki and D. Kastler, in roughly chronological order. For more details see chapter 5. 13 For more details on the issue of axiomatization versus contact with physics see chapter 5. Moreover, see Haag’s remark on page 58 of Haag (1996).

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55

AQFT as well and, on the other hand, the techniques in standard QFT for getting finite quantities by means of renormalisation procedures became ever more refined and systematic so that the unease decreased. In conclusion one can say that AQFT, in the sense of a thorough reformulation of QFT in an axiomatic manner, did not fully satisfy the initial expectations. And this is another similarity between the ideal language project and axiomatic reformulations of QFT. Both programs turned out not to be as easily translatable into action as one was hoping after the early successes. In both cases it had to be realized that it is extremely difficult to construct a perfectly systematic and clear language or formalism which fulfils all the needs that an ordinary language or grown formalism does. It is possible to build such a system but at the risk of loosing contact with reality. In contrast to the philosophical ideal language project, however, AQFT should not be considered a failure. AQFT is an area of ongoing and successful research which is, at least partly, due to a fundamental difference between axiomatically oriented reformulation programs regarding QFT and the aim of an ideal language in analytical philosophy. AQFT can and I think should be seen as more than a systematic and concise formalism in the sense of a language.14 On a realistic reading, AQFT says something about the ultimate ‘building blocks’ and the structure of the world. Unless one takes a purely instrumentalist attitude a physical theory is more than a mere machinery for the prediction of measurement results. One expects the formalism to somehow represent how nature works. The hope is to understand the underlying entities and mechanisms which produce what we measure.

6.3

QFT Versus AQFT

So far I have only spoken about the way how AQFT is structured as a theory. Before I will come to my main thesis in this chapter, which rests on the above considerations, I wish to address the objection that AQFT as 14

Wittgenstein is an exception in this respect since his pledge in favour of an ideal language is based on ontological considerations, too. In his atomistic conception the world is solely composed of facts, where a fact can be defined as that which makes a true proposition true.

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a physical theory is not sufficiently corroborated and maybe even empty of application. The following facts do, I think, supply at least a part of such a corroboration. Although AQFT is not meant to replace standard QFT these two formulations still are in certain rivalry. While standard QFT is well corroborated by its enormous empirical success AQFT has a poor record in this respect. However, new and not yet commonly accepted theories, like AQFT, do aquire a boost of probability to be true if they prove to be successful in contexts for which they have not been deployed in the first place. In the case of AQFT, I want to argue, one such context is quantum statistical mechanics where AQFT can be successfully applied in the treatment of so-called KMS states.15 It turned out that in the thermodynamic limit there is a fruitful use of unitarily inequivalent representations, whose existence and interpretation is one of the main starting points for AQFT.16 The second physical context that I wish to mention where AQFT is used with great success is QFT in curved space-time. The reason for the successful, and possibly even unavoidable, employment of the formulation of AQFT in the context of curved space-time lies in the fact that certain contingencies of the formalism that is used in standard QFT become visible and turn out to be a great obstacle when the special framework of using flat Minkowski space-time has to be left.17 In conclusion one can say that AQFT proves to be successfully applicable in a wide range of physical contexts. Not surprisingly, these contexts are usually such that certain generalizations are necessary which render the contingent framework of standard QFT to narrow and thus call for AQFT. For this reason I think it is justified to say that AQFT is a successful physical theory. Now I get to my main point in this chapter. In ontological perspective, I wish to argue, AQFT is to be preferred over QFT in the sense that AQFT permits more than standard QFT a clear view of the most general onto15

See the entry ‘KMS condition’ in glossary D.3. A comprehensive account is given in Bratteli and Robinson (1981). 16 L. Ruetsche presents a very interesting philosophical view regarding the significance of unitarily inequivalent representations in quantum statistical mechanics in her paper Ruetsche (2003). 17 For some words about the approach of QFT in curved space-time see footnote 34. A helpful introduction to QFT in curved space-time is Wald (1994) with an emphasis on the motives for the way it is formulated. Regarding the significance of AQFT for QFT in curved space-time see in particular sections 1.1 and therein.

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57

logical structure of the fundamental level of nature. Although ironically AQFT was initially meant as a strictly positivist theory in which only measurable quantities occur, gradually it was realized that there is no need to stick to that attitude.18 Moreover, apart from attitude it is questionable whether the positivist aim could actually be carried out in AQFT.19 In my view AQFT is more than rigorous formalism and the positivist attitude of various people working in AQFT, in particular in the early stages of its development, can be separated from the theory itself. AQFT is a serious attempt to reformulate QFT by putting, in an axiomatic fashion, fundamental physical notions and entities first and by trying to limit as much as possible the number of classes of quantities in terms of which everything else is expressed. Since the principles in formulating AQFT are thus very similar to the ones in ontology AQFT suggests itself more than standard QFT as an object for ontological investigations. AQFT as well as ontology are not concerned with practical purposes but aim at an understanding of the most general aspects of what the world is made of. This is quite different with standard QFT where one of the important goals has always been to calculate physical quantities most effectively. It is no coincidence that Richard Feynman is one of the most celebrated people in the research on QFT although conceptual rigour was never among his main concerns. One could argue that the expressed emphasis on ontology in QFT and AQFT and the actual successes in carrying it through are crosswise. On the one hand, whereas elementary particle physicists, using and establishing QFT, proclaimed to be “In Search for the Ultimate Building Blocks of Nature” t’Hooft (1996), the advocates of AQFT stressed, in a tone that is often adverse to ontology, to merely talk about measurements in finite regions of 18

To my knowledge Simon Saunders was the first philosopher of physics who explicitly pointed out the possibility of a realist interpretation of AQFT in his rich article Saunders (1988). See in particular sections 0., 1.9. and 1.10. therein. Other philosophers of physics with a realistic attitude towards AQFT are, I think, Michael Redhead, Rob Clifton, Dennis Dieks and probably Hans Halvorson, to name just a few. 19 An even more explicitly positivist approach was the so-called S-matrix-program which turned out to be a failure. The S-matrix-program was initiated by Heisenberg and had its heyday in the early sixties in particular through the work of C. F. Chew. Heisenberg’s initiative was explicitly grounded on positivist arguments as it can already be read off from the title “The observable quantities in the theory of elementary particles” of his two first papers Heisenberg (1943a) and Heisenberg (1943b) on this topic.

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space-time.20 On the other hand, it was in QFT that the efficient prediction of measurable quantities always played a very important role while AQFT, even if this was not initially intended, has a great deal to say about the most general nature of the ultimate ‘building blocks’. For this reason it seems to me that there is room for a natural coexistence of standard QFT and AQFT. Standard QFT is definitely the first choice for the working physicist, not just as far as empirical matters are concerned but also for most theoretical questions. In the most general ontological perspective, however, AQFT should be the leading object of study, supplemented by investigations about standard QFT.21

6.4

The Philosophical Interest in (A)QFT

In the last section I have argued that one can get a clearer picture of nature at its most basic level by looking at AQFT rather than at standard QFT—at least in some respects. This does not mean that one can neglect standard QFT in a philosophical analysis. First, AQFT is not (yet) as rich as standard QFT since it only captures some very general—but also very basic and conceptually important—structures. And second, since both approaches, standard QFT and AQFT clearly have a preliminary status, 20

That Haag proposes an ontology for AQFT, namely an event ontology, has, as far as I can judge, not been greeted with great applause in the AQFT community, and, again as far as I can tell, not because of Haag’s specific conception but rather because of his very interest in philosophical, in particular ontological, considerations (see, e.g., section VII.3 of his book Haag (1996)). Unfortunately, from the perspective of philosophy of physics Haag’s position is not very convincing, first, because it is a muddle of operationalism (and thereby anti-realism) on the one side and then, surprisingly, ontological thought’s about AQFT on the other side, and, second, because it seems that his event-ontological conception is incompatible with the very approach of AQFT where space-time regions are primary, and not derived from an event structure via ordering relations. For the second point of critique see footnote 46 in Landsman (1996), a very knowledgeable, independent and detailed review of Haag’s book. 21 In Kuhlmann (2010) I argue this point out in more detail. In his comprehensive exposition of AQFT, Halvorson and M¨ uger (2007), philosopher of physics H. Halvorson also stresses the significance of AQFT for foundational questions. In contrast, Wallace (2006) has a critical assessment of the relevance of AQFT. Recently, D. Fraser (2009) dismissed Wallace’s point of view, arguing that consistency plays a central role in choosing between different formulations of QFT since they do not differ in their respective empirical success (see Fraser (2009)).

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it would not be wise to fade out potentially valuable information. It is not sure which structures will survive. Accordingly, both standard QFT as well as AQFT will be taken into consideration in this work. Since QFT and AQFT are not, at least not any more, seen as competing theories there seems little risk to shed more light on one or the other depending on the context. It is only since the second half of the eighties that QFT and AQFT came into the focus of the philosophy of physics community.22 Some of the most fruitful discussions were stimulated by reexaminations of physical theorems from the sixties and seventies, in particular the Reeh-Schlieder theorem. Further results of interest are Haag’s theorem, a lemma by Borchers and a number of closely connected investigations by Hegerfeldt. I will introduce and evaluate these issues in more or less detail in some of the following chapters, particularly in chapter 8. The properties of the relativistic vacuum often play a central role in these discussions.23 A mayor reason for this interest is that the vacuum displays features which seem to be in conflict with standard ontological conceptions about what things are and how things and properties are related to each other. Some central issues in the philosophical debate are questions about locality, localization and causality. Since N-particle states can be ‘built up’ from the vacuum state it is not just one exotic state with exotic properties. Therefore, ontological considerations about the vacuum have a more general bearing which explains why this issue attracts so much philosophical attention. One problem and threat for the significance of ontological considerations about QFT is the preliminary status of this most fundamental theory of physics. The lack of a quantum theory of gravitation, the questioned legitimacy of the unavoidable renormalization procedures and the still unsolved 22

To my knowledge is was Simon Saunders’ deep insight into and his intense interest in AQFT which to a good part initiated the philosophical debate about AQFT. Already in the sixties Gordon Fleming was investigating questions of locality and covariance in the construction of position operators, e.g. in his articles Fleming (1965a) and Fleming (1965b). I consider his earlier work as part of the physics debate although later on Fleming became active in discussions among philosophers of physics with contributions on the connection of superluminal signaling and hyperplane dependence, e.g. in Fleming (1988), and together with Jeremy Butterfield on localization and Lorentz-invariance in Fleming and Butterfield (1999). 23 Cf., e.g., Redhead (1995b), Redhead (1995a) and the monograph on The Philosophy of Vacuum Saunders and Brown (1991).

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inconsistencies in connection with the measurement problem are the most prominent examples. QFT as it stands cannot be the final theory. How can further thought about its interpretation then be justified before the final consistent version is found? First, if one were to wait for this completion it is very likely that a philosophical analysis of QFT would never even start. Besides some “Dreams of a Final Theory” there is nothing which suggests that the basic theories of physics will be completely discovered in the near future. Second, interpretational reflections on the foundations of physics and its inconsistencies might help in the search for the final theory. Third, some quantal structures have been very ‘steady’ for more than 70 years now and lead to strikingly good predictions so that the belief is well-grounded that at least a good part of these structures will remain in all improved theories.

Part II

Classical Ontologies

Chapter 7 Classical vs. Revisionary Ontologies I now come to the main parts of the investigation about different ontological conceptions of QFT which is subdivided into classical and revisionary ontologies. Before starting to consider different ontologies this chapter will reflect upon the division itself into classical and revisionary ontologies. I will go into two connected aspects. One aspect is the historical forerunner of this division in content as well as terminology, namely Strawson’s distinction of descriptive and revisionary metaphysics. I will deal with the concept of descriptive metaphysics in the following introduction 7.1. The other aspect to be explored about the classical (or descriptive)/revisionarydistinction is the fact that classical ontologies, the term used in this thesis, as well as Strawson’s descriptive metaphysics are related to the concept of substance. It is an important link between the classical/revisionary- and descriptive/revisionary-distinction that for both classical and descriptive ontologies the notion of substance plays a central role which it either does not or in a highly non-standard fashion in revisionary ontologies. In sections 7.2 and 7.4 I will consider some diverging ways to understand the notoriously elusive notion of substance.

7.1

Introduction

In his book Individuals - An Essay in Descriptive Metaphysics P. F. Strawson introduced the notion of descriptive metaphysics as opposed to revisionary metaphysics Strawson (1959). He describes his basic idea as follows: Descriptive metaphysics is content to describe the actual structure of our thought about the world, revisionary metaphysics is 63

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concerned to produce a better structure.1 Without much further justification and discussion Strawson then proceeds to present his own contribution to the corpus of descriptive metaphysics. One part of his investigation are detailed studies about how we conceive of everyday “things” like sounds and persons. Although Strawson hardly uses the expression ‘ontology’, what he actually does is ontology albeit in the tradition of analytical philosophy of language.2 The term ‘descriptive metaphysics’ seems to display a contradiction in itself. Is it not that metaphysics tries to go beyond mere description? Although Strawson’s concept of metaphysics looks modest it is provocative in its context.3 It entails the reproach that various other metaphysicians were at least presumptuous if not misguided. Strawson names Descartes, Leibniz and Berkeley as historical examples for revisionary metaphysicians while he places himself in the tradition of Aristotle and Kant. Nevertheless, in 1959 it was obvious that Strawson’s “descriptive metaphysics” was a repudiation of certain strands in contemporary analytical philosophy. To make things even more controversial, Strawson adds a little later in his book about the metaphysician’s work: The structure he seeks does not readily display itself on the surface of language but lies submerged.4 Quite naturally, this last qualification of descriptive metaphysics arouses suspicion about the very possibility of descriptive metaphysics thus conceived. If descriptive metaphysics has to neglect certain traits of how our conceptions of the world appear to be in favour of structures which are then against our first intuition how are we to distinguish descriptive metaphysics from metaphysics? Provided that this suspicion is legitimate the distinction between descriptive metaphysics and revisionary metaphysics would be a matter of degree rather than being fundamental. 1

Strawson (1959), p. 9. In his more recent article “Semantics, logic and ontology” Strawson (1975) he explicitly elaborates on the close connection between semantics and ontology beyond the existence of merely structural resemblances. 3 P. Simons’ recent “Against modesty: claims of revisionary metaphysics.” Simons (1999) is one of the latest reactions Strawson has provoked. 4 Strawson (1959), p. 10. 2

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Criticism of this sort can be met at least partly. Strawson’s question is why we look at the world in the way we do. Strawson wants to describe the structures which yield an answer to this question and not the immediate surface of how we speak about everyday things. Descriptive metaphysics construed in this sense is descriptive relative to a certain level of what it tries to describe.5 The division of the following investigation into classical and revisionary ontologies has two reasons. The first reason is that each ontological conception within either group can be understood much better with an eye on this classification than in isolation because the historical and systematical background of the respective ontologies is very similar. Process and trope ontology, for instance, can both be seen as reactions to problems with classical substance ontologies. Whereas they agree, at least partly, about their negative diagnoses of classical ontologies, they differ on the positive side, i. e. concerning the appropriate remedy. Quite naturally, this common background leads to a certain direct competition between process and trope ontology. The second reason for dividing the following investigation into two parts is that it makes the discussion of different ontological approaches to QFT more effective. Motivations and problems are far more homogeneous within each group than across groups. One could almost say that the real competition begins only once you have chosen your group. For instance, particle and field interpretation of QFT are here taken together into the group of classical ontologies. They are the two standard options for the ontology of QFT. Accordingly, many investigations are taken to be of importance for this pair of alternatives rather than for one or the other alternative in isolation. It is convenient, therefore, to have the respective arguments close together. However, since the classical/revisionary-grouping still has, as many other distinctions, a certain degree of arbitrariness this cannot be the whole story.6 It is only meant as a first approximation which has to be refined 5

Tugendhat (1967) is an excellent early classification and evaluation of Strawson’s place in philosophy. 6 Wayne (2008), for instance, proposes an interesting combination of field and trope ontology, which lies diagonally to my partitioning. This shows that the four main options I discuss don’t need to be understood as being mutually exclusive.

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and corrected. To this end a comprehensive evaluation and discussion of all the considered ontological alternatives for QFT will conclude this study. I have preferred to speak of classical versus revisionary ontologies rather than speaking of descriptive versus revisionary. One reason are the abovementioned problems with the notion of descriptive metaphysics. Another reason has to do with one of my later results in the context of QFT. This result indicates that it is possible that an ontological approach which is commonly considered to be revisionary can turn out to be descriptive in certain cases. “So why don’t you put this ontology into the first group of classical or descriptive ontologies then?” you could ask. The answer is simple: because besides the possible appropriateness of the label ‘descriptive’ the above-mentioned reasons speak against this classification, i.e. because its background and its problems are similar to the ones of the other ontologies in this group. To be more concrete, I will argue that with respect to a certain formulation of QFT trope ontology appears rather descriptive while it is commonly rated as a revisionary ontology. As I mentioned already the concept of ‘substance’ is intimately connected with classical (or descriptive) ontologies. It is often taken to be the philosophical counterpart of the everyday notion of material things including ourselves. This makes it understandable why it is rooted at the centre of classical ontologies. Because “substance ontology” is so deeply rooted in classical ontologies it is often the negative background for revisionary ontologies. Some versions of process ontology even make the negation of all “substance-ontological presuppositions” their very starting point 7 Since the notion of substance is thus central for both descriptive and revisionary metaphysics the next two sections will deal with the question of how the concept of substance is to be construed.

7.2

Aristotle’s Theory of Substances

Although the notion of substance plays a central role in metaphysical writings from ancient times up to now there is little agreement among philosophers about its meaning. There is no single clear-cut meaning of the concept of substance to which even most philosophers would agree. 7

See Seibt (1996), for instance.

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Nevertheless, the everyday use of ‘substance’ yields a first approximation. When we say that something, say a business, has got a lot of substance we want to express that it has a particularly strong basis which renders it to be in a better position to survive rough times than another business with less substance. A business with a lot of substance is more likely to survive change since it is more independent from the ups and downs of the market due to its substantial basis. Even if it changes some features of its habit and some of its appearance it can stay basically the same. This illustration gives a first rough-and-ready idea of what is meant by substance.8 I will now proceed with the philosopher who is more than any other connected with the idea of substances, namely Aristotle. The notion of substance was introduced by Aristotle using the already existing term ousia (which goes back at least to Plato). Although Aristotle’s reflections about substances are a pivotal part of his metaphysics and there is ample material about this issue in his writings, the totality of Aristotle’s statements is apt to increase rather than diminish the despair about the lack of a uniform understanding. There are at least three aspects which contribute to this situation. It is not immediately clear what Aristotle’s view was and whether he had one consistent view at all. To make things worse, there is a plethora of features which different philosophers consider to be at the heart of the concept of substance. And finally, the relation of later versions of the concept of substance to Aristotle’s idea(s) is opaque. In order to cope with this situation there is a tendency to dispose of the problem by using ‘Aristotle’ as a mere label without claiming that a conception which is called ‘Aristotle’s view’ is exactly matching Aristotle’s actual view. On the one side this is a legitimate and fruitful way to get beyond philological debates about Aristotle. On the other side this procedure entails the risk that non-viable notions of substance ontology are constructed for the purpose to be refuted elegantly. In order to meet this risk I will discuss three contrasting current points of view. The first one is a modern and benevolent reconstruction of Aristotle’s view as a consistent and convincing theory of substance put forward by M. Frede and G. Patzig Frede and Patzig (1988). It is the main issue of this section. The other 8

A very accessible introduction to various historical and systematical questions about the concept of substance is the monograph Hoffmann and Rosenkrantz (1997).

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two accounts of the notion of substance have a negative diagnosis about its consistency and fundamentality and argue from the point of view of process ontology Seibt (1996) and trope ontology Simons (1998a) respectively. I will elaborate and comment on these accounts in the end of this chapter as well as in 10, 12 and 13. There are a number of interesting and important questions about Aristotle’s theory of substances which - in favour of other questions that are more important in the context of the present investigation - I shall not even touch. To what extent and in which sense is Aristotle’s theory a reaction and correction of Plato’s theory of forms? How is the fact that substance is one of Aristotle’s categories besides nine others related to his view that substances are those entities which have primary existence? Is there a development and change of ideas to be found in Aristotle’s writings so that one can say, for instance, that the books of his Metaphysics display his mature ontology while his Categories must be seen as the result of an early stage? How well is the term ‘substance’ suited as a translation of Aristotle’s term ‘ousia’ ? Detailed discussions can be found in the works cited in this section and the respective references therein.9 In what follows I will restrict myself to the discussion of the final results of the above-mentioned three contemporary authors about ‘Aristotle’s theory of substances’ in the sense of the most fundamental entities in the world. The first interpretation to be considered here is the one put forward by M. Frede and G. Patzig in their introduction, translation and commentary of the famous book Z of Aristotle’s metaphysics Frede and Patzig (1988). In the following paragraphs I will summarize their exposition and interpretation of Aristotle’s view.10 When we consider the existence of concrete particulars like horses or human beings, asks Aristotle, what is it that has primary existence. ‘Primary’ here means that everything else which is not an entity of primary existence is dependent on these primarily existing entities. Is the matter which a human body is composed of primary or is it its form? Of the two 9

A good and neither very technical nor philological recent account of Aristotle’s conception of substances can be found in the section “Aristotelian substances” (p. 123-135) of Loux (2002). A more detailed study by the same author is Loux (1991). A comprehensive and accessible authoritative introduction to Aristotle’s philosophy is Ackrill (1981). 10 Frede (1987), chapters 2-6 will be helpful for readers who are not familiar with German, in particular ‘Substance in Aristotle’s Metaphysics’.

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candidates for primary existence—form and matter—matter can be excluded since matter is always matter of an object. The existence of matter always presupposes something concrete as well as some form. One could argue now that something similar holds for forms as a well. The form of human beings or of horses is something which can be realized many times just as universal properties like redness can. However, according to Aristotle’s stance in opposition to Plato, universals do not exist before and apart from individual things in a separate realm of eternal ideas. Due to his own doctrine about universals Aristotle is thus blocked to say that universal forms have a primary existence. For Aristotle universal forms depend on the existence of concrete things and hence on matter. Both matter and universal forms are therefore dependent entities so that the search for entities with primary existence, which are not dependent on anything else, has not yet been successful. As to the choice between matter and form one has reached a stalemate. On the one hand neither matter nor form seem to be viable candidates for primary existence since both somehow depend on the other. On the other hand there seem to be no further alternatives. Frede and Patzig offer the following solution. They argue that with respect to the choice between matter and form not all alternatives have been considered yet. The arguments against the primary existence of form which were put forward so far refer only to forms taken as universals. This leaves room for the further alternative that forms are not construed as universals but as individuals. According to this interpretation by Frede and Patzig Aristotelian substances (or better ousiai) are individual forms. Only in a wider sense substances would be concrete things like horses or human beings where we have individual form and matter together.11 One can get an idea about this conception when thinking of a human body. Due to the natural metabolism every single molecule of a human body is replaced by a new one after some seven years, to my knowledge. Despite of this fact we still think that we are dealing with the same human 11

A concise but rather technical study of Aristotle’s theory of forms is Nortmann (1997). Chapter 11 is an illuminating evalutation from a modern point of view leaving all philological considerations aside. A very helpful short list of references to discussions about the question whether or not, according to Aristotle, forms should be conceived as particulars or universals can be found in Moreland (2002), footnote 19, p. 159.

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being. Since the permanence of a human being’s identity through this kind of change can thus neither be attributed to the material out of which he is composed nor to the general fact that he is a human being it must be his individual form that accounts for his persisting identity. I finish this section with some comments and critical remarks. For my first point I need to anticipate some later terminology and ideas whose explanation can be found in chapter 11 on trope ontology. It seems to me that one could call the interpretation of the Aristotelian notion of substance by Frede and Patzig something like a ‘singular trope theory’. From the point of view of trope theory an Aristotelian individual form could thus be characterized as a bundle of tropes which happens to consist of just one trope, namely the comprehensive form trope. Note that this is a characterization of the individual forms interpretation of substances in terms of trope ontology. It is a ‘perverse reading’, however, since one of the main goals of the individual forms view consists in getting away from somehow bundling properties. It is the very opposite of a bundle theory. Again, there will be more on bundle theories in later chapters on revisionary ontologies. For a first introduction to bundle theories consult the general chapter 3 on ontology. The characterization in terms of trope ontology brings me to something which to me seems like a weak point of the individual forms interpretation of substances (or ousiai). To use an expression coined by D. Armstrong Armstrong (1989) the construal of substances as individual forms is a “blob theory”. The advantage of blob theories is that they are very simple and afford little ontological expenditure. However, these advantages bring about an unpleasant disadvantage. The price blob theories have to pay is a certain shortage of explanatory power since there are a number of undebatable facts about the world which they cannot grasp. In the case of the individual forms interpretation of substances, it seems to me at least, that it is hard to explain what it means that two substances resemble each other. If the individual form of an object is on the most fundamental level already I cannot see what is left to analyze the resemblance of two substances. It is not satisfactory to say that two human beings resemble each other more closely than two others and to take this as a brute fact. One would like to refer to certain features of these people which are somehow aspects of the whole individual form. In an individual forms theory of substances,

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however, there are no entities one could refer to below the level of the individual forms. Even apart from the mentioned criticism of the individual forms interpretation of substances we are left with the problem that this construal of the Aristotelian theory of substances might be too closely tailored for living organisms. Aristotle’s prime examples for substances refer to living organisms like human beings or horses. Their physical parts in an everyday sense, e.g. the kidney, are not substances since they are not independent of the organism as whose organs they function. Parts in a scientific sense, e. g. electrons and quarks, are only potential substances as long as they are a part of a living organism. For ontological investigations regarding physical theories it seems to be more appropriate to consider other characterizations of substances.12 It is important for the context of the present investigation how the notion of substance has been construed by those philosophers who try to use it as a matrix from which to construct alternative conceptions either to the traditional notion of substance or as an alternative to substance ontologies altogether no matter of which kind. However, it should be clear by now that argumentations against purported traditional notions of substance should be handled with care. In the end, there might not be anything like the traditional notion of substance. Nevertheless, I believe that it is legitimate and fruitful in this situation to argue that certain construals of substance cannot be maintained and to try more viable or even completely diverging options. Section 7.4 is devoted to some of these attempts. They are of particular importance for part III of this study since they are the starting point for revisionary ontologies. 12

In his paper “Substances, physical systems, and quantum mechanics” E. Scheibe has argued that it is “the modern concept of a physical system which comes nearest to the traditional notion of substance” Scheibe (1991), p. 215. He discusses four aspects of this “traditional notion of substance”, independent existence, monadic predication, completeness and individuality.

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7.3

Substance Ontologies

The plural in the heading of this section is meant to emphasize that there is no single conception that could be called the substance ontology.13 However, since Aristotle’s Categories and Metaphysics there are a number of so-called category features which are discussed as essential for the notion of a substance. While categories (e.g. substances, properties, events, etc.) are meant to give us the most general classification of everything there is, the term ‘category features’ refers to the defining characteristics of these categories such as dependence, individuality, universality etc. Obviously two categories cannot have the same set of category features since otherwise they would be one and the same category. Characterizing ontological categories by making a list of category features is a typical, and I think very powerful, method of the analytical approach to ontology14 which in this respect takes up the Aristotelian tradition. Two main goals of (the general or formal part of) ontology are the identification and characterization of the basic categories under which all things can be classified. This section deals with the second goal. What follows is a list of category features which are commonly taken to be central for the notion of a substance or even as the core of its meaning. Whether all these features or certain combinations should or even can be appealed to at the same time is a question I postpone to the next section. Independence of Existence The main reason for the outstanding position of substances above other categories is their independence of existence, i. e. their ability to subsist alone. One of Aristotle’s examples for a substance is a horse whereas the leg of a horse would not meet the strong criterion of subsistent existence.15 However, the main contrast to the independent existence of a horse is not the less independent existence 13

The terminology as well as the style of exposition in the current section is very much inspired by the writings of J. Seibt and P. Simons, in particular Simons (1998a). Further relevant references to these authors can be found in section 7.4. A very lucid methodological reflection on ontology, or metaphysics respectively, as category theory can be found in the introductory chapter of Loux (2002). 14 For an introduction to analytical ontology see section 3.3. 15 Note that there is a certain ambiguity or maybe even shift of view in Aristotle’s writings—from Categories to Metaphysics—whether, e.g., a horse as a whole is an example for a substance or rather the unchanging and individual “core” of it.

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of a horse’s leg but the case of completely dependent entities like properties, states or boundaries. These entities are not independent because a property is always the property of something, a state is a state in which something is and a boundary is a boundary by which something is limited. Properties, states and boundaries, it seems, could not exist if there was nothing else.16 Certainly, this is true of a horse as well. However, it is at least conceivable that there is nothing but one horse whereas a boundary could not even be imagined without something else whose boundary it is. The boundary depends on that which it limits. In order to catch the above-mentioned differences it is helpful to distinguish different senses or degrees of dependence.17 Since the exact formulation of different forms of dependence is a fairly subtle and technical business I refer the reader to the relevant literature. The discipline which is primarily concerned with these distincions is mereology which deals with the relation of wholes and their parts.18 Ultimate Subjecthood of Predication A further feature which is often taken to be characteristic of substances is their being the ultimate subjects of predication. A substance is something to which predicates can be ascribed but a substance cannot itself be ascribed as a predicate to anything. One can say that Luigi Pastore from Bari likes rumpsteaks but it makes no sense to say that someone is a Luigi Pastore unless one refers to a kind of person rather than one particular person. Note that as a category feature being the ultimate subject of predication is by itself a characteristic of particulars rather than substances. Aristotle derived the category feature of subjecthood of predication from the importance of subject-predicate discourse. Aristotle thought that this way of talking and thinking reflects an ontological hierarchy of the things about which we talk and think. In modern times it has been repeatedly critized—looking at non-western languages—that subject-predicate 16

The situation is different of course for the radical position of Platonic realism about universals. Here properties can even exist when they are not instantiated. 17 Simons, e.g., distinguishes weak and strong dependence. Note for later purposes that a trope can be defined as a dependent concrete particular in contrast to a substance which is an independent concrete particular. These definitions are used, e.g., in Simons (1994), p. 557. 18 One of the best-known publications on mereology is Simons (1987).

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discourse should not be taken for granted. This observation is one motivation for revisionary ontologies. Individuality Another very important category feature of substances is their role as individuators. Something is needed which is responsible for an individual’s being an individual. A collection of universal properties could always be realized twice no matter how specific these properties are.19 An individuating factor is needed. There are several notions of individuality such as Aristotle’s prime matter and Duns Scotus’ haecceitas or thisness. In modern times the early Bertrand Russell and a bit more recently Gustav Bergmann defended bare particulars as individuators.20 And as Strawson argued in his book Individuals (1959) spatio-temporal location plays the individuating role. For later purposes I will concentrate on the bare particular theory of individuality. According to the standard account, bare particulars are supposed to be— due to their bareness—without any attributes. However, one can argue now that individuation cannot be the only role of bare particulars since otherwise their individuating role would be like saying that bare particulars individuate because they individuate.21 Hence, they have to play at least one further explanatory role and that is commonly seen to be their capacity to be bearers of properties—which is caught by the ancient expression ‘substratum’).22 This route spells trouble for the bare particular (or substratum) theorist, however, because a supposedly bare particular seems to have, besides individuating power, at least one further attribute, namely its being bearer 19

The possibility of multiple realization of “bundles of universals” is the main argument against an ontology in which particulars are analyzed as bundles of universals which stand in the so-called compresence relation to each other. An important ingredient of this argument against the tenability of the bundles of universals conception of things is Leibniz’ principle of the identity of indiscernibles. It states that two things which are not discernible by any property are in fact one and the same thing. Good discussions of the bundles of universals view—which was proposed by Russell in the forties—and further references can be found, e.g., in Armstrong (1989), chapter 4, and in Loux (2002), chapter 3. 20 See Bergmann (1967) as cited in Loux (2002), p. 103 and footnote 6, p. 135. 21 The argument I present here is a modification of Seibt’s argument, see p. 71, section 3.2 in Seibt (2002), regarding primitive thisness. 22 This explains why the modern expression ‘bare particular’ is often used synonymously with the ancient but still widespread expression ‘substratum.’

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of properties. Bare particulars thus do not seem to be quite as bare as they were supposed to be. And things might be even worse because as bearers of properties bare particulars have properties while at the same time this is exactly what their bareness is meant to exclude. Thus bare particulars have no attributes, due to their bareness, and they do have attributes, due to their being bearers of properties. The notion of a bare particular as an individuating bearer of properties seems to lead to an immediate contradition. There are some alternative ways for how to defend a bare particular (or substratum) view. One defense concerns the understanding of ‘bare’. If ‘bare’ means having no attributes essentially then the above-mentioned contradiction dissolves because bare particulars have and do not have attributes in two different ways. Due to their bareness bare particulars have no attributes (essentially) and as bearer of attributes they have attributes but only non-essential ones.23 Although this line of defense might meet the one part of the above-mentioned contradiction (viz. not having properties as a bare particular and having properties as a bearer of properties), the other part remains in place because the very attribute to be a bearer of properties is obviously an essential property since it catches the very essence of the notion of a substratum (or bare particular). Thus bare particulars would have at least this one essential property although they are meant to be bare of any essential properties. A possible counter-argument against this objection could result from questioning that being a bearer of properties is itself a property. The to-ings and fro-ings of the debate about the tenability of a substratum (or bare particular) theory show that the immediate objections do not go through as easily as it first appeared. Whereas it appears difficult if not impossible to settle the debate by purely philosophical reasoning I think one could get important impulses from a conceptual analysis of quantum physics. This is one of the reasons why I have discussed individuality a bit more intensely than the other category features of substances. I think it 23

Attributes of something are rated as essential if loosing them entails that this something looses its identity. An electron, for instance, can undergo various “harmless” changes regarding its spatio-temporal attributes but if it was to loose, e.g., its attribute of having one unit of negative electric charge, which is one of its essential attributes, then it would not be an electron any more. Thus the notion of essential attributes is closely linked to the issue of natural kinds, like electrons, or human beings.

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is in this respect more than in any other that quantum physics may be in conflict with substance ontological conceptions. Persistence Through Change One can only conceive of a concrete particular to change when something survives the change so that it can be said to have changed. Otherwise one gets the presocratian dilemma that either everything is constantly changing (Heraclitus) or that nothing can change (Parmenides)—leaving aside the controversial question whether or not this gloss does justice to Heraclitus as well as Parmenides. Aristotle’s solution to this puzzle is that there are substances which are survivors of change while, during a change, some (non-essential) attributes are replaced by new ones. As Simons stresses considering substances primarily as that which survives any change entails that the ultimate substances are indestructible.24 Today ‘persistence’ is commonly used as a neutral term for continuing existence of concrete things through time, covering both the so-called endurance as well as the perdurance view.25 According to the endurance view continuing existence is to be understood as identity of a whole thing through change in time, i. e. as diachronuous identity. The perdurance view is that material things are aggregates not only of their spatial parts but of their so-called temporal parts as well. Thus according to a perdurantist ontology things are, just like events, four-dimensional entities so that the alleged identity through time should rather be understood as a continuation of temporal parts. Looking back now at possible definitions of individuality one can see that reidentifiability can only be a sufficient condition for individuality since it depends on the view one adopts regarding different understandings of persistence. Determinateness Determinateness is the analytical extraction of one aspect—besides unity—of what Aristotle might have meant by his less technical expression tode ti, “a this”. Something has determinateness if it is via particular properties and a particular behaviour characterized as 24

See Simons (1998a), p. 238. The endurance-perdurance distinction was introduced by David Lewis in Lewis (1986) and elaborated in particular by Mark Johnston (1987). As the source for these pieces of information and a further discussion confer chapter 6 in Loux (2002), in particular footnote 1, p. 247. For a reference to Johnston’s work see footnote 5, p. 248, again in Loux’s book. 25

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an specimen of a (natural) kind. For this reason these properties and behaviour are called essential.

7.4

Substances Under Attack

Generally, ontologists who rate themselves as revisionary26 repudiate the view that the notion of substance is indispensible and basic for our ontological thinking about the world, even on the everyday level. The elusiveness of the concept of substance makes it particularly hard to evaluate the legitimacy of this repudiation, however. Since there are various different ways to construe the concept of substance it is difficult to argue for or against its applicability. These problems led to the following strategy on the side of revisionary ontologists. In a first step they sort out a certain set of characteristics which they consider to be indispensible for the concept of substance. In a second step they show that substance when understood in this way either leads to contradictions or has no or almost no cases which would fall under this concept. Argumentations against ‘substance ontology’ do mostly not contend that each single ingredient of the notion of substance has to be dropped. It is rather that certain sets of ingredients are either contradictory in themselves or have a poor applicability. Peter Simons, in his article “Farewell to substance: a differentiated leavetaking” Simons (1998a), concedes that substance is legitimate and “harmless” as an everyday notion. However, he argues that substance forfeits its status of being primitive, i. e. unanalysable, after metaphysical reflection in general as well as with respect to particular sciences. Simons’ main point is that substances do not have the fundamental status that is commonly ascribed to them since they can be further analysed in terms of tropes. Thus Simons’ point against the notion of substance is not that it was inconceivable or even inconsistent. He only claims that substances are not basic. I will come back to this point several times during my investigation. Simons’ stance is of particular value for the present ontological investigation of QFT because he considers the results of particular sciences as a 26

Note that the self-assessment does not always match with the mutual assessment. For instance, process-ontologist Johanna Seibt rates trope ontology as conservative whereas trope-ontologists see themselves as revisionary Simons (1999).

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pivotal source and guideline for metaphysics. Since I will set out trope theory in general in chapter 11 and Simons’ version of it in chapter 13 I will not go into any details here. J. Seibt’s critique of substance ontology is much more radical than Simons’. She does think that the notion of substance is inconsistent and that there is no remedy. Seibt concentrates on three basic ontological problems, the problem of individuality, the problem of universals and the problem of persistence. According to her view, any ontology that adheres to the so-called “substance-ontological paradigm” fails in the face of these problems.27 One key issue in Seibt’s argumentation is the relation between individuality and particularity which is a subtle and, if reflected upon at all, a controversial matter. While these two notions are often effectively equated, Seibt points out that there are concrete individuals which are multiply occurent, for instance types, activities and stuffs.28 An example would be a smile that someone has in certain situations. The existence of multiply occurent individuals shows, according to Seibt, that it is conceivable to have individuals that are not particulars so that individuality would not entail particularity, as it is often assumed without further comment. To break up the unquestioned linkage between individuality and particularity is in fact a central step in Seibt’s process-ontological approach. At this stage the brief mentioning of one of Seibt’s points is only meant to show the variety of ways how substance ontologies can be challenged. Processontologically motivated objections to substance ontologies in general as well as Seibt’s approach in particular will be taken up again in detail in chapter 10. There, quantum physical considerations will have the primary impact as far as the clash between substance- and process-ontological conceptions are concerned. 27

For details see, e.g., Seibt (1995), Seibt (1996) and Seibt (2002). See p. 69 (section 3.2) as well as p. 82f (section 3.5), in particular footnote 53, in Seibt (2002). 28

7.5. SUBSTANCE ONTOLOGY AND QUANTUM PHYSICS

7.5

79

Substance Ontology and Quantum Physics

While there have been a number of philosophers who attacked substance ontologies, one if not the most serious threat for the tenability of substanceontological thinking roots not in philosophy itself but in conceptual reflections about modern physics.29 One of the most famous of these attacks was launched by Ernst Cassirer and his arguments have been taken up repeatedly. However, in analyses of Cassirer’s arguments it became evident that one has to be careful with a premature diagnosis of an incompatibility of quantum physics and substance ontologies. Since Cassirer presupposes one particular notion of substance which goes back to Leibniz any conclusion has a very limited range of validity. For that reason I wish to make a systematical investigation of various issues in quantum physics where a conflict with substance-ontological conceptions might occur. Mostly these conflicts refer only to certain (potential) features of substances and not to a particular notion of substance. In order to show the more ground preparing nature of my investigation I will give three examples of potential problems between category features of substances on the one side and an array of relevant aspects of quantum physics on the other side. 7.5.1

Incompatible Observables

One crucial novelty of quantum physics in comparison to classical physics consists in the impossibility to simultaneously ascribe determinate values for incompatible (or incommensurable) observables like position and momentum to a quantum object. Whether or not this impossibility precludes the application of the notion of substance for quantum objects depends not only on the selection of category features but also on how one understands these features. In the context of incompatible observables the most immediately problematic category feature is determinateness. One view is that for something to be a substance all its properties must be determinate at any given time. Understood in this way the category feature can also be called completeness since it requires substances to be completely 29

In Kuhlmann (2006) I measure to what extent quantum physics prompts a revision of traditional substance ontologies. To this end I explore several conflicts between (potential) features of substances and some outstanding conceptual peculiarities of quantum physics.

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determined. This view—which roots in Leibniz’ work—is often taken as a basis for showing that the notion of substances is inappropriate in the case of quantum objects. However, as Soler Gil has shown in his extensive investigation ”Aristotle in the Quantum World” (2003) this conflict only arises for the Leibnizian conception of substance but not the Aristotelian one. According to Aristotle, says Soler Gil, a substance is determinate in the sense that it is determined to which kind it belongs and to this end it is only necessary to specify its essential properties. Position and momentum, however, are relational properties and thus do not belong to the essential properties of a quantum object. And the same applies to all other pairs of incompatible observables. Essential properties of quantum objects, on the other side, like mass and charge, are not subject to any uncertainty relations. The determinateness of only the essential properties of quantum objects could thus suffice to avoid a conflict with the general tenability of the category feature determinateness. Nevertheless, due to a potential conflict with another category feature, namely individuality, the impossibility to simultaneously ascribe determinate values for incompatible observables to a quantum object might still undermine the application of the notion of substance. However, again this situation hinges on how one understands the category feature individuality. If one adopts the view that continuous spatio-temporal location, i.e. a trajectory, guarantees the individuality of a substance then the best-known incompatibility of a pair of observables, namely of position and momentum, makes it impossible to understand quantum objects as substances. Due to Heisenberg’s uncertainty relations a quantum object can never be in a state where both position and momentum have determinate values— if the position is sharp, then the momentum is completely indeterminate and vice versa. And this is the point where the discussions of potential conflicts between peculiarities of observables in quantum physics on the one side and the category features of determinateness and individuality on the other side are linked. Although the inevitable indeterminateness of the values for either position or momentum (or both) is not necessarily in conflict with the category feature of determinateness, if understood in an Aristotelian sense, there might still be, due to the same indeterminateness, a conflict with another category feature, namely individuality. Nevertheless, again there is only a conflict on the basis of one particular conception

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of individuality. In this conception it is a continuous spatio-temporal path which accounts for the individuality of (impenetrable) substances. But as it became evident in the above discussion of category features this is just one way to understand individuality. In conclusion one can say that the impossibility of a simultaneous ascription of determinate values for incompatible observables does contain a threat for the applicability of the notion of substance in the range of quantum physics. However, all the actual conflicts with category features of substances I have discussed in this section depend on certain readings of these features. Since the investigation of incompatible observables does not settle the issue of the tenability of a substance-ontological categorizing in quantum physics in an uncontroversial way it is necessary to explore further potential points of conflict. 7.5.2

Non-Vanishing Vacuum Expectation Values

Non-vanishing vacuum expectation values cause trouble for the idea that substances are ultimate subjects of predication. Unless one is willing to say that the relativistic vacuum of quantum field theory is itself a substance, or consists of substances—which is not quite as absurd as it might first seem—one gets the following problem. Since expectation values come about by averaging over all possible measurement outcomes according to their respective probabilities, non-vanishing (i. e. with a value other that zero) expectation values seem to indicate that there is something happening in the vacuum without there being anything to which this activity could be predicated. This argumentation has to be taken with care, however. Instead of simply saying that the vacuum is the state with no particles present—since the expectation value of the number operator is zero—it should be noted that by definition the vacuum is the energy ground state, i. e. the eigenstate of the energy operator with the lowest eigenvalue, and for the relativistic vacuum of quantum field theory this value is not zero as in the classical case. Moreover, due to the inherently relativistic Unruh effect, the statement that no particles are present is contingent on the choice of the inertial frame. Nevertheless this fact does not immediately solve the potential conflict between the occurrence of non-vanishing vacuum expectation values and the notion of substances as ultimate subjects

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of predication. The surprising features of the relativistic vacuum speak against the idea that particles are the substances in quantum physics but that does not by itself rule out other candidates for substances. Summing up one can say that either the idea of substances is not applicable in the realm of quantum physics or the viable candidates are much less classical than one might wish. Eventually, it must be emphasized that it is in particular the potential loss of the category feature individuality that may endanger the applicability of substance-ontological thinking to quantum objects. Problems concerning the individuation and reidentifiability of particles, i.e. the distinguishability of particles in its diachronous and in its synchronous aspect respectively, have notoriously caused trouble for the idea of individual traceable particles. These problems arise in QM already and lose nothing of their importance in QFT. I will discuss these issues at length in the appendix section B.2. Summing up one can put on record that, first, quantum physics clearly exerts revisionary pressure on substance ontologies. However, and this is my second result, the revisionary pressure depends very much on the type of substance ontology one is considering, or in other words, which category features one ascribes to substances.

Chapter 8 Particle Interpretation of QFT The particle interpretation of QFT is not only the oldest ontological attitude towards QFT, it seems almost impossible to think of QFT without thinking of particles. Learning that the top quark has been “observed” seems to leave little doubt that we are finally dealing with particles even when these particles have “strange colours and flavours.” Why should billions of dollars have been spent on particle accelerators when there are no particles to accelerate? Although it seems undeniable that modern physics is to a large extent making theories and experiments involving particles it is this very interpretation which has the most fully developed arguments against it. Why not simply dismiss the particle interpretation then? I can see at least two reasons. Firstly, the immediate evidence speaks in favour of the existence of particles. Secondly, it turned out to be a difficult task to say what the indispensable characteristics of the particle interpretation are. The dismissal of the particle interpretation is not the only way to react to the arguments against it. There still is the option to say that our classical concept of a particle is too narrow and that we have to loosen some of its constraints. Allowing for these options, various arguments against a particle interpretation of QFT will be explored and evaluated in this chapter. Before doing so, the first section is reserved for some general reflections about the particle concept itself.

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8.1 8.1.1

The Particle Concept General Features

The notion of particles is at the core of all classical corpuscular theories of matter. But even in classical physics the concept of an (elementary) particle is not as unproblematic as one might expect. On the one side, conceiving of a classical particle as akin to a tiny golf ball immediately prompts the further question for its parts. On the other side, the assumption of point particles leads to physical problems when charged particles are considered: If the whole charge of a particle was contracted to a point, an infinite amount of energy would be stored in this particle since the repulsive forces become infinitely large when two charges with the same sign are brought together. The so-called self energy of a point particle is infinite. These reflections may suffice to indicate that the particle concept has to be construed before it is put to use. Since it might not be the most appropriate way to start with a rigid definition of a particle, I will, in this section, only collect and reflect upon some possible ingredients of the particle concept. At this stage, no final decision is attempted as to the question what constitutes necessary conditions, whether one can find a sufficient condition and which features should be rated as only contingent. Moreover, I think it is advisable to keep an eye on experience, i. e. on empirically corroborated scientific theories. Probably the most immediate trait of particles is their discreteness. Particles are countable individuals in contrast to a liquid or a mass. Obviously this characteristic alone cannot constitute a sufficient condition for being a particle since there are other things which are countable as well without being particles. Money is countable but one would not say when three hundred Euro are paid into an account that three hundred discrete individuals have been added to the other individuals on that account. A physical example are the countable maxima and minima of the standing wave of a vibrating string. It seems that primitive thisness or haecceity1 is missing to make up a sufficient condition for a particle. In addition to being countable it must be 1

The notion of haecceitas or thisness has been introduced by Duns Scotus around 1300.

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possible to say that it is this or that particle which has been counted. The qualification ‘primitive’ indicates that the ‘thisness’ cannot be analysed any further, there is nothing else which is responsible for it or which it explains it. An example from physics may serve to underline what is meant. Disturbances of a medium can result in the propagation of a regular pattern of ups and downs, e. g., after a stone has been thrown into a pond. It is a characteristic feature of such a wave motion that different disturbances which are propagating in opposite directions (e. g. when two stones have been thrown) produce additive results when they meet and retain their undistorted shape when the waves have passed each other. Although the ups and downs of the displacement of water can be counted, one would hesitate to say that we are counting the same discrete individuals, say before and after the two wave patterns have passed each other. There seems to be a fundamental difference between ups and downs in a wave pattern and particles. Particles seem to have primitive thisness in addition.2 There is still another feature which is commonly taken to be pivotal for the particle concept, namely that particles are localizable in space. As was argued in the first paragraph of this section, localizability need not refer to point-like localization. However, it will turn out in section 8.3 that even localizability in an arbitrarily large but still finite region can be a strong condition for quantum particles.3 Eventually, I wish to mention possible ingredients of the particle concept which are explicitly opposed to the corresponding (and therefore opposite) features of the field concept. Whereas it is a core characteristic of a field that it is a system with an infinite number of degrees of freedom, the very opposite holds for particles. A particle can for instance be referred to by the specification of the coordinates x(t) that pertain, e. g., to its center of mass. The number of degrees of freedom of the particle is then given by the number of these coordinates. In contrast to a particle, a field φ(x, t) has to be specified by its value for each point (x, t), where the components of x as well as t range over all real numbers (i. e. xi , t ∈ IR) so that the 2

In Teller (1995), p. 103f and p. 112f, primitive thisness as well as other possible features of the particle concept are discussed in comparison to classical concepts of fields and waves as well as in comparison to the concept of field quanta. 3 The introduction in Wightman (1962) is a good account of the first period of research about problems with localizability in quantum physics. References to more modern investigations will follow in section 8.3.

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field takes independent values at an infinite number of space-time points. A further feature of the particle concept is connected to the last point and again explicitly in opposition to the field concept. In a pure particle ontology the interaction between remote particles can only be understood as an action at a distance. In contrast to that, in a field ontology, or a combined ontology of particles and fields, local action is implemented by mediating fields.4 Finally, there are two further features to be mentioned in this section which are commonly associated with particles. Classical particles are massive and impenetrable, again in contrast to (classical) fields.5 And, to use the above-mentioned example again, wave fields can be superimposed without any mutual disturbance. Nevertheless, although massiveness and impenetrability are distinguishing features of classical particles versus classical fields, the contrast is blurred in the quantum regime. The quantum light particle, i. e. the photon, is massless (referring to its rest mass) and infinitely many photons can be packed into the same region of space. We are here confronted with manifestations of the wave-particle duality. After this brainstorming of possible features of a particle let me say something about the purpose of proceeding in such an open way. The concept of particles has been evolving through history in accordance with the latest scientific theories. In Newtonian physics momentum could only be associated with matter and, in an atomistic view, ultimately with elementary particles. However, the completed theory of electromagnetism in the 19th century informed us that the ascription of momentum and energy is not quite as exclusive. One can ascribe energy and momentum just as well to an electromagnetic field in an area with no particles. This is only one example how the distinction of genuine particle features and genuine field features has been changed by new scientific findings. Carrying energy and momentum are no longer considered as distinctive characteristics of particles. Therefore, considering the tenability of a particle interpretation for QFT is not quite such a straightforward issue as one might think. There is no fixed concept of particles available so that there are always two options 4

In Haag (1996) considerable emphasis is put on this feature of fields. See for instance p. 7f. 5 For a discussion of the relation between impenetrability and synchronic as well as diachronic identity see Lowe (1998), pp. 112f.

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when one is confronted with features of QFT that are in conflict with the above-mentioned features of particles. Either one abandons the particle interpretation or one adjusts the particle concept, as has been done before. I think that this choice can sometimes be quite hard and involves the consideration of different clashing principles. 8.1.2

Wigner’s Analysis of the Poincar´ e Group

This section deals with a much recognized and, at least partly, successful attempt by Eugene Wigner (1939) to give a more rigorous account of particles. Some see this approach as supplying the best, and within its range of validity, complete answer to the question ‘What is a particle according to modern physics?’. Nevertheless, however illuminating the next section may be, as such it will not answer the question ‘Is a particle interpretation of QFT possible?’. It will rather put flesh on the bones of a particle interpretation—provided that such an interpretation should be tenable in the first place. I will elaborate on this issue after having introduced Wigner’s approach. Relativistic quantum mechanics (RQM) is an intermediate step between ordinary (non-relativistic) quantum mechanics and (relativistic) quantum field theory.6 On the one hand RQM was - like ordinary quantum mechanics - initially meant to be a particle theory. On the other hand the reconciliation of ordinary quantum mechanics with special relativity theory is a very important step towards (relativistic) quantum field theory. Since relativistic covariance is the only new ingredient in RQM as compared to ordinary quantum mechanics it seems reasonable to expect that an investigation of the relevant symmetry group of special relativity theory, namely the Poincar´e group, can yield information about invariant entities. The general idea about the connection between properties of physically relevant symmetry groups and ontological questions is the following.7 A symmetry group is a set of transformations which do not change the (structure of the) laws of nature. Symmetry transformations can be understood either as transformations of the coordinate system, on a passive interpretation, or of all the actual physical objects, on an active interpretation. 6 7

See the physics glossary and chapter 4 Symmetry considerations are, for instance, of central importance in Auyang (1995).

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Such transformations can be continuous ones like spacetime translations or rotations or they can be discrete ones like spatial reflections. Depending on the physical theory one is dealing with—e. g. classical mechanics, electromagnetism, special relativity theory, quantum mechanics—one has different symmetry groups, like the Galilei group for classical mechanics or the Poincar´e group for special relativity theory. Changes in the basic symmetry groups generally reflect a major change in what one believes to be the right physics. Since symmetry operations change the perspective of an observer but not the physics an analysis of the relevant symmetry group can yield very general information about those entities which are unchanged by a change of perspective. Such an invariance under a symmetry group is a necessary (but not sufficient) requirement for something to belong to the ontology of the considered physical theory. While symmetry transformations act on coordinates one can have corresponding transformations that act on the states of a physical system. In group-theoretic terms one says that the symmetry transformations can be represented by operations on a set of states, which is therefore called a representation space. Often one denotes an element of a symmetry group G by g and the corresponding operation on a set of physical states by Tg . For a set of such operations on a space of states to be a representation of the group, the basic requirement is that for any two g1 , g2 ∈ G the composition rule (8.1) Tg1 Tg2 = Tg1 g2 is fulfilled. In words, this requirement says that an operation on the state space which corresponds to the succession of two group transformations g2 and g1 is the same as the successive operations corresponding to the separate group transformations. I. e. directly working with operations in the representation space gives you the same result as first performing successive group transformations and only then representing the result on the state space. A representation of a group thus preserves the group structure, which explains the term ‘representation.’ Elementary particle physics belongs to the domain of special relativity because the velocities of elementary particles are of the order of the velocity of light. The relevant symmetry group is the Poincar´e group which is the invariance group of the 4-dimensional Minkowski spacetime M , i. e. the spacetime of special relativity, in contrast to the classical 3-dimensional Eu-

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clidean space and the separate 1-dimensional time that are used in Galilean Physics. The statement that the (proper) Poincar´e group, denoted by P+↑ , is an invariance group means that the physics, i. e. the laws of physics, must not change under transformations that are specified by this group.8 An element of the Poincar´e group is a transformation g := (Λ, a) acting on Minkowski spacetime where Λ denotes Lorentz transformations (boosts and rotations), and a denotes translations in 4-space, i. e. in Minkowski spacetime.9 The Poincar´e group is therefore a 10-parameter group.10 While Poincar´e transformations act on Minkowski space-time one can have corresponding transformations, e. g., on a Hilbert space. As explained above, if a set of such transformations fulfills a group multiplication rule similar to 8.1, it is called a representation of the Poincar´e group on that Hilbert space. Since the representing entities, or short ‘representors’, are unitary operators on that Hilbert space they are denoted by U (Λ, a) where the Λ’s are Lorentz transformations and the a’s spacetime translations, as above.11 An important example for the later analysis is how a spacetime translation in Minkowski space is represented by an operator in the representation space, i. e. the Hilbert space. It is given by U (a) = e−iP 8

µ

aµ

(8.2)

The symbols ‘↑’ and ‘+’ indicate that spatial reflections as well as changes in the direction of time are excluded. The exclusion of these discrete transformations is caught by the adjective ‘proper’. 9 While the homogeneous Lorentz group consists only of Lorentz boosts and rotations, the inhomogeneous Lorentz group (or Poincar´e group) has spacetime translations in addition. Thus in order to take full account of special relativity theory one has to consider the inhomogeneous Lorentz group (or Poincar´e group). 10 Correspondingly, the Poincar´e group has 10 generators, i. e. three Ks (“kick”) for boosts in three directions, i. e. a change of the Lorentz frame, three Js (angular momentum) for rotations about three axes and four P s (4-momentum) for translations in space-time. 11 Strictly speaking one is dealing with a projective representation of the Poincar´e group. This further complication is due to the occurrence of phase factors because quantum states are not represented by vectors but rays in a Hilbert space. In order to avoid the use of a composition rule different from the one given in 8.1 one can swallow the phase factors by going over to the covering group of the Poincar´e group, of which the unitary operators U (Λ, a) form an ordinary representation. See theorem 3.1.2 as well as the relevant explanations in section I.3.1 of Haag (1996) for a few more details or section 2.7 in Weinberg (1995) for the whole story and an interesting comment on the connection with superselection rules at the end of that section.

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where P µ is the energy-momentum operator which is the infinitesimal generator of spacetime translations. Moreover, I have used the abbreviation U (1, a) ≡ U (a). The elements of a representation of the Poincar´e group by unitary operators U (Λ, a) on a Hilbert space effect a transition |ψ → |ψ
 = U (Λ, a)|ψ

(8.3)

in the state space, i. e. in the Hilbert space which is the representation space of this representation of the group. The elements of this kind of representation, i. e. the unitary operators U (Λ, a) thus lead from one state to another. Therefore, to each representation there corresponds a certain set of states which are connected by these representors of the underlying symmetry group. There are some representations which have a feature that differentiates them from the general case and that is irreducibility. In the special case of an irreducible representation there is no proper subset of the representation space of states which is invariant under the action of a corresponding set of representors, i. e. here the unitary operators U (Λ, a). In other words, the elements of an irreducible representation transform the representation space of states into itself but not do so for any proper subset of it. The notion of irreducible representations of a symmetry group is the key ingredient of Wigner’s famous analysis of the Poincar´e group from 1939.12 The main idea that links up the so far mostly mathematical issues with physical and possibly ontological questions is the supposition that each irreducible (projective) representation yields the state space of one kind of elementary physical system. The prime example of an elementary system is an elementary particle which has the more restrictive property of being structureless.13 The physical justification for linking up irreducible representations with elementary systems is the requirement that “there must be no relativistically invariant distinction between the various states of the 12

See his paper Wigner (1939) “On unitary representations of the inhomoneneous Lorentz group”. While this paper is comprehensive and detailed it is clearly not a particularly accessible introduction to these issues. For this purpose consult the literature cited footnotes 15 and 16. 13 Here I am using Wigner’s (and T. D. Newton’s) expressions and categorisations. The distinction between elementary systems and elementary particles is further discussed in Newton and Wigner (1949).

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system.”14 In other words this means that the state space of an elementary system shall have no internal structure with respect to relativistic transformations. Put more technically, the state space of an elementary system must not contain any relativistically invariant subspaces, i. e. it must be the state space of an irreducible representation of the relevant invariance group. If the state space of an elementary system had relativistically invariant subspaces then it would be appropriate to associate these subspaces with elementary systems. The requirement that a state space has to be relativistically invariant means that starting from any of its states it must be possible to get to all the other states by superposition of those states which result from relativistic transformations of the state we started with. The main part of Wigner’s analysis from 1939 consists in finding and classifying all the irreducible representations of the Poincar´e group. Doing that involves finding relativistically invariant quantities that serve to classify the irreducible representations. It turned out that the crucial quantity which is conserved by relativistic transformations is p2 = pµ pµ = m2 c2

(8.4)

which is derived in appendix A and where the definition of the energymomentum 4-vector pµ is given. Using the convention c = 1 (see the entry ‘Natural Units’ in the physics glossary) one gets the first classification parameter p2 = m2 . Since we are here dealing with a relativistically conserved quantity different values of p2 correspond to different irreducible representations. In order to bring more physically relevant structure into the spectrum of values for p2 and hence into the array of irreducible representations we only need one further consideration before giving a complete classification table of irreducible representations. According to Galilean relativity, when we pick out one time, the present, we get a partition of spacetime into three different regions, the present, the past and the future. The situation is very different in special relativity. Here picking out one point, or an ‘event’, in Minkowski spacetime, say x, (and one frame of reference) leads to a partition into six different regions. First one has the present, second, the region of points that can be causally influenced by something happening at x (the forward light cone), and, third, the region that can have causally influenced x (backward light cone). 14

Introduction of Newton and Wigner (1949).

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Condition(s)

Standard interpretation

Geometrical description

(i) p 2 = m2 > 0, p 0 > 0

massive particles

hyperboloid in forward light cone

(ii) p 2 = m2 > 0, p 0 < 0

unphysical

hyperboloid in backward light cone

(iii) p 2 = 0,

p0 > 0

massless particles surface of forward light cone

(iv) p 2 = 0,

p0 < 0

unphysical

(v) p µ ≡ 0

(→ p 0 = 0) vacuum

(vi) p 2 < 0

virtual particles

surface of backward light cone point space-like hyperboloid

Table 8.1: Classes of irreducible representations. So far the situation is similar to Galilean relativity. But in the case of special relativity there are still quite a few points left over due to the fact that there is an upper bound for the propagation speed of a signal, namely the velocity of the speed of light. The causal past and the causal future have boundaries that are only accessible by light rays and points beyond these boundaries can neither be causally effected by x nor can they have had a causal effect on x. So we have three further physically distinguished regions besides the present, the causal past and the causal future. These three regions are the surface of the forward light cone, the surface of the backward light cone and the points that are spacelike separated from x. The complete list of classes of irreducible representations of the Poincar´e group is displayed in table 8.115 This table is generally seen as the classification of one-particles states with respect to their transformation behaviour under the Poincar´e group. As one can see classes of irreducible represen15

The given list is a combination of corresponding lists and commentaries on p. 60 in Ryder (1996), p. 29 in Haag (1996) and on p. 66 in Weinberg (1995). Note that there is a typographical error in Haag’s table since it has to be p0 > 0 and p0 < 0 for the third and fourth class.

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tations with p 0 < 0, i. e. with negative energy, are seen as probably not corresponding to any physical systems since the lack of a lower bound of the energy would render a physical system unstable. The next classification parameter is the spin s. This second parameter allows for a further distinctions between irreducible representations that correspond to the same mass. It is thus the pair (m, s) that catches the complete classification of irreducible representations. To each pair (m, s) there corresponds a different irreducible representation and hence a different kind of elementary system of which elementary particles are the most important examples. Eugene Wigner’s pioneering identification of types of particles with irreducible unitary representations of the Poincar´e group is exemplary until the present.16 One success of Wigner’s approach is that relativistic wave equations for all possible types of free particles, such as the Klein-Gordon equation or the Dirac equation, can be derived in a systematic fashion without heuristic ad hoc moves. For instance, one can get the Dirac equation by studying how spinors transform under the (homogeneous) Lorentz group. Concluding the account of Wigner’s analysis of the Poincar´e group I will address the question whether Wigner has supplied a definition of particles as it is often put in the literature. I wish to argue that although Wigner has in fact found a highly valuable and fruitful classification of particles his analysis does not contribute very much to the question what a particle is and whether a given theory can be interperted in terms of particles. What Wigner has given is a sort conditional answer. If relativistic quantum mechanics can be interpreted in terms of particles then the possible types of particles correspond to irreducible unitary representations of the Poincar´e group. However, the question whether, and if yes in what sense, RQM can be interpreted as a particle theory at all is not even addressed in Wigner’s analysis. For this reason the discussion of the particle interpretation of QFT is not finished with Wigner’s analysis as one might be tempted to say. For instance the pivotal question of the localizability of particle states is still open. 16

The content and significance of Wigner’s paper Wigner (1939) is discussed, e. g., in Haag (1996), p. 28ff, Streater (1988), p. 144ff and in the introduction of Buchholz (1994). A very short, fairly non-technical introduction as well as a critical evaluation can be found in Auyang (1995), section 3.6.

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I will end this section with some hints and references on further more technically involved issues connected with the particle concept. Various extensions of the particle concept have been considered in AQFT over the past decades. Some keywords are infra- and quasi-particles as well as the particle content of a quantum field theory and asymptotic particle states.17 To give just one example, B. Schroer introduced the notion of infraparticles Schroer (1963) in order to cure the restricted applicability of Wigner’s particle concept to particles which do not carry electrical charge.18 The next section deals with the best-known scientific context in which elementary particles are investigated. The point of concern will be the tension between this experimental basis of QFT on the one side and the conceptual investigations about the corresponding theory on the other side.

8.2

Theory and Experiment in Elementary Particle Physics: Is a Particle Track a Track of a Particle?

In today’s most basic theory of the material world, quantum field theory (QFT), there seems to be an insurmountable hiatus between two apparently incompatible conceptions of the fundamental entities: fields and particles. On the one hand there is a long and successful tradition of scattering experiments in particle accelerators. The observed ‘particle traces’, e. g. on photographic plates of bubble chambers, seem to be a clear indication for the existence of particles. On the other hand, however, the theory which has been built on the basis of these scattering experiments, viz. QFT, turns out to have considerable problems to account for the observed ‘particle trajectories’. Not only are sharp trajectories excluded by Heisenberg’s uncertainty relations for position and momentum coordinates which hold for non-relativistic quantum mechanics already. More advanced theoretical 17

A non-trivial account of these developments can be found in Buchholz’ overview article “On the manifestations of particles” Buchholz (1994). A decisive early study was published in the article “When does a quantum field theory describef particles?” Haag and Swieca (1962). 18 The term infra-particles refers to the fact that the infrared problems are avoided which made the extension of Wigner’s particle concept to electrically charged particles impossible.

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examinations in AQFT19 , which will be described and scrutinized in the present chapter, show that quantum particles which behave according to the principles of relativity theory cannot be localized in any bounded region of space-time, no matter how large. This result excludes even tube-like trajectories which are allowed (provided their boundaries are unsharp20 ) when only the constraints of Heisenberg’s uncertainty relations are taken into account.21 From this theoretical point of view it thus appears to be impossible that our world is composed of particles when we assume that localizability in any region of space-time is a necessary ingredient of the particle concept. Surprisingly, the very theory which excludes localizability is remarkably good in predicting experiments which apparently involve localizable particles. For the working physicist this contradiction is not an important issue because it does not cause any problems, neither for the theoretical nor the experimental physicist, as long as conceptual questions as such are not at stake. In the last few decades Rudolf Haag and his colleagues, a group of theoretical physicists which puts much emphasis on conceptual clearness in their often pioneering work, have tried to fill this longstanding gap between theory and experiment. Within the framework of AQFT they proposed a mathematical model for ‘almost localized’ particles as they appear in scattering experiments22 . The main ideas are firstly to describe scattered particles in terms of measurement results of a certain arrangement of particle detectors and secondly to assume only approximate localizability. Before coming to more details of this model the next few paragraphs will give some necessary background information about experiments in High Energy Physics and the sense in which there is a gap between the achieved results and their theoretical description. 19

A basic account of algebraic quantum field theory, in short AQFT, was given in section 5.3. Section 6.2 dealt with the relation between QFT and AQFT. The present chapter is, together with chapter 13, the most important reason why I have introduced AQFT. At least some parts of section 8.3 presuppose these accounts. 20 The requirement that the tubes are to have unsharp walls is due to Jauch’s theorem. For details consult Jauch (1974). 21 The blurredness of real particle tracks, however, is much larger than the minimum which is required by Heisenberg’s uncertainty relations. 22 Haag (1996), pp. 75-94, 271-289, is an introduction in two steps with an increasing degree of complexity.

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All the experimental information which was used to test QFT comes from particle detectors. These are employed in target regions of particle accelerators where very fast and therefore very energetic elementary particles can hit each other and sometimes give rise to new particles which emerge from the scattering process. Since the aim of these scattering processes is to break the involved particles apart, one has to use very high energies and therefore very high velocities in order to exceed the binding energies. Since the velocities can be of the order of the velocity of light (though of course smaller) one has to take relativity theory into account for an appropriate theory of these processes, whereas ordinary quantum mechanics is non-relativistic. The founders of QFT proceeded in a somewhat conservative fashion: They used the formalism and the methods of classical Lagrangian mechanics and only modified it where necessary which is in particular due to the fact that some formerly scalar- or vector-valued functions had to be replaced by operators (see physics glossary). The result of this procedure were operator-valued quantum fields corresponding to different kinds of elementary particles and certain quantum states which these quantum fields can be in. A surprising result was that particles themselves no longer appear in this theory. Although there are entities like “N-particle states” (see physics glossary) among the possible states of QFT it is not clear how these states relate to N particles. This is not only due to problems with the lack of individuality in systems with superposed identical particles. There is another essential characteristic of the particle concept which gets lost in QFT, viz. localizability. Since this topic is mathematically involved and not easy to display separately I wish to demonstrate the problem together with the way it is addressed in advanced QFT by using a mathematical model which is directly linked to modern detection devices for elementary particles. The well-known cloud chamber photographs show particle tracks which are e. g. split after collisions or after a creation of new particles or which are curved due to a magnetic field. This detection method from the early days is visually compelling but has disadvantages in the numerical analysis because the only data it is based upon are graphical. Today one uses much more elaborate detection devices which directly supply the elementary particle physicist with electrical signals that can be processed by computers.

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This procedure allows of various possibilities to improve the exactness and value of the experimental data. A serious problem for the selective detection of elementary processes consists in the fact that there is always a large proportion of signals which are irrelevant for the process in which one is interested. A method to suppress background signals lies in the use of energy thresholds which have to be exceeded before the detector responds. The more intricate task is the discrimination of different elementary processes which occur in the same region at a similar time. A very successful way of achieving this aim is a coincidence arrangement of detectors: Only those signals are assumed to originate from one and the same process which were detected at exactly the same time. In AQFT this detection method has been employed to tackle a conceptual question by modeling the described experimental situation in a mathematical way. The detector model is meant to demonstrate that a relativistic N-particle state after a scattering process can be understood as a state of N “singly localized” particles at least in the asymptotic limit, i. e. when time goes to infinity. The coincidence arrangement of N detectors is described by a product of N operators. Due to the Reeh-Schlieder theorem (see section 8.3.1) these operators can only be “almost localized”, since strict localization is incompatible with the condition that the detectors must not respond to the vacuum. Operators are said to be almost localized when they are smeared out with test functions which vanish quickly when their arguments go to infinity. The significance of the described detector model and in particular of the notion of almost localized operators for the tenability of a particle interpretation will be discussed in subsection 13.3.5. The reason why I do not consider this question here already is that I will have arguments why an answer depends crucially on certain philosophical presuppositions which are investigated somewhat better in the context of chapter 13.

8.3

Localization Problems

So far there is no unquestioned proof against the possibility of a particle interpretation for QFT. The pieces of circumstantial evidence seem to be strong, however. The core of these pieces consists in problems to localize “particle states” in any sensible way. This will be the issue of the following

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section. Section 8.1 showed that the very definition of a particle is more involved than one would expect at first glance. There are various features which are commonly connected with the particle concept. However, which of them taken together make it up to a sufficient condition and, even more importantly, which of them are necessary? The advantage of clear-cut necessary conditions is that it allows for the possibility of no-go theorems to the effect that a conception which rests on these conditions can be ruled out. Exactly this has been investigated intensively with respect to the particle interpretation of relativistic quantum mechanics. The results are generally considered to be among the best-established ones on the ontological foundations of relativistic quantum physics. Reeh and Schlieder, Hegerfeldt, Malament and Redhead all gained mathematical results, or formalized their interpretation, which prove that certain sets of assumptions lead to contradictions. They all purport to be in the position to exclude the corresponding interpretations which are, to say the least, closely connected with the particle concept. Up to now it is a point of debate what exactly has been shown and how the different results relate to one another. The clarification of these questions is pivotal for an enquiry of the ontology of QFT since the particle interpretation is, besides the field interpretation, the most widespread one. In the following subsections I will analyse the relation between two of the above-mentioned no-go theorems, namely the one resulting from Michael Redhead’s interpretation of the Reeh-Schlieder theorem and a nogo theorem by David Malament. Two reasons render the ensuing analysis particularly fruitful. First, it supplies a firm mathematical ground to start from and, second, a thorough comparison necessitates a very close look at the exact presuppositions and the legitimacy of the conclusions drawn by either author. In the light of these two points it is almost of secondary importance what the analyis will finally tell us about the relation of the theorems.

8.3. LOCALIZATION PROBLEMS 8.3.1

99

The Clash of Causality and Localizability

The Consequences of the Reeh-Schlieder Theorem The Reeh-Schlieder theorem23 (1961) is a central analyticity24 result from algebraic QFT (AQFT), the axiomatic reformulation of QFT which I have introduced in section 5.3. From a physical point of view the Reeh-Schlieder theorem is based on vacuum correlations. Although the theorem stems from an analysis of the vacuum state, the ‘0-particle state’, it can easily be extended to other ‘N-particle states’ with N = 0.25 This already demonstrates the scope of its importance. In short the upshot of the ReehSchlieder theorem is that local measurements can never decide whether one observes an N-particle state. Or one can express the result alternatively by saying that local measurements do not allow for a distinction between an N-particle state and the vacuum state. I begin with a technical statement of the theorem. With Ω being the vacuum state and R(O) the von Neumann algebra of local observables acting on the Hilbert space H, the following result can be derived on the basis of the axioms of AQFT (see appendix D): Reeh-Schlieder Theorem: For any bounded open region O in space-time the set {AΩ : A ∈ R(O)} is dense in Hilbert space H. The definition of dense as well as of other technical concepts can be found in glossary D.3. For a rough-and-ready explanation one can say that one set lies dense in a second set if its elements are so finely distributed over the whole second set that for any given element in this second set and any given distance one can find an element in the first set which lies closer to this element in the second set than expressed by the given distance. The set 23

Although Reeh and Schlieder (1961) is the original source later accounts render an easier access. For references confer the titles which are cited in the following footnotes. 24 A function is called analytic if it can be represented by a Taylor series. Note the difference with the notion of differentiability. For instance, the elements of the Schwartz space are infinitely often differentiable but they are not analytic since their Taylor series would just be 0 everywhere. 25 See Haag (1996), p. 102. Moreover, in the rest of this text I will drop the ‘...’s which are meant to indicate that, e. g., ‘N-particle state’ only labels a certain state. Whether or not it is legitimate to actually understand such a state as a state of N particles is the very question at stake here.

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{AΩ : A ∈ R(O)} is said to be generated from the vacuum state Ω by the von Neumann algebra R(O) of local observables associated with O because it is the set which you get when all the operators A in R(O) are applied to the vacuum state. With these explanations the content of the ReehSchlieder theorem can be expressed as follows: choosing suitable elements of R(O) and acting with them on Ω any vector in H can be approximated arbitrarily closely. The statement of the Reeh-Schlieder theorem is different and much stronger than the well-known fact that the Hilbert space H can be spanned by eigenstates of the number operator which can be build up from the vacuum state by applying suitable creation operators.26 The strength of the Reeh-Schlieder theorem becomes clearer when one considers that O can be a small neighborhood of a point in space-time. What the Reeh-Schlieder theorem now asserts is that acting on the vacuum state Ω with elements of R(O) one can approximate as closely as one likes any state in H, in particular one that is very different from the vacuum in some space-like separated region O
. The Reeh-Schlieder theorem is thus clearly exploiting long distance correlations of the vacuum. The consequences of the ReehSchlieder theorem for the issue of entanglement are discussed in Clifton and Halvorson (2001). Even though the Reeh-Schlieder theorem is an astonishing result it is not immediately obvious what the conceptual consequences actually are. To this end one needs the following Corollary of the Reeh-Schlieder theorem: Ω is a separating vector for R(O), i.e. two elements A1 , A2 ∈ R(O) which yield the same result when acting on Ω must be one and the same operator, or in short AΩ = 0 ⇒ A = 0, where A ≡ (A1 − A2 ). The use of this corollary yields an interesting interpretive result. Again, the assumptions are the standard axioms of AQFT. It will be shown that: Local measurements can never decide whether one observes an Nparticle state.27 26

It can be shown that local algebras R(O) - about which the Reeh-Schlieder theorem talks - never contain pure creation (or annihilation) operators. 27 Cf. Redhead (1995a).

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Let us call this statement ‘Redhead’s claim’ because he expressed this conclusion more explicitly than anybody else, at least to my knowledge. The underlying mathematical result is the fact that a projection operator PΨ which corresponds to an N-particle state Ψ can never be an element of a local algebra R(O). Since the proof of ‘Redhead’s claim’ is very instructive and comparatively easy I will restate it here. Proof Given the Reeh-Schlieder theorem and its above-mentioned corollary the proof is a straightforward reductio ad absurdum. Consider an arbitrary Nparticle state Ψ (with N = 0). Since Ψ is orthogonal to the 0-particle state Ω the corresponding projector PΨ satisfies PΨ Ω = 0. Now consider whether Ψ is an element of a local algebra R(O) corresponding to a bounded region O. If this were the case then one could decide by a local measurement, restricted to the region O, whether we have an N-particle state Ψ or not, or, to be more cautious, whether we will find such a state or not when we perform such a measurement. Assume now as a trial that PΨ is an element of a local algebra R(O). In this case the corollary is applicable so that PΨ Ω = 0 would imply PΨ = 0. This, however, contradicts our assumption that PΨ is the projection operator corresponding to the N-particle state Ψ, which cannot be 0 unless Ψ itself is 0. We are forced, therefore, to drop q.e.d. our assumption that PΨ is an element of a local algebra R(O). The exact meaning and range of Redhead’s claim will become clearer when one compares it with the ensuing no-go theorem by David Malament. The comparison itself will follow the exposition of the two results to be compared and makes up the core of this section. I will conclude the section with some critical remarks about the legitimacy of the respective interpretations. Malament’s No-go Theorem Malament’s no-go theorem (1996) is another consequence of analyticity which rests on a lemma of Borchers (1967).28 In short it says that a relativistic quantum theory of a fixed number of particles, satisfying in particular a localizability condition for measurements in disjoint spatial sets 28

Both papers can be found in the references.

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at the same given time, predicts a zero probability for finding a particle in any spatial set. Malament’s no-go theorem rests on four conditions for the following setting. It is assumed that there are projection operators P∆ on a Hilbert space H, each of which represents the proposition that a particle detector would respond if a position measurement were performed in the spatial set ∆. ∆ is taken to be a bounded open subset of a spacelike hyperplane in Minkowski space-time M. Furthermore it will be assumed that there is a strongly continuous, unitary representation U (a), a ∈ M (in H) of the translation subgroup of the Poincar´e group in M. Malament’s conditions now are the following: (i) Translation Covariance Condition: P∆+a = U (a)P∆ U (−a)

(8.5)

for all a in M and all spatial sets ∆. ∆ + a denotes the set ∆ after a translation by a. (ii) Energy Condition: The spectrum of the Hamiltonian operator H(a) is bounded below (see glossary D.3) provided that H(a) satisfies U (ta) = e−itH(a) for all unit vectors a in M which are future directed and timelike. (iii) Localizability Condition: in the same hyperplane, then

If ∆1 and ∆2 are disjoint spatial sets

P∆1 P∆2 = P∆2 P∆1 = 0.

(8.6)

(iv) Locality Condition: If ∆1 and ∆2 are spatial sets (not necessarily in the same hyperplane) that are space-like related, then P∆1 P∆2 = P∆2 P∆1 .

(8.7)

What do these conditions mean? Condition (iii) is the essential ingredient of the particle concept: A particle - in contrast to a field - cannot be found in two disjoint spatial sets at the same time. P∆1 and P∆2 must therefore be orthogonal. The condition is very weak since it does not set any finite limit to the traveling speed of a particle. Condition (iv) is the relativistic part of Malament’s assumptions. It requires that the statistics for measurements in ∆1 must not depend on whether or not a measurement has been performed in the space-like related ∆2 .

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P∆1 and P∆2 must commute therefore. This condition again is very weak since it does not require for P∆1 and P∆2 to be orthogonal which would rule out that the particle can travel at superluminal speed. How does Malament’s no-go theorem work? Using a lemma of Borchers and the four conditions above Malament derives that P∆ = 0

for any spatial set ∆.

(8.8)

This means that the probability for finding a particle in any finite region of space is 0 no matter how large ∆ is. Since this conclusion is not acceptable Malament’s proof has the weight of a no-go theorem provided that we acccept his four conditions as natural assumptions for a particle interpretation. What exactly does this say about the possibility of a particle interpretation? A relativistic quantum theory of a fixed number of particles, satisfying in particular the localizability and the locality condition, has to assume a world devoid of particles (or at least a world in which particles can never be detected) in order not to contradict itself. Malament’s no-go theorem thus seems to show that there is no middle ground between QM and QFT, i. e. no theory which deals with a fixed number of particles (like in QM) and which is relativistic (like QFT) without running into the localizability problem of the no-go theorem. One is forced towards QFT which, as Malament is convinced, can only be understood as a field theory. Nevertheless, whether or not a particle interpretation of QFT is in fact ruled out by Malament’s result will still be a point of concern in the subsequent investigations. At least prima facie Malament’s no-go theorem alone cannot supply a final answer since it assumes a fixed number of particles, an assumption that is not valid in the case of QFT. 8.3.2

Locating the Origin of Non-Localizability: A Comparative Study

Although both Redhead’s interpretation of the Reeh-Schlieder theorem as well as Malament’s no-go theorem are concerned with the non-localizability of relativistic N-particle states they seem to be two different results on a related matter. This is not to say that there is any contradiction between

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the two. It only seems that we have two more results which enrich our knowledge about the possibilities and restrictions for an ontological interpretation of QFT. I wish to show that there is, in fact, just one result. Even though Redhead and Malament draw two different conclusions and use two different theorems for their considerations I will demonstrate that the mathematical machinery employed is the only difference from an interpretive point of view. The assumptions made as well as the result one gets under these circumstances are exactly the same with respect to the question of localizability of relativistic N-particle states. In order to prove this claim I will compare Redhead’s and Malament’s work part by part using one common mathematical language. Since Malament’s no-go theorem is already very clearly and explicitly structured I will use his theorem as the standard for comparison. Redhead starts from the Reeh-Schlieder theorem with its very general assumptions and content. It is only at the end of his considerations that Redhead gets more specific with respect to the question of the localizability of relativistic N-particle states. Malament, in contrast to this, addresses the question from the very start. Accordingly, his assumptions, in particular his four conditions, are more specific from the outset of his argumentation. In order to see this I will start from the end where the parallels can be seen most easily. For this purpose I will compare the two final results which have direct impact on the leading question of particle localization. In the second step I will examine whether the assumptions of the two results are identical so that one can legitimately say that we have effectively just one result from an interpretive standpoint. The Proofs Malament, on the one hand, shows that, given his assumptions for any spatial set ∆, (8.9) P∆ = 0 where P∆ is the projection operator onto the set ∆ which can be any subset of a spacelike hyperplane of Minkowski space-time. This means that the result applies to every possible inertial observer, i. e. for any given inertial observer and any given time we get P∆ = 0 no matter where and how large ∆ is chosen. Redhead, on the other hand, concludes that local measurements can

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never decide whether we observe an N-particle state29 . In Redhead’s consideration of the Reeh-Schlieder theorem there is no explicit reference to subsets of spacelike hyperplanes, however. His conclusion, therefore, seems to be more general in that respect already since it applies, due to the ReehSchlieder theorem, to any bounded open set in space-time. However, there are implicit restrictions to these space-time sets in the context of localizability. In this case one is concerned with position measurements which should be as general as possible as long as the area where the measurement takes place stays finite. Nevertheless, however general the position measurements we consider are, each single measurement has to take place at some time in some observers frame of reference. Or, to put the same thing in other words, the ‘bounded open set in space-time’ we consider has to be a bounded open subset of a spacelike hyperplane, just as in Malament’s no-go theorem. So the first check in the comparison of Redhead’s interpretation of the Reeh-Schlieder theorem and Malament’s no-go theorem leads to the result that they refer to the same set of measurement regions. Now compare what Malament and Redhead actually say about position measurements in those space-time regions we discussed above. Malament, on the one hand, concludes that “there cannot be a relativistic quantum mechanics of (localizable) particles”.30 He derives his conclusion by showing that P∆ = 0 for any spatial set, given his assumptions. Since this result is a striking contradiction to all experimental facts Malament can claim to have a no-go theorem provided that one accepts his assumptions. Redhead, on the other hand, concludes, given his assumptions in turn, that “it is not a local question to ask “are we in an N-particle state?””.31 Formally he shows that PΨ can never be an element of a local algebra, where Ψ is an N-particle state. He proves this claim indirectly by showing that PΨ would have to be 0 if it were an element of a local algebra. One can immediately see that there is at least a certain superficial similarity between Malament’s and Redhead’s results. In both cases some kind of localizability is assumed and shown to lead to contradictions by deriving that a certain class of projection operators, which are directly linked to localizability, would have to vanish in that case. Since, for different reasons, 29

See section 8.3.1. See the title of his paper Malament (1996). 31 See Redhead (1995a), p. 127. 30

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it is not acceptable for these projection operators to be zero the possibility of localizability can be excluded for certain states under certain conditions. In order to see whether there is more than just this superficial similarity between Malament’s and Redhead’s results it is necessary to compare the respective classes of projection operators P∆ , ∆ bounded open set in spacelike hyperplane

(Malament)

and PΨ ∈ R(O), O bounded open set

(Redhead)

more closely. Whereas P∆ is explicitly linked to localizability32 this is true for PΨ only in an implicit way via the assumption that it is an element of a local algebra R(O). Let us compare P∆ and PΨ for the same bounded open region ∆ of space-time, i. e. compare P∆ with PΨ ∈ R(∆). Surely P∆ should be an element of R(∆) since it refers to measurements in space-time region ∆. As we have seen, however, P∆ can only be the trivial element 0 since it vanishes under those assumptions we stated above.

32

The assumption of localizability is complete together with the localizability condition which we will discuss in the next paragraph.

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We can thus take Malament’s result P∆ , ∆: arbitrary spatial bounded open set Conditions for translation covariance, energy, localizability and locality (“Mal.-assumptions”) are fulfilled. (Borchers’ th.)

=========⇒ P∆ = 0 in contradiction to experiments 

 =⇒ Mal.-assumptions can not all be maintained.

(8.10)

and reformulate it as follows: P∆ ∈ R(∆), ∆: arbitrary spatial bounded open set + localizability assumption (Borchers’ th.)

=========⇒ P∆ = 0 in contradiction to experiments 

 =⇒ Localizability assumption cannot be maintained for relativistic N-particle states. (Impossible since localizability is an indispensible ingredient of the particle concept) or P∆ ∈ R(∆) (8.11) where “+”denotes the logical conjunction of propositions. Redhead’s result can now be formulated in an absolutely similar fashion as PΨ ∈ R(∆), ∆: arbitrary spatial bounded open set, Ψ: N-particle state (∗) (Reeh-Schl. th.)

==========⇒ PΨ = 0 in contradiction to (∗)  

=⇒ PΨ ∈ R(∆).

(8.12)

Since in both cases we are dealing with N-particle states (implicitly in Malament’s case, explicitly in Redhead’s case) propositions about P∆ and PΨ amount to the same thing: the impossibility to measure N-particle states when they are assumed to be localized. Evaluation The results about non-localizability which have been explored in this section may appear to be not very astonishing in the light of the following

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facts about ordinary QM: Quantum mechanical wave functions (in position representation) are usually smeared out over all IR3 , so that everywhere in space there is a non-vanishing probability for finding a particle. This is even the case arbitrarily close after a sharp position measurement due to the instantaneous spreading of wave packets over all space. Note, however, that ordinary QM is non-relativistic. A conflict with SRT would thus not be very surprising although it is not yet clear whether the above-mentioned quantum mechanical phenomena can actually be exploited to allow for superluminal signaling. QFT, on the other side, has been designed to be in accordance with special relativity theory (SRT). The local behaviour of phenomena is one of the leading principles upon which the theory was built. This makes non-localizability within the formalism of QFT a much severer problem for a particle interpretation. Only very recently Malament’s reasoning has come under attack in Fleming and Butterfield (1999) and Busch (1999). Both argue to the effect that there are alternatives to Malament’s conclusion. The main line of thought in both criticisms is that Malament’s ‘mathematical result’ might just as well be interpreted as evidence that the assumed concept of a sharp localization operator is flawed and has to be modified either by allowing for unsharp localization Busch (1999) or for so-called “hyperplane dependent localization” Fleming and Butterfield (1999). I fully agree to the extent to which the conclusiveness of Malament’s interpretation is concerned. However, one problem is that the proposed alternatives are not sufficiently worked out to allow for a final evaluation. The discussion in subsection 13.3.5 has a direct bearing on this issue. The threads will come together in chapter 15.2.1. In his article “A dissolution of the problem of locality” Saunders (1995) Saunders draws a different conclusion from Malament’s (as well as from similar) results. Rather than granting Malament’s four conditions and deriving a problem for a particle interpretation Saunders takes Malament’s proof as further evidence that one cannot hold on to all four conditions. According to Saunders it is the localizability condition which might not be a natural and necessary requirement on second thought. A short word on terminology: Saunders calls Malament’s localizability condition “weak-placing condition” since it does not, as Saunders “strong -placing condition” does, entail microcausality which Malament postulates

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separately in his locality condition. Saunders “strong -placing condition” reads as follows: ∆1 , ∆2 spacelike related ⇒ P∆1 P∆2 = P∆2 P∆1 = 0. Stressing that “relativity requires the language of events, not of things” Saunders argues that the localizability condition loses its plausibility when it is applied to events: It makes no sense to postulate that the same event can not occur at two disjoint spatial sets at the same time. One can only require for the same kind of event not to occur at both places. For Saunders the particle interpretation as such is not at stake in Malament’s argument. The question is rather whether QFT speaks about things at all. Saunders considers Malament’s result to give a negative answer to this question. A kind of meta paper on Malament’s theorem is Clifton and Halvorson (2002). Various objections to the choice of Malament’s assumptions and his conclusion are considered and rebutted. Moreover, Clifton and Halvorson establish two further no-go theorems which preserve Malament’s theorem by weakening tacit assumptions and showing that the general conclusion still holds. One thing seems to be clear. Since Malament’s ‘mathematical result’ appears to allow for various different conclusions it cannot be taken as conclusive evidence against the tenability of a particle interpretation of QFT and the same applies to Redhead’s interpretation of the ReehSchlieder theorem.

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8.4

Further Problems for a Particle Interpretation of QFT

Problems Arising From the Standard Formalism of QFT The standard definition for the vacuum state |0 is that it is the energy ground state, i. e. the eigenstate of the energy operator with the lowest eigenvalue. Now recall the notable result in ordinary non-relativistic QM that the ground state energy of e. g. the harmonic oscillator is not zero in contrast to its analogue in classical mechanics. The same is true for the vacuum state in QFT. The relativistic vacuum of QFT displays even more striking features. The expectation values for various quantities do not vanish for the vacuum state. The label “|0” does not indicate that the energy is zero in the vacuum state. It rather stems from the interpretation that there are no particles present in the vacuum state: an N-particle state can be built up from the vacuum state by the N-fold application of a creation operator (see chapter 4). Non-vanishing vacuum expectation values prompt the question what it is that has these values or gives rise to them if the vacuum is taken to be the state with no particles present. Since the vacuum state |0 is closely linked to N-particle states where N is not zero, properties of the vacuum state have a great impact on the particle interpretation as a whole. If particles were the basic objects about which QFT speaks how can it be that there are physical phenomena even if nothing is there according to this very ontology? An even greater but related challenge for a particle interpretation of QFT is the Unruh effect which is the topic of the following subsection. “Nothing” can be a lot for a Fast Observer: The Unruh Effect The Unruh effect is a surprising result which seems to show that the concept of a particle is observer dependent. The Unruh effect (1976) is the striking phenomenon that a uniformly accelerated observer in a Minkowski vacuum (the standard vacuum |0) will detect a thermal bath of particles, the socalled Rindler quanta.33 To be more specific, an accelerating observer in 33

See Unruh (1976) and Unruh and Wald (1984) for details. Teller discusses the Unruh effect on pages 110-113 of his book Teller (1995). He tries to show that it is not a

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111

flat space-time feels himself immersed in a thermal bath of particles at temperature kT = a/2π (
≡ 1), (a: acceleration of the observer) when the quantum field is in its vacuum state as determined by inertial observers. Whereas the number of particles in the Minkowski vacuum is 0, an accelerated observer suddenly detects a thermal bath of particles. A mere change of the frame of reference thus leads to a change of the number of particles. Since basic features of a theory should be invariant under transformations of the referential frame the Unruh effect constitutes a severe challenge to the concept of particles as basic objects of QFT. A Lesson from QFT in Curved Space-Time Studies of QFT in curved space-time34 show that the particle concept hinges on Poincar´e symmetry. This result indicates that the existence of a particle number operator might be a contingent property of the flat Minkowski space-time. In flat space-time Poincar´e symmetry is used to pick out a preferred representation of the canonical commutation relations which is equivalent to picking out a preferred vacuum state. This leads to a well-known definition of the notion of a particle. However, neither the existence of global families of inertial observers nor the Poincar´e tranformations which relate between these families can be generalized to curved space-time. QFT in curved space-time can actually teach us something about standard QFT (in flat space-time). Since QFT in flat space-time is a special case of QFT in curved space-time, QFT in curved space-time can help us to see what is contingent in QFT in flat space-time. fundamental problem for a particle interpretation. 34 In ‘QFT in curved space-time’ one treats gravitation classically (as in General Relativity Theory) and the matter fields propagating in this classical space-time as quantum fields. QFT in curved space-time should have a limited range of validity and break down when the space-time curvature approaches Planck scales and be replaced by a proper quantum theory of gravitation coupled to matter. See the authoritative monograph Wald (1994) for a detailed introduction to QFT in curved space-time.

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8.5

Results

On the one side, the adoption of a particle interpretation of QFT would make the importance of particle experiments and the predominance of speaking in terms of particles comprehensible. It could explain why charge only exists in discrete amounts which is a typical feature of particles and not continuously which is characteristic for field quantity On the other side, we saw that there are various problems for a particle interpretation. Some results indicate that particle states cannot be localized in any finite region of space-time no matter how large it is. Other results show that the particle number might not be an objective feature. Nevertheless, it turned out that most arguments need to be seen in relative terms. At this stage of research it can only be recorded that there are various potential threats for the tenability of a particle interpretation. However, before one can take these arguments as conclusive evidence against a particle interpretation of QFT alternative explanations have to ruled out at first. A more comprehensive and detailed evaluation of possible arguments against a particle interpretation will be carried through in subsection 15.2.1 of the conclusion when we have a better background for this discussion.

Chapter 9 Field Interpretations of QFT Many textbooks on QFT include in their introduction some remarks about the term ‘quantum field theory’ and the entities about which it is a theory. Some textbooks stress that QFT is just as much a particle theory as it is a field theory. Others stress that it is even more a particle than a field theory and that the term ‘quantum field theory’ is somewhat misleading. Still other textbooks say that the term is fully justified since the incorporation of relativity theory into quantum physics leads to the inevitable field character. One thing one can learn from this is that there is obviously no agreement among physicists and that the situation is by no means clear. Another thing one can learn is that these two possibilities, particles or fields, are the standard options for the kinds of entities to which QFT refers. Accordingly, particle and field ontology are the first two approaches which are under investigation in this study. I considered the particle ontology first because it is the most immediate option for that theory which is the theoretical basis for electrons, quarks and protons after all. Nevertheless, the field interpretation of QFT is arguably the kind of ontology to which most physicists would subscribe if pressed for a decision. In this chapter I will investigate how well-founded and viable this choice actually is.

9.1

The Field Concept

Classical Newtonian mechanics is formulated as a theory about bodies and forces with pure “action at a distance”. It is only stated which force bodies exert on each other. Nothing is said about how these effects are mediated 113

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since it is assumed that there is an instantaneous interaction between two massive bodies. It turned out that the electromagnetic interaction between charged bodies cannot be described within this framework. A mediating field, the electromagnetic field, had to be introduced which accounts for the local transmission of electromagnetic forces. The systematic and efficient formulation of the theory of electromagnetism with Maxwell’s equations at its core revealed another famous feature. There is a limiting velocity for the transmission of signals, namely the velocity of light. In classical electromagnetism the existence of a limiting velocity for the transmission of signals simply emerged from this theory which rests on observed electromagnetic phenomena. It was Einstein who established this feature as a requirement for any physical theory. Hence the term ‘Einstein causality’ which was introduced in section 5.3 already. Before describing how this principle was put to use in the formation of QFT I will say a little more about the notion of a field in general. While the introduction of fields as mediators for the transmission of forces is a good starting point for getting an intuitive idea, the standard definition of a field is somewhat different. A field is generally defined as a system with an infinite number of degrees of freedom for which certain field equations must hold. A comparison of the specification of a field to the one of a point particle makes it clear what this definition means. A point particle can be described by its position x(t) which changes as the time t progresses. In a three-dimensional space there are three degrees of freedom for the motion of a point particle corresponding to the three coordinates x1 - x3 of the particle’s position. In the case of a field the description is more complex. The field is represented by the specification of a field value φ for each point x in space where this specification can change as the time t progresses. A field is therefore specified by φ(x, t), i.e. a (time-dependent) mapping from each point of space to a field value. As I indicated already the formal specification φ(x, t) is not enough for something to be a field. Certain field equation need to be fulfilled. Without giving any further details I wish to point out just one extreme case why the formal specification φ(x, t) cannot be sufficient. Consider φ(x, t) where ˜ with x ˜ being a particular point in space. In this case φ(x, t) = 0 ∀x = x φ(x, t) would just describe an ordinary point particle instead of a proper field.

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One further information about fields should be supplied in order to make it understandable how one can come across the idea to think of fields as being the basic entities in the world. The intuitive notion of a field is that it is something transient and fundamentally different from matter. However, in physics it is perfectly normal to ascribe energy and even momentum to a pure field where no particles are present. This surprising feature shows how gradual the distinction between fields and matter can be.

9.2

Fields as Basic Entities of QFT

There are two lines of argumentation which are often taken to show that an ontology of fields is the appropriate construal of the most fundamental entities to which QFT refers. The first argumentation rests on the fact that so-called field operators are at the base of the mathematical formalism of QFT. The other line of argumentation is indirect. Since various arguments seem to exclude a particle interpretation, the only alternative, namely a field interpretation, must be the right conception. 9.2.1

The Role of Field Operators in QFT

It is well known that the basic variables describing the kinematical behaviour of a particle, position and momentum, are of a peculiar nature in QM. In the early days of quantum theory they were called quantum numbers (‘q-numbers’) as opposed to classical numbers (‘c-numbers’). The peculiarity of q numbers is the fact that they do not commute in general, a fact whose details are condensed in the canonical commutation relations (CCRs). This peculiarity is in fact so characteristic that these relations are a sufficient information about the behaviour of a quantum particle. Everything can be derived by specifying these relations. In mathematical terminology the reason for the general non-commutation of q-numbers is that they are operators and not ordinary numbers. The order in which operators act on something does matter in general. In order to denote this difference operators get a hat. The transition from classical to quantum mechanics can thus be described as ˆ (t) x(t) → x

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and correspondingly for the momentum, where certain CCRs hold for their components. Without going into the details let me just state that in a similar fashion the transition from a classical field theory (like electromagnetism) to quantum field theory can be characterized by the transition ˆ t) φ(x, t) → φ(x, for the field and a corresponding transition for its conjugate field for both of which a certain specification of CCRs holds. In difference to a classical ˆ t) of QFT are called operator-valued fields field φ(x, t) the basic fields φ(x, since to each point of space and time an operator is attached. As one could see there is a formal analogy between classical and quantum fields. In both cases field values are attached to space-time points where these values are real-valued in the case of classical fields and operatorvalued in the case of quantum fields. In technical terms the analogy reads as one between the mappings x → φ(x, t), x ∈ IR3 and ˆ t), x ∈ IR3 . x → φ(x, This formal analogy between classical and quantum fields is one reason why QFT is taken to be a field theory. However, it has to be examined now whether this formal analogy actually justifies this conclusion. In his paper “What the quantum field is not” Teller (1990) which became a central chapter of his later book An Interpretive Introduction to Quantum Field Theory Teller (1995) Teller puts considerable emphasis on a critique of this conclusion. He comes to the conclusion that ‘quantum fields’ lack an essential feature of all classical field theories so that the expression ‘quantum field’ is only justified on a “perverse reading” of the notion of a field. His reason for this conclusion is that in the case of quantum fields - in contrast to all classical fields - there are no definite physical values whatsoever assigned to space-time points. Instead, the assigned quantum field operators represent the whole spectrum of possible values. They have, therefore, rather the status of observables (Teller: “determinables”) or general solutions. Something physical emerges only when the state of the system or when initial and boundary conditions are supplied.

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I think Teller’s criticism of the standard gloss about operator-valued quantum fields has one justified and one unjustified aspect. The justified aspect is that quantum fields actually differ considerably from classical fields since the field values which are attached to space-time points have no direct physical significance in the case of the quantum field. However, and here I disagree with Teller, this fact is not due to the operator-valuedness of quantum fields as such. It was not to be expected anyway that one would only encounter definite values for physical quantities in QFT. QFT is, like QM, an inherently probabilistic theory, after all. Nevertheless, even taking the probabilistic character of QFT into account there still is the problem that we need quantum fields as well as state vectors in order to fix probabilistic properties. But I do not think that one can therefore conclude that quantum fields are not physically significant at all. It seems to me that physical significance of field quantities cannot be judged along classical distinctions. I think that there is more physical information encoded in quantum fields than Teller ascertains. But I agree with Teller that the field character of QFT is by no means as obvious as it first seems. The formal analogy between classical and quantum fields as such is not a fully convincing argument for a field interpretation of QFT. If a field interpretation should actually yield the appropriate ontology for QFT than it seems that those objects which are called “quantum fields” are not already the fundamental entities one is looking for, at least not alone.1 9.2.2

Indirect Evidence for Fields

The indirect evidence for a field interpretation consists in the reasoning that all arguments against a particle interpretation are tantamount to ar1

Teller’s own proposal is an ontology of QFT in terms of field quanta. Teller argues that the “Fock space representation” or “occupation number representation” suggests this conception with objects (quanta) which can be counted or aggregated but which cannot be numbered. The number of objects is given by the degree of excitation of a certain mode of the underlying field. Particle labels like the ones in the Schr¨odinger many-particle formalism do not occur any more. I think that it is questionable whether it is legitimate to draw such far-reaching ontological conclusions from one particular representation. In addition to that the Fock space representation cannot be appropriate in general since it is only valid for free particles.

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guments in favour of a field interpretation. Examples are results about peculiar features of the vacuum, non-localizability and non-local correlations. However, just as in the argumentation for a field theory in the last section, indirect arguments do not equip us with an explicit idea about what these fields are. What are the fundamental entities in a field interpretation of QFT? Shall we think of a field as a singular object or are the field values, or better: the mappings from spacetime points to field values, the basic objects? This alternative is similar to the one for the spacetime substantivalist. Should we, as spacetime substantivalists, be realists about spacetime as a whole or rather realists about spacetime points? The best way to show that indirect arguments for a field ontology are not sufficient is to present an alternative reaction. In the next section I will deal with a point of view that attributes a certain function to fields, however not only without supporting a field ontology but explictly defying to take fields as basic entities.

9.3

Fields Versus Algebras

In his book Local Quantum Physics: Fields, Particles, Algebras Haag argues (p. 105) that the “rˆole of “fields” is only to provide a coordinatization of this net of algebras”, referring to the system of nested algebras that I have introduced in section 5.3 on the algebraic approach to QFT. As an argument in favour of this view he points to an early result of algebraic QFT concerning so-called Borchers classes.2 Different fields can lead to the same S-matrix (the scattering matrix from section 4.5) provided that the fields belong to the same Borchers class. Due to the equivalence of fields within the same Borchers class it does not matter physically, says Haag, which of these equivalent fields or field systems we choose. The only difference the choice makes is how the net of algebras is coordinatized. Haag’s point here is that the physically basic information is encoded in the way observable algebras are associated with bounded spacetime regions. The introduction of fields only serves to translate this information into a mapping from spacetime coordinates to field values, i. e. what Haag’s calls a coordinatization. 2

See section II.5.5 in Haag (1996) for details.

Part III

Revisionary Ontologies

Chapter 10 Process Ontology 10.1

The Strands of Process Ontology

Independently of the discussions on conceptual problems of QFT process ontology has been thought about in philosophy since ancient times, for instance by Heracleitus and Aristotle.1 It is still not clear what a process ontology looks like in detail and where and how it can help to overcome unsolved problems. The philosophical discussion has got new impulses in recent years: J. Seibt and others argued for a radical revision of the very foundations of ontological theories.2 Instead of ontologies which are based on the ‘myth of substance’ Seibt proposes a process-ontological approach. Seibt’s process ontology presents itself primarily as a rejection of some deeply rooted presuppositions of ontological thinking so far. Seibt labels this bundle of presuppositions ‘substance ontology’, thereby including a large variety of ontologies, e. g. ontologies based on tropes as well. The fact that even trope ontologies are subsumed under the term ‘substance ontology’ already indicates that the traditional and well-known categorical dualism of ‘substance’ and ‘attribute’ does not lie at the heart of Seibt’s criticism. It is rather the dualism of universals versus particulars that she considers to be at the core of substance ontology.3 1

See Rescher (2001) for a historical survey. See Seibt (1995), Seibt (1996) and Seibt (2004) for a very ambitious and sophisticated argumentation as well as Rescher (1996) for a more elementary treatment. 3 Schurz’ article on “A quantum mechanical argument for the existence of concrete universals” (Schurz (1995)) is an interesting example of what it means and how it can help in physics to soften the dichotomy of universals and particulars: Schurz demonstrates that the problems with identical particles (see section B) appear in a very different light when particles are seen as concrete universals rather than particulars. Particles of one 2

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The central idea of the process ontology according to J. Seibt is to view non-countable rather than countable entities as the most fundamental ones. A countable concrete particular entity (e. g. a particular table) is then taken as a “minimally homoeomerous” non-countable entity. A ‘homoeomerous’ (homoeo-merous = like-parted, Seibt (1996) p. 167) entity is one which has in all its parts the same intrinsic properties as the whole down to some minimal parts. The minimal part of a homoeomerous entity is called ‘minimally homoeomerous’. The term ‘homoeomerous’ has in some respects the same meaning as the popular expression ‘self-similar’ which is used to describe the scale-invariance of fractal structures. I will start my investigation from the fundamental theories of modern physics and explore physical motivations for a process ontology as well as some consequences for the interpretation of the formalism.

10.2

Why Process Ontology in QM and QFT?

The most convincing philosophical arguments for a process ontology are arguments against a substance ontology. This fact is reflected in considerations about the adoption of a process ontology for specific scientific theories: With respect to QM and QFT the strongest motivation for thinking about a process ontology are severe foundational problems which might stem partly from an unquestioned substance ontological background upon which most interpretations are based. There are a couple of problems in QM and QFT which have always defied a satisfactory solution suggesting that a loosening of certain deeply rooted ontological beliefs and restraints might be unavoidable in the end. I have already discussed one complex concerning difficulties with microparticles as individual objects. The next paragraph on “Consequences of a Process Ontology for the Interpretation of QM and QFT” will show how process ontology sheds a different light on old problems. At the same time it will help to clarify what a weakening of the dichotomy of universals sort are all identical even though they are not numerically identical. Universals like ‘electron’ are repeatedly instantiated and the number of electrons is the cardinality of this instantiation. The problem of the distinguishability of electrons dissolves since it makes no sense to speak of different individual electrons any more.

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123

and particulars means in concrete cases and how this might help to tackle problems. The second complex of problems has not been discussed so far since it is not obvious that these problems are connected primarily with ontology: Microparticles or systems of microparticles are described by the quantum mechanical state function whose best-known representation is Schr¨odinger’s wavefunction in coordinate space. Describing systems with state functions is a very powerful and general method. Even though it has been equally successful in predicting and explaining physical phenomena, however, the exact connection between the state function and the system which it is meant to describe poses some unsolved conceptual problems. To be more precise the problem consists in the attribution of properties to quantum mechanical objects. There is just one easy case: If the state of a quantum system is an eigenstate of the observable to be measured (e. g. energy, spin etc.) the outcome of the measurement will with certainty be the corresponding eigenvalue. In this and only in this case there is no problem in saying that the system possesses this value of the observable as a well-defined physical property.4 If the quantum system is not in an eigenstate of the relevant observable there are, with respect to that observable, severe difficulties to ascribe any property to the system before as well as after the measurement.5 A special instance of these general difficulties with the attribution of properties appears in the very important case of incompatible or incommensurable observables. Examples of such pairs are position and momentum or different spin components (e. g. in xand z-direction). Mathematically, the incompatibility of two observables is reflected by the fact that the operators A and B that represent these observables do not obey the law of commutativity. In this case their so-called ‘commutator’ [A, B] := AB − BA (10.1) does not vanish.6 The corresponding physical interpretation simply states that it is not possible to perform joint measurements of such ‘noncommut4

This is what David Albert calls the ‘Eigenstate-Eigenvalue Rule.’ See for example his recent publication Albert and Loewer (1996). 5 See Mittelstaedt (1998) for a detailed discussion. 6 For the sake of simplicity I will drop the distinction between observable and the operator that represents it from now on.

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ing’ observables. The consequences are even stronger than that, however: It is not even possible to assign values of both observables to one object at the same time not to speak of their measurement. Such strong statements seem to stand in contrast to Heisenberg’s uncertainty relations which allow measurements of noncommuting observables to a certain degree of accuracy at least. In order to overcome this gap simultaneous measurements of noncommuting observables have been explored extensively in the last decades.7 This led to the concept of ‘unsharp observables’ which can be measured simultaneously and whose unsharp values can be assigned to the quantum system while this is impossible for the sharp counterparts of these observables. On the one hand the concept of unsharp observables enables us to describe many physical situations a lot more realistically than with the older too narrow and too rigorous schemata of QM.8 On the other hand we get new conceptual problems: To speak of unsharp observables and unsharp values suggests the idea that a quantum system possesses unsharp properties. It has to be stressed that the claim to be considered is not just that certain properties can be measured only in an unsharp fashion when measured simultaneously. The claim goes further to the point that the system actually incorporates unsharp properties. The considered problems all hinge on the assumption that we can only understand physical quantities and measurements in terms of basically invariant object systems which in some way possess properties that can change in time. If this assumption were dropped things would appear in a very different light. In some sense the problems mentioned above could not even be stated any more. Since the situation is so desperate otherwise it is this vague hope which constitutes one motivation for a physicist to think about process ontology. In addition to this hope for a radically new solution for old problems the mathematical formalism of QM smoothly fits to the view to have properties without definitely assigning them to any underlying substances. One further motivation for a process ontology arises from an interesting feature of QFT which will be the background for the case study in the following section as well. In QFT we have an important piece of formalism 7 8

Busch (1982) is one of the first sources where these issues are investigated intensively. Cf. Busch et al. (1995) for many elaborate examples.

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125

which is interpreted as a description of the creation and destruction of particles.9 This fact is often seen as a characteristic difference between QFT and classical QM. In our context one could say that classical QM is, in this respect, closer to the concept of a conservation of substance than QFT where we have the possibility of transitions between states with different numbers of substances. These transitions are described by creation and destruction operators which act on the respective states. Maybe a description in terms of processes is more suitable to the otherwise strange idea of a creation of substances. The next section will explore further aspects of the creation and destruction of particles with respect to the ’substance versus process ontology-debate’.

10.3

A Case Study: Consequences of the Ontological Hypotheses for the Interpretation of Feynman Diagrams

So far there is no worked-out process ontology for QFT and it seems difficult to achieve this aim (see the last section in this chapter on “Remaining Problems”).10 In the meantime one can only use general outlines of such a conception. In order to investigate the consequences of an approach with processes as basic objects of QFT, I want to focus on just one of the problems I mentioned in chapter 4, namely the ‘creation and destruction of particles’ as they appear in either naive or preliminary descriptions of Feynman diagrams because of the occurrence of creation and destruction operators in certain mathematical terms which stand behind Feynman 9

See chapter 4 for some aspects of the mathematical background which led to this interpretation. 10 At the present time, H¨attich (2004), partly building on ideas by H. P. Stapp from the 1970ies, is the most comprehensive proposal for a process-ontological interpretation of QFT. Unfortunately, in my view, H¨attich has chosen the Whiteheadian version of process ontology which is suffocated by idiosyncratic terminology. For this reason, I find it very hard to assess H¨attich’s proposal in a sufficiently competent way. Nevertheless, one point I wish to emphasize is that H¨attich links structures in (the Whiteheadian) ontology to structures in the algebraic formulation of QFT, as I have also been advocating for the last decade (with respect to trope ontology). A quite different proposal for a processontological interpretation of QFT is Seibt (2002). Also see my criticism in Kuhlmann (2002).

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diagrams. The purpose of the case study is to uncover how the chosen ontological approach can effect the interpretation of the formalism of QFT. As a representative part of formalism I shall look at the treatment of scattering processes in perturbation theory and especially at the visualisation of S-matrix elements by Feynman diagrams. The procedure of the case study is to examine how the interpretation of Feynman diagrams changes when we go over from the conventional substance ontology to a process ontology.11 Even though a major part of chapter 10 deals with Feynman diagrams they are - for our purposes - not interesting in themselves but merely in order to illustrate some consequences of a process ontology. I speak of a process ontology because instead of any characteristics of a detailed conception of process ontology I shall only use very general features which result primarily from a loosening of constraints of substance ontology. The case study should not be misunderstood as giving arguments either for or against process ontology. The sole purpose is to explore some effects of a process ontology for QFT which cannot be seen immediately. The following list of arguments against a realistic interpretation of Feynman diagrams is the first step of the case study. I call the interpretation of Feynman diagrams ‘realistic’ or ‘literal’ when it is assumed that the diagrams correspond to something in the outer physical world in a one-toone fashion. This first set of arguments constitutes the conventional view on Feynman diagrams which is grounded on the equally conventional and therefore usually unquestioned substance ontology. In the second step of the case study this list will serve as the background for a counter argumentation within a process ontology. 11

In Kuhlmann (2000) I present a self-contained account of this case study and its significance in the search for an ontology of QFT.

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Substance ontology and arguments against a realistic interpretation of Feynman diagrams a) Arguments concerning the origin of Feynman diagrams as parts of a sum (i) Feynman diagrams have been invented for the sole purpose of classifying and calculating a sum of very complex terms which is otherwise extremely hard to handle systematically. No ontological obligations (i.e. statements about which objects exist) are connected with this procedure. One simple example is the scattering process of two electrons whose states may be specified by their respective 4-momenta p and q. If before scattering the state of the two electrons is | p, q the probability of finding the two-electron-state | p
, q
 after the scattering between the electrons has taken place is given by

probS|p,q (|

p , q ) = |p , q | S | p, q| = |











2

∞


p
, q
| S i | p, q| , 2

i=0

where S is the so-called S-Matrix which describes all the possibilities of the scattering process in terms of incoming and outgoing states. p
, q
| S | p, q is one element of the S-Matrix with respect to the complete orthonormal set {| p, q} of basis vectors and the summation over i indicates an expansion of the S-Matrix.12 Feynman diagrams are now used to calculate an S-Matrix element in successive orders which in turn are given by the expansion index i. To each value of i, i.e. to each order, corresponds a certain number (one or more) of Feynman diagrams. Each Feynman diagram looks like the sketch of a scattering process which is ever more complicated - i.e. 12

In perturbation theory we use the so-called ‘interaction picture’ in which the equation of motion for the time-dependent state vector of the object system is governed by the interaction part of the Hamiltonian. Solving this equation of motion for an initial condition iteratively leads, in the limit t → ∞, to an expansion of the S-Matrix by comparison with its definition. The contribution in each order of the expansion index i consists of a sum of multiple integrals where i determines the number of integrations. For details see, for example, Mandl and Shaw (1993) p. 98-102 (ch. 6.2).

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CHAPTER 10. PROCESS ONTOLOGY contains more vertices - the higher i is. This can, for example, lead to diagrams which look like sketches of multiple internal scattering processes. It is common talk amongst particle physicists to call higher order diagrams or processes respectively ‘corrections’ to the ‘main processes’ which are given by the lowest non-vanishing order. The advantage of using Feynman diagrams (instead of direct calculation of the above sum) lies in the fact that there are some simple graphic rules which are sufficient to determine all diagrams that have to be taken into account for each order whereas in a direct calculation it happens very easily that some relevant terms are missed out. The usage of Feynman diagrams is graphically represented in figure 10.1 (page 128).

Figure 10.1: Illustration of the connection between Feynman diagrams (here only of the lowest order) and the probability for a certain scattering process. (ii) The division into Feynman diagrams has a purely conventional character which results from the ‘artificial’ construction of interaction Hamiltonians in perturbation theory. (iii) A realistic interpretation of an infinite series of Feynman diagrams (which is used to calculate the probability of an overall scattering process) commits - at least in a particle view - to the assumption of an infinite amount of particles that contribute a single scattering process. (iv) Only the sum over all relevant Feynman diagrams of all orders leads to an observable quantity. Therefore the individual diagrams have no physical significance (cf Schrader-Frechette (1977)).

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(v) To suppose that a scattering process is built up by the superposition of independent subprocesses which are given by the relevant Feynman diagrams produces false probabilities: The transition probability for a scattering process is not the sum of the probabilities of the individual processes that are depicted in the Feynman diagrams which are associated with this transition, as in equation (10.3). The reason for this is that the transition amplitude for a scattering process is given by the sum of the ‘transition amplitudes’ of the alleged subprocesses, as in equation (10.2) (cf Weingard (1982), p. 239). probS|p,q (| p
, q
)





2

= |p , q | S | p, q|

∞
2 = | p
, q
| S i | p, q|

=

i=0 ∞


|p
, q
| S i | p, q| , 2

(10.2) (10.3)

i=0

where terms in the first sum with two different indices are called ‘interference terms.’ Exactly these terms constitute the difference to the second sum.

b) Arguments concerning single Feynman diagrams (vi) Feynman diagrams consist of sharp trajectories which are excluded because of the Heisenberg uncertainty relations for position and momentum. (vii) If Feynman diagrams were visualisations of real processes the number and kinds of virtual particles in these processes would be sharp which is not possible (cf Weingard (1982), p. 240). (viii) A Feynman propagator between two points x and x
with space-like separation would describe a particle traveling with a speed greater than the velocity of light (cf Mandl and Shaw (1993), p. 56 and K¨all´en (1958), p. 256). (ix) Single second order diagrams of covariant perturbation theory are actually combinations of two time-ordered diagrams. These combinations

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CHAPTER 10. PROCESS ONTOLOGY are necessary in order to get covariance since the division into timeordered processes with t > t
and t
> t is not Lorentz-invariant if (x − x
) is a space-like separation (cf Mandl and Shaw (1993), p. 55 and Weingard (1982), p. 240).

The last nine arguments are supposed to show that we must not interpret individual Feynman diagrams as representing any real scattering processes in the physical world. Some of these arguments are explained in more sophisticated textbooks on physics. It is never stated, however, that some of them are grounded on implicit presuppositions - namely the substanceontological presuppositions mentioned above - and some arguments are problematic for other reasons. We will therefore explore these two things in the next section.

Process ontology and counterarguments to the arguments against a realistic interpretation of Feynman diagrams The following set of arguments in our case study are counterarguments to the first set of arguments against a realistic interpretation of Feynman diagrams. While the first set of arguments was mainly grounded in a substance ontology the second set contains arguments against these arguments which can partly only be raised from the point of view of a process ontology. In particular, we will have to examine whether the first set of arguments can still be maintained when we have no substance-like bearers of properties any more. (i’) The point of discussion here is whether a realistic interpretation of Feynman diagrams is possible not whether it is necessary. The origin of a certain piece of theory or formalism is of minor importance to its interpretation. (ii’) The counterargument again runs like the first one: Even if there were mathematical or practical reasons for a specific construction or representation originally these facts cannot be used against a further ontological interpretation.

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131

(iii’) Argument (iii) is only relevant in a substance ontology where something is conserved. In a process ontology ‘particles’ (or the like) themselves are seen as composed of processes! A superposition of an infinite amount of processes is not problematic since there is nothing like a conservation of processes. (iv’) Argument (iv) can be boiled down to the point that there is no direct empirical evidence of the processes depicted in Feynman diagrams. If this was a good argument it could be used to rule out many other entities as well, like quarks and electromagnetic field strength. (v’) Subprocesses need not be independent. There is no reason why the relevant subprocesses should not influence each other when taking place together. In this case the interference terms describe the mutual influence. (vi’) Arguments (vi) - (viii) are again only relevant in a substance ontology. In a process ontology there is no object traveling from x to x
. (vii’)

“

(viii’)

“

(ix’) There is no need to take every part of the mathematical formalism ontologically seriously even in a process ontology. This is the counterpart to the first two arguments in this list: We are only examining which parts of the formalism could be given a direct ontological meaning in a process ontology.

10.4

Evaluation of the Case Study

The case study revealed that the interpretation of Feynman diagrams depends crucially on the ontological hypotheses which form the background of the interpretation. Nine arguments against a realistic interpretation of Feynman diagrams were discussed. All these arguments could be shown to rest on a substance-ontological background because to each argument we could find a counterargument when a process-ontological background is assumed. We can therefore record that a realistic interpretation of Feynman diagrams is excluded in a substance ontology while it seems possible in a process ontology.

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We have thus found one point where the decision between substance and process ontology makes a difference to the interpretation of the formalism of QFT, namely of Feynman diagrams. This only shows, however, that these alternative ontological assumptions do have concrete interpretive consequences. At this stage of the argumentation, our result is neither an argument for nor against a process ontology. This would only be the case if we had independent reasons to prefer (or to disapprove of) a realistic interpretation of Feynman diagrams. Although the case study constitutes no direct reason against a substance ontology or for a process ontology it could hopefully help to show the significance of this question for the interpretation of physics. At the same time it turned out, at least as far as our case study is concerned, that the difference between substance and process ontology lies only in the interpretation and not in any empirical consequences. There remains a lot of detailed work to be done, however, in order to examine how much a process ontology would contribute to the solution for instance of the problems I have sketched in section B.

10.5

Remaining Problems

The most outstanding deficiency of current conceptions of a process ontology is the lack of a satisfactory positive description and definition of the assumed basic processes. For mathematically minded physicists there is the immediate question for a mathematical definition and a concise description of the mathematical structure of the set of processes. A first idea would be to understand a process as the triple of two events and a unitary time evolution operator. A good starting point could be to explore where conventional conceptions of processes differ from the kind of processes which a process ontology postulates. An interesting subquestion to the first one is the connection of process ontology to recent theories of the structure of space-time (e. g. geometro-dynamical models).13 The second set of remaining problems is concerned with explanations for 13

David Finkelstein made some interesting proposals in his older papers Finkelstein (1973), Finkelstein (1974) and Finkelstein (1979) as well as in his recent book Finkelstein (1996).

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phenomena which are natural for a substance ontologist while they call for a lot of effort on the side of the process ontologist. Whereas the substance ontologist has a hard time explaining how change in time is possible even though the things which change supposedly keep their identity, the process ontologist has the opposite problem: Why do we have the strong impression that many things are more or less static if everything is composed of processes? On top of that the process ontologist has to explain why many conservation laws seem to be fulfilled by nature as it appears to us. One old explanation is to assume counterprocesses which exactly balance other processes with the overall effect of the appearance that nothing happens. The last point to be mentioned here is the question whether the adoption of a process ontology leads to any changes of scientific theories and the connected formalism. If the answer is ‘yes’ this could show a way to experiments which actually do make an observable difference between substance and process ontology.

Chapter 11 Trope Ontology I: The Ontological Status of Properties The last candidate for an ontology of QFT to be considered in this study is trope ontology. The fact that the trope-ontological approach to QFT comes after the event- and the process-ontological approaches in the ‘revisionaryontologies-section’ is not to indicate that trope ontology would be the most radical revisionary ontology. The reason is rather that it is the latest revisionary ontology which has been proposed for quantum physics.1 It is hard to appreciate trope ontology without knowing the context in which it originated. Taken in isolation trope ontology might easily appear either trivial or inconceivable. In order to prepare the right setting for it I will hence describe this context. It should become clear why many analytical ontologists find trope ontology so appealing. I shall begin with a short characterization of the so-called ‘problem of universals’ and the two main lines of response, realism about universals and nominalism. The focus will be on nominalistic positions since it is a variant of nominalism, namely trope ontology, which I shall primarily deal with in the context of QFT later on. 1

With respect to QM probably the first serious argumentation showing the fruitfulness of trope ontology for the solution of some problems in the conceptual foundations can be found in Simons (1994). Special emphasis is given there to the so-called “problem of the identity of indiscernibles” in connection with many-particle systems in QM. With respect to QFT the author of this book was the first, to my knowledge, to argue for a trope-ontological account of (the algebraic formulation of) QFT in a number of talks and one publication (Kuhlmann (1999)).

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The Problem of Universals

Most of our talking and thinking involves grouping things according to certain classifications such as green, circular or beautiful. Some of the classifications such as ‘beautiful’ hinge on our own evaluation and thus reflect something about ourselves rather than features of our experience or of an external world. However, grouping things as being liquid or circular, for instance, seems to reflect an objective similarity among these things which does not depend upon us. Now the question arises whether objective similarities among things have an existence on their own. Do e. g. properties exist besides those things which have these properties? Are properties real entities on their own? Although it first appears absurd to say that properties as such exist a second thought shows that it is not easy to deny their existence. The key problem is how to explain that two distinct things can have the same property, e. g. that they are both blue or have the same charge. One cannot deny that they ‘have something in common’ although they are completely distinct. But if a property was another entity on its own how is it to be conceived that such an abstract entity has an effect on those two things? Since the extensive medieval controversy of this issue it is common practice to call something which can be said about many things a universal.2 Due to this medieval terminology the controversy about the ontological status of properties is often called the problem of universals.3 Another way of introducing the problem of universals is by studying the reference of terms in the standard subject-predicate discourse. Looking at a sentence like ‘This apple is red.’ it is clear that the term ‘apple’ is meant to refer to a particular apple. But to what does the predicate ’red’ refer? On the one hand, just saying that it refers to the apple as well seems unsatisfactory since this would imply that the term ‘red’ has a different reference in each context it is used. On the other hand, claiming that it refers to a separate entity redness gives a stable account of the references but it prompts the question what the sentence ‘This apple is red.’ means. 2

Following for instance Wolterstorff (1970) one can make a further distinction between two kinds of universals, substance universals and predicable universals (p. 65). An example for a substance universal is being an electron, examples for predicable universals are all ordinary properties like being charged. 3 A very accessible and non-technical introduction to the problem of universals as well as to various approaches for its solution can be found in chapters 1 and 2 of Loux (2002).

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Obviously it is not the identity of the apple and redness what is meant. Understanding the apple as a proper part of redness seems not intelligible either since in contrast to an apple redness is not a concrete object in space and time. Not less inconceivable sounds the opposite, namely that redness is a proper part of the apple. For this would entail that redness would be a part of this apple and a part of all other red things as well. But how can it be that one and the same entity exists as a whole in different things? It seems that whichever way one tries to go one runs into absurdities. And it is this situation which is called the problem of universals. One driving force behind ontological inquiries in general and the debate about universals in particular is the aim of simplicity which is famously connected with Ockham’s razor. Although the metaphor cannot be found in Ockham’s work the principle of parsimony played an important role in his whole thinking. Today this guideline is widely accepted among ontologists. The idea is that we should take as few entities and in particular as few categories as possible into our ontological inventory. One can say that the core of the debate about universals is the question whether we need a separate category of universals or whether we can do with particulars alone. The problem of universals is almost as old as philosophy itself. One reason why it has such a long history is that it is in fact a bundle of problems rather than just one well localized debate. Besides its ontological dimension which I have outlined above it has interesting aspects in epistemology, semantics and philosophy of science. However, here I will concentrate on the ontological aspect of the problem of universals which is concerned with the ontological status of properties.

11.2

The Traditional Responses

The traditional positions within the debate on the ontological status of properties are realism about universals and nominalism. Realism about universals claims that a property is something real which, as a whole, is repeatable in many different individuals. This is why the realist calls properties universals as well, that which can be said about many things. The realist about universals claims that when two different things have

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the same property this is a matter of strict identity.4 Armstrong determines strict identity via the principle of the indiscernibility of identicals dating back to Leibniz. Being the converse of the different and independent principle of the identity of indiscernibles which has been formulated by Leibniz as well the principle of the indiscernibility of identicals states that “Sameness of things gives sameness of properties” (p. 3). Two strictly identical things thus have exactly the same properties. Applying the notion of strict identity to the characterization of realism about universals means that when different things have the same property there exists an entity, i.e. a universal, which is exactly the same in these different things. What the realist about universals claims is that it is not only the case that there are similarities in certain aspects. He claims that with respect to the property in question there is no difference whatsoever. It is one and the same property. Each case of an occurrence of the same property is called an instantiation of the universal. In its best-known and most extreme version realism about universals takes a universal to be an abstract entity, i.e. one which is neither in space nor in time. Plato is most famous for this so-called universalia ante res-position. A salient consequence of this position is that universals can exist even when there is nothing which exemplifies them. Such universals are called uninstantiated universals. The very opposite position to realism about universals is called nominalism. A nominalist denies that there actually are any properties in the world or if he admits properties he denies the possibility of strict identity of properties across different things. The nominalist not only denies the existence of properties separate from concrete things, i.e. entities which are at least localizable in time. This point of view would still be compatible with realism about universals in its version of universalia in rebus (e. g. Aristotle). The nominalist claims that properties as something which different things can have are merely our construction. No extra entities in the world besides particulars correspond to these allegedly repeatable properties. We only have the impression that there are properties that are distinct from concrete individuals which have these properties in the sense that the same property occurs in different concrete individuals. 4

Armstrong points out Armstrong (1989) that using the notion of strict identity as opposed to identity in a loose and popular sense yields a “deeper view of the Problem of Universals” (p. 5).

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Different versions of nominalism have been put forward.5 The simplest one is class nominalism which assumes that everything falls into natural classes where the naturalness is taken to be a primitive ontological ingredient which cannot be analysed any further. To say that a has property F is understood by the class nominalist to mean that a is a member of the natural class of things which, conventionally speaking, all have property F. The immediate reaction to class nominalism is to question the primitiveness of natural classes. One should think that it is the very property, say F, that was meant to be avoided which makes the natural class a coherent class. On the one hand there is something right about this objection against class nominalism. On the other hand a caution is appropriate here. Most if not all ontological conceptions have to take more than one kind of entities as basic or primitive. Even when one starts by assuming one (or more) kind(s) of entities after some reflection it can turn out that there is at least one more uninvited kind of entity which has to be acknowledged. Often this additional kind of entity causes at least some unease. Whether one is willing to accept the respective situation depends primarily on how convincing it seems to stop asking at a certain point. After all there always is a point where one has to start so that this fact is not problematic in itself. There are various strong arguments against the standard forms of nominalism. As an example for one kind of problems that nominalism faces I will consider only one simple form of nominalism (viz. class nominalism) and discuss one specific problem. Nominalists have serious difficulties to 5

One encounters a confusing plethora of labels for nominalistic positions since various classifications have been put forward. In his earlier book Armstrong (1978a) Armstrong distinguishes five forms of nominalism, namely predicate nominalism, concept nominalism, class nominalism, mereological nominalism and resemblance nominalism. Armstrong mentions “Ostrich or Cloak-and-Dagger Nominalism” (p.16) as well which he ascribes to Quine and considers to be simply begging the question. In his later book Armstrong (1989) he retains only class nominalism and resemblance nominalism of these five (or six) forms of “regular nominalism”, probably in order two give enough room to trope nominalism which he now considers to be the most sustainable form of nominalism and most serious competitor to his version of realism about universals.. Three forms of nominalism can be found in Loux (2002), austere nominalism, metalinguistic nominalism and trope theory. Loux considers Quine, Sellars and D. C. Williams as the classical representatives of the respective forms while Ockham embraced all three forms depending on the kind of terms he was discussing.

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account for more complex statements involving relations and higher order predicates. Investigating one special case may suffice to indicate why. According to class nominalism a universal (or type) is to be identified with the class of particulars (or tokens) that correspond to it. Redness is identified with the class of all objects which are red. Now consider the following example of a statement with a three-place relation: Redness is more like orange than it is like yellow.6

( )

Since nobody would doubt that this is a perfectly sensible statement the class nominalist has to make sense of it too. He has to analyse it by using classes and unrepeatable particular things instead of universals. This means that he could translate the above sentence to something like For each element in the class of red things it holds that it is more similar to each element in the class of orange things than it is to each element in the class of yellow things. But what have we got now? Although the last statement is true for a few things it is blatantly wrong in all other cases. A red flower is not more like an orange curtain than it is like a yellow flower. As a remedy one could think of comparing the whole classes by saying that The class of red things is more similar to the class of orange things than it is to the class of yellow things. This again is wrong, however, or at least one can easily imagine a state of affairs of the world which renders this statement wrong while at the same time “Redness is more like orange than it is like yellow” is true. Just imagine a world in which red and yellow only occur as colours of the vegetation and orange only for pieces of furniture. I will now point to some problems which realism about universals faces. In order to make things as easy as legitimately possible and in order to be fair to nominalism let us consider the most extreme version of realism about universals, the universalia ante res-version. Other than nominalism, realism about universals has no general problem with relations or higherorder predicates. Nevertheless, realism about universals has a problem specifically with the relation of similarity between universals. An infinite 6

See Armstrong (1989), p. 34.

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regress is threatening if this similarity could not be analysed any further. One would be faced with an infinite number of similarity relations, similarity relations of similarity relations and so on. I will come back to this problem for the universals theory a little later. Instead, let me mention one further problem which the “universalia ante res-version” realism about universals faces, namely the “problem of instantiation”. How are we to explain that universals from a realm of abstract things outside of space and time play a role in our concrete world? How is the transgression from Plato’s heaven of universals to our spatio-temporal world to be conceived? In short, one can say that whereas nominalism has the problem to understand the world solely in terms of concrete particular things, extreme realism about universals has the problem to explain how abstract universals can have any relevance for concrete particular things.

11.3

A New Solution: Trope Ontology

In the last section I gave an example in order to show why one of the standard forms of nominalism has a problem to account for more complex but undeniably sensible statements like “Redness is more like orange than it is like yellow.” I have demonstrated why such statements involving relations and higher order predicates cannot be explained or represented when properties are identified with natural classes of particulars as the class nominalist wants to do in order to avoid properties. I will now come to a more refined and somewhat exotic version of nominalism which is able to solve many problems which the standard versions of nominalism are confronted with.7 The alternative version of nominalism I am talking about is called trope ontology where tropes in this context should not be associated with exotic vegetation or with ancient sceptics which have used this term for the classification of arguments. The ontological notion of tropes, although not the expression ‘trope’, can already be found in the 7

I am leaving aside David Lewis’ possible worlds version of class nominalism Lewis (1986) which can be seen as a refinement of the standard form of class nominalism. Although Lewis’ realistic view of possible worlds in fact creates a ground from which various problems of plain class nominalism can be coped with I have the impression that the ontological extravagance of his “modal realism” is in a dangerous tension to the nominalist’s pursuit of ontological parsimony.

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work of Ockham besides at least two other versions of nominalism.8 The punchline of trope ontology is to take properties as particulars and not as universals. Properties taken as non-repeatable particulars are called tropes. In the trope-ontological scheme of the world tropes are taken as the simple and basic items of which everything is composed. ‘Trope ontology’ is a 20th century term for a philosophical position which in its content can perhaps even be traced back to ancient philosophy. There are various famous philosophers who are (sometimes) considered to have been trope ontologists or who at least in some respects had ideas which can be considered as trope-ontological. In some cases this evaluation refers to the explicit content of their respective philosophies, in other cases only to the effective picture which emerges when certain statements are taken together to form one consistent view.9 The most well-known examples of historical philosophers are Aristotle, William of Ochkam, John Locke, George Berkeley, David Hume, Edmund Husserl10 and Rudolf Carnap. Again, note that the mentioning of these philosophers in the context of trope ontology does neither refer to their whole oeuvre nor is this evaluation uncontroversial. This applies in particular to Aristotle.11 Partly because of its somewhat mysterious name trope ontology appears to be very exotic and in a sense it certainly is. Since it diverges considerably from everyday ontological thinking it is commonly classified as a revisionary ontology, besides e. g. event ontology and process ontology. Trope ontology is a popular position nowadays partly because a number of problems vanish most easily once this point of view is adopted. In fact, trope ontology gives such a convenient solution to some problems that Johanna Seibt has stressed the “comfortable conservativeness of the 8

See p. 85 and footnote 21 in Loux (2002). John Locke is an example of an ‘effective’ trope ontologist in this sense. A short argument for this claim can be found on p. 63f in Armstrong (1989), a modern defense of a Lockean-style version of trope ontology appeared in the much discussed paper Martin (1980). 10 A detailed discussion of Husserl’s de facto trope theory can be found on pp. 555-563 in Simons (1994). Husserl’s own term for tropes is ‘moments’. Note that Husserl recognizes universals as well! Husserl’s theory thus includes tropes without being a (tropeontological) answer to the problem of universals. This possibility indicates that the problem of universals is not the only aspect of the significance of tropes although it is probably the best-known and most illuminating one. 11 See p. 554 in Simons (1994) and p. 85 in Loux (2002) for an evaluation. 9

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approach” which, by the way, she considers to be not without costs.12 In our times Keith Campbell is probably the best-known advocate of the standard theory of tropes which he expounded in his 1990 monograph Abstract Particulars Campbell (1990). Earlier explicit trope ontologists are G. F. Stout in the twenties and thirties and D. C. Williams in the sixties. It was Williams who introduced the term ‘trope’ (p. 7) in his seminal set of papers Williams (1953a) and Williams (1953b). Campbell’s expression ‘abstract particulars’ goes back to Williams as well (p. 7). In his “alphabet of being” Williams distinguishes two kinds of parts which an individual object has, gross parts which are concrete and finer or diffuser parts which are abstract. The following short argument in favour of the need to accept tropes as an ontological category (given by a realist about universals!) should attract the attention even of those readers who are not sympathetic with nominalism.13 Perception is a causal relation, namely between the perceiver and that what is perceived. What we perceive of everyday objects, for instance, are not the objects in their whole but we perceive some of their properties, e.g. the orange colour of my cup. But only particular items can be causally efficacious, that is I perceive this orange colour of my cup and not the universal orangeness which, as a universal cannot have a causal effect. Thus we need to include tropes into our ontological scheme. As I have mentioned above adopting a trope-ontological point of view makes it possible to address problems like the ‘red-orange-yellow-problem’ which the standard versions of nominalism could not solve. Trope ontology enters the debate about the ontological status of properties with a compromise between realism about universals and standard nominalism. The trope ontologist says: Don’t worry about acknowledging properties like red or yellow, but take them as particulars and not as universals. It is acceptable to say that this flower has the property of being red, as long as its redness is taken to be something which occurs just once in this particular flower. We have this particular red trope of this particular flower and another particular red trope of a particular book and so on. How is trope ontology to solve the red-orange-yellow-problem in sentence ( ) which I have been discussing in section 11.2? Since statement ( ) is 12 13

Seibt (2002), paragraph 4. Lowe 2006, p. 15.

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meant to be a general statement about red, orange and yellow, the trope ontologist has to think about an analysis catching on the one hand the generality of the statement while using on the other hand only properties taken as particulars, i.e. tropes. On this trope-ontological background nominalistic moves like analysing properties in terms of natural classes of particulars awake to new life which is a major reason for the persuasiveness of trope ontology. While the classical nominalist cannot carry his program through when it comes to explaining more complex statements the very same ideas survive in a trope-ontological environment. In order to see how this is possible consider the combination of trope ontology and class nominalism. The trope ontologist using natural classes will analyse the statement ( ) by saying that (each element of) the class of red tropes is more like (each element of) the class of orange tropes than it is like (each element of) the class of yellow tropes. The above argumentation against this analysis of the statement ( ) in terms of classes of particulars is no longer valid. Even if we lived in a world where all the vegetation is red or yellow and all pieces of furniture are orange the class of red tropes is still more like the class of orange tropes than it is like the class of yellow tropes. In the context of trope ontology we are no longer forced to take a whole red everyday thing into the class of red things but only its red trope. I postpone the question about the exact relation between e. g. a red flower and its red trope until next chapter where I will focus on this and related questions. In a similar fashion the trope ontologist is able to analyse all kinds of complex statements which the realist about universals can explain and the classical nominalist cannot. Following Armstrong one can classify this third alternative between realism about universals and classical nominalism as ‘moderate nominalism’. Trope ontology is moderate since it includes properties and relations into its ontological scheme. It is still closer to nominalism, however, than to realism about universals because only particulars are admitted since properties are taken as particulars and not as universals. We thus almost have a middle ground between nominalism and realism about universals and this fact makes trope ontology so strong with regard to its explanatory power. I wish to conclude this section by mentioning that the bundle theory of tropes is not without difficulties. What makes a bundle of tropes a bundle?

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What ensures the identity of a bundle at two different times? What insures the identity of a trope as a trope of a certain object? Before presenting my own evaluation I will give an account of one, if not the, single most distinguished overall assessment of the pros and cons for various positions in the debate about the problem of universals.

11.4

An Evaluation of the Debate

David Armstrong offers a comprehensive evaluation of the debate about the problem of universals in his 1989 book Universals - An Opinionated Introduction. Although Armstrong himself is a well-known realist about universals he acknowledges that it is not possible to strictly rule out all other positions, at least not at the current stage of research. In view of this situation Armstrong has given a balanced evaluation of a large number of arguments for and against various positions concerning the ontological status of properties. In contrast to his earlier books Armstrong (1978a) and Armstrong (1978b) on universals Armstrong now thinks that a serious competitor to his own universals theory can be found in one special form of nominalism, namely one version of trope theory. I will briefly summarize the main points of Armstrong’s final evaluation which has found much attention in all camps of analytical ontologists. After a thorough examination Armstrong sees just two viable options left over, the trope resemblance theory on the one hand and a moderate universals theory on the other hand. All other options are faced with knock-down arguments with no hope for a solution. Thus according to Armstrong’s evaluation neither nominalism nor realism about universals are out of the race, although in both cases only a moderate version can survive. On the one hand we have trope theory which is moderate, says Armstrong, since it does acknowledge the existence of properties in contrast to ‘immoderate nominalism’ which denies their existence altogether. Trope theory still is a nominalistic position, however, since it refutes to take properties as repeatable entities. It accepts properties only as particulars, namely tropes, but not as universals. Like the standard versions of nominalism trope theory as well has to explain general statements about properties. As we saw the idea of natural classes is far more convincing

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in the context ot trope ontology than in its original form of natural class nominalism. There is still another form of nominalism, namely resemblance nominalism, which on its own and again much more so in the context of trope ontology rates even better. The idea of resemblance nominalism consists in the assumption that there are natural classes of things which are characterized by objective resemblances among the members of a natural class. The relation of exact resemblance is an equivalence relation, logically speaking. An equivalence relation divides (or partitions) a field of things into mutually exclusive classes, so-called equivalence classes. This fact makes classes of exactly resembling particulars an appropriate tool for the reduction of properties to particulars. In contrast to class nominalism where the naturalness of classes is taken to be primitive, resemblance nominalism gives an analysis of their naturalness. Only the resemblances are taken to be primitive facts about the world which cannot be analysed any further. Resemblance nominalism thus stops asking one step later than class nominalism. Armstrong calls resemblance nominalism on a tropeontological background “trope resemblance theory” and considers it to be the strongest combination of trope ontology with traditional versions of nominalism. Corresponding to trope ontology as a moderate form of nominalism there is a moderate form of realism about universals which is the one that Armstrong himself favours. Unlike the immoderate Platonic version of realism about universals it does not assume that universals can exist independently of concrete particulars. Moderate realism about universals denies, for instance, the possibility of the existence of uninstantiated universals like being a unicorn which is arguably the most bizarre consequence of the immoderate Platonic realism about universals. The reason for Armstrong’s above-mentioned conclusion that for the time being two contradictory positions seem to be tenable, viz. trope resemblance theory and a moderate universals theory, consists in the fact that both options are faced with difficulties which have a similar probability to be tackled in the future. I shall begin with Armstrong’s view of the difficulties for the trope resemblance theory. One difficulty is the conceivability of an exchange of exactly resembling tropes between two different objects. Even though the exchange of exactly resembling tropes between two objects would not

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change anything the trope ontologist is forced to say that the two objects have changed. Another problem is the irreducibility of resemblance. The ontological status of the relation of resemblance of two tropes is unclear. If it is not analysable it has to be taken as a primitive notion besides tropes. It cannot be explained as an identity of certain constituents of these tropes since then we have to presuppose the relation of identity which can only be done in a universals theory since the relation of identity would have to be taken as a repeatable entity. The universals theory is faced with problems which are partly the counterparts of the difficulties for the trope theory. While the trope theory has problems with exactly resembling tropes the main problem for the universals theory is the case of less than exact resemblance of universals. Two universals cannot resemble each other exactly. Otherwise they would just be one and the same universal. Obviously two universals can, however, resemble each other less than exactly so that they are rightfully called two universals. The question now is, whether and how this relation of less than exact resemblance of universals can be analysed. If it should turn out that it cannot be analysed in terms of universals the relation of less than exact resemblance would have to be taken as a primitive notion besides universals. And taking less than exact resemblance as primitive would mean that a new entity would have to be accepted for each case of less than exact resemblance of two universals. Now some cases of less than exact resemblance will resemble other cases of less than exact resemblance less than exactly and thus further less than exact resemblance primitives enter into the ontology so that we get an infinite regress. Even if this infinite regress would not be fatal it would at least lower the economy of the universals theory considerably. Since the time of Armstrong’s evaluation trope theory has established a firm place as a viable new competitor in the field of ontology, and one can indeed observe that, nowadays, there is hardly any comprehensive survey of ontology that does not give trope theory a serious consideration.

11.5

Conclusion and Outlook

Despite Wittgensteins verdict that there is no problem of universals to be found it is a fact that there still is an ongoing debate about it. The

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debate it not only very lively in itself but stands in an intense interaction with other philosophical enterprises. Not surprisingly the debate about universals is most intimately connected with other ontological enquiries. Leaning on Husserl one can distinguish two different parts of ontology. One part is formal ontology and the other one is material or regional ontology. Formal ontology is concerned with the establishment of a consistent categorial structure of being qua being. An important guideline in this field is the mostly accepted principle of parsimony which urges to assume as few basic ontological categories as possible. Let me give an example. If it is possible to reduce every sensible statement about universals to statements including just particulars then dispense with universals. It is interesting to notice that a certain position in regard to the problem of universals pretty much fixes the whole categorial structure one holds. This is the reason why many monographs on metaphysics have an extensive discussion of the problem of universals at their core. The other part of ontology where the problem of universals plays an important role is material or regional ontology. Here one is not concerned with a categorial structure as such but only in connection with empirically founded results. Categorial structures are now tested against the state of the art of our scientific theories. The categorial structure is only acceptable if it is possible to place any element of our best empirical theories in its scheme of categories. Peter Simons calls this demand the “integration requirement”. It is obvious that the fulfillment of the integration requirement is not an easy task and that it is only possible in collaboration with specialists in various fields of science. Nevertheless, most current ontologists, at least in the analytical tradition, would not be willing to give up this requirement and do without checking how well different categorial scheme fit to our empirical world. In the view of the delineated understanding of ontology, I would evaluate the state of the debate about the problem of universals in the following way. There is a broad spectrum of positions which have been proposed in regard to the ontological status of properties. Corresponding to this broad spectrum of positions there is a large variety of arguments for and against all of these proposals. There is not a single position where even the upholders themselves would claim not to have any problems. In the light of this situation it would not be adequate, on the one hand, to draw a final

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conclusion in favour of one of the proposed positions. On the other hand, it would be equally inadequate to put all proposals on a par. Although there is no need nor the possibility to make a final decision at this moment it seems advisable to be clear about where and how to carry on in the future. In this respect there are certain hints to be found already now. Not all arguments in the debate about the ontological status of properties have the same weight. On the one hand there are some knock-down arguments which in fact seem to be apt to exclude one or the other of the alternatives. This applies for instance to pure class nominalism. There are various kinds of statements which the class nominalist cannot make sense of while hardly anyone is willing to simply mark these statements as senseless. On the other hand there are arguments which raise unease about the respective position but which do not demonstrate its untenability. For instance, it might turn out that one is forced to include certain resemblances into ones categorial scheme, e.g. less than exact resemblances of universals. The need to include new categories into the list of accepted categories makes the categorial scheme less elegant but it does not prove it to be untenable. One might conclude that certain resemblances have to be accepted as brute facts which do not allow further analysis. Although the acknowledgment of unexpected primitive facts should not be easily accepted it is no straight counterargument. The important question is whether the point where one has to stop asking for further analysis seems adequate. Summing up one can say that there are at least four reasons for adopting a trope-ontological conception of the world. First, the gap between universals and concrete particulars is bridged by admitting properties as basic entities in the ontological scheme. This is achieved by letting properties enter as particulars and not as repeatable universals. Second, a middle ground between nominalism and realism about universals is found in virtue of which the problems of either side can be avoided while their respective achievements are retained. Third, as properties are empirically more basic and accessible than everyday things it seems natural to put these entities at the bottom of the ontology and not those more complex things which can be seen to be made up by them. Forth and last, the trope ontological scheme is very economical because there is just one main class or category of basic entities, viz. tropes.

Chapter 12 Trope Ontology II: Properties and Things The leading general question of ontology, understood as explicated in chapter 3, is which fundamental categories of entities there are in terms of which everything in the world can be analysed or to which it can be reduced. One of the most important subquestions for any ontological theory is how things in the usual sense are to be conceived. This topic is a particularly salient one in the case of trope ontology since although tropes are properties of things the central claim of trope ontology is that properties are all there is to things. Thus the trope ontologist has to give an account of thinghood exclusively in terms of properties, i. e. tropes. This is the question I will address in the present chapter. Although the issue of the relation of thinghood and properties is of course closely connected to the question about the ontological status of properties it is helpful to keep these two issues apart in order to have a rough classification of the problems which trope ontology is faced with. I think that this first classification makes the handling of the respective problems more effective. The last chapter dealt with the long-standing and still unresolved discussion about the ontological status of properties, the so-called problem of universals. The point of disagreement in that discussion was whether or not a property, for instance solidity, exists as an irreducible kind of entity which is repeatable in many distinct things, namely those particulars which ‘have the same property’, like this keyboard and that telephone. While realists about universals answer this question in the affirmative, nominalists deny that there are any additional entities besides those particulars which ‘have the same property’. Nominalists defend their position by trying to show 151

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that it is possible to analyse all statements involving properties solely in terms of particulars. Since the attempt of a complete avoidance of properties, for instance by class nominalists, is confronted with severe problems a third alternative position besides realism about universals and traditional nominalism emerged, namely trope theory, and found various supporters over the last two decades. On the one hand trope ontologists maintain the nominalistic intuition not to accept properties as an extra kind of entities which are characterized by being multiply instantiable in distinct things. On the other hand trope ontologists do not dispense with properties as an ontological category altogether. They achieve this compromise by accepting properties only as particulars and not as repeatables. The claims of trope ontology as a comprehensive ontological position are much more far-reaching, however, than just holding that properties are to be conceived as particulars, i.e. as tropes. Trope ontology does not merely say that there is a category of tropes. The claim is that tropes make up the only fundamental category which is needed. According to this view everything else can be reduced to tropes. Although most ontologists who hold that there are tropes will make the further claim that there are only tropes on the fundamental level there seems to be no necessary connection between these two claims.1 In its standard form trope ontology is conceived as a so-called bundle theory. Everyday objects as well as objects in scientific theories like elementary particles, molecules and genes are considered to be bundles of tropes, or bundles of bundles of tropes. In order to illustrate the strength of this assertion I will give an example which catches the general idea although the situation is in fact more complicated. According to the bundle theory of tropes, this particular cup is the bundle of this green, this roundness, this consistency, this gloss, etc. It is not said that we first have some kind of a ‘bare cup’ and and then we can add its colour trope, its shape trope and so forth. There is nothing to an object besides its tropes. This is the main reason for using the term ‘bundle’ which should otherwise not be taken too figuratively. In particular, tropes of one bundle obviously do not necessarily occupy different spatial positions as, for instance, in a bundle of wires. Another way of expressing the strong claim of trope ontology is 1

Lowe’s position is the case of an ontology where tropes are included but only as one fundamental category besides some others. See Lowe (1998) and Lowe (2006).

153 to say that it is a one-category theory. There is just one category of basic entities, namely tropes, and everything else can be reduced to tropes. Trope ontologist Peter Simons stresses that the tropes of a trope bundle should not be thought of as proper parts of this bundle.2 Simons agrees with some other authors that to think of tropes as “ways something can be” catches the nature of some natural kinds of tropes better than thinking of them as things which are at the same place as parts of an object. However, Simons has various cautions to add. One of them is that these ways would have to be particularized ways since a trope ontologist cannot accept ways which are considered as universals. The other and more important caution is about the view not to think of tropes as real entities at all. Simons argues, and I think legitimately, that tropes are in the end meant to make up substantial objects. If tropes are not somehow real entities, even if they cannot exist independently like substances, then how can something real arise by bundling tropes together? To put the bundle view of objects in classical terms, we have no split-up into an underlying persisting substratum on the one side and changeable attributes on the other side. The advantage of abandoning this division is that it allows to overcome the artificial discrimination of substratum and attributes, which seems, among other things, to derive from the misguiding structure of most western languages. Although it is convenient to speak of something permanent and its changing properties it is unclear what this something is, according to this line of criticism. It seems that the substratum is a fictitious product of abstraction from all properties, a “something I know not what” (Locke) which is ultimately inconceivable. This makes it understandable why many ontologists find it is appealing to dispose of the idea of a substratum. 3 Trope ontology and the bundle conception of objects come together naturally. Most ontologists who acknowledge tropes also subscribe to a bundle theory of objects. However, this connection does not apply to those few philosophers who assume both tropes and universals (for instance E.J. 2

For the details of his emphasis that tropes are not proper parts of a trope bundle see Simons (1994), pp. 561-565, as well as his book Parts. A Study in Ontology (Simons (1987)). 3 See Morganti (2009a) for a recent defense of the received view that the substratum theory and the bundle theory are in fact enemies.

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Lowe) or both tropes and substances (for instance C. B. Martin and possibly Ockham). In other words it is a one-category theory of tropes that is naturally connected with a bundle conception of objects. In particular, if one has (independent) reasons to believe that a bundle conception of objects is the best view, one is more or less forced to embrace a trope ontology since the only alternative, namely a bundle theory of universals, as defended by Bertrand Russell, turned out to be indefensible. In my later proposal for a new ontology of (A)QFT this connection, that is from a bundle view towards trope ontology, will play an crucial role.

Part IV

The Trope Bundle Interpretation of QFT: A New Ontology for Fundamental Physics

Chapter 13 Dispositional Trope Ontology 13.1

Introduction

The last section of part III dealt with trope ontology in general without any significant reference to physics. As in the case of process ontology it is primarily for philosophical reasons that trope ontology has been established and it is mostly discussed in this context. In this chapter I wish to propose a trope-ontological interpretation of QFT building on this philosophical foundation. I start by elaborating on an argument by trope ontologist P. Simons. He argued Simons (1994) that his “nuclear theory of tropes” can be used for the solution of a conceptual problem in quantum mechanics, namely the problem of the individuality of ‘identical particles’. My own proposal differs from and goes beyond Simons’ ideas in a number of aspects. Simons’ trope ontological attitude has arisen out of philosophical considerations (I suppose) and, in the last decade or so, he has been checking the applicability and appropriateness of his approach in various particular sciences, with physics being one among them.1 I am proceeding just the other way round. I start from a particular theory of one special science, namely QFT, which - as I have argued in part I - I consider to be ontologically more fundamental than any other theory of natural sciences. In a second step I am looking for an ontological construal of nature which best fits this theory (i.e. QFT). So one can say that whereas Simons 1

Confer footnote 6 on page 20 on my use of the expression ‘particular sciences’. Moreover, note that in recent years Simons has considered an alternative approach, the so-called ontology of invariant factors, explicated in Simons (2002), for instance. Nevertheless, his method of constructing a general philosophical scheme and then checking it against particular sciences seems not to have changed.

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starts off being biased philosophically but neutral regarding single sciences, I start off being biased regarding the priority of single sciences and neutral with respect to ontological conceptions. Of course Simons and my biases have their respective basis. I go beyond Simons in two aspects. First, I try to be more explicit in specifying which are the fundamental tropes in the case of quantum physics. Second, my aim is to give this specification with respect to QFT, and not QM as Simons does. I briefly recapitulate the main ideas of trope ontology. I introduced trope ontology as a diplomatic way to handle the ‘problem of universals’ which mediates between extreme nominalism and extreme realism about the ontological status of properties. The treatment of properties is the salient feature of trope ontology. Properties are acknowledged as entities which really exist outside of our mind and which are not merely mental constructions or even just words. In this sense trope ontology deviates from classical or extreme nominalism and makes a concession to realism about universals by accepting the reality of properties. However, the trope ontologist still shares a (if not the) basic skepticism of the nominalist against realism about universals. Both classical nominalists and trope ontologists deny that properties are universals. The trope ontologist keeps the nominalistic attitude that it is inconceivable that properties are entities which exist outside of space and time and which can be multiply exemplified or instantiated. Both classical nominalist and trope ontologist think that all there is are concrete particulars, i.e. entities which occur only once (particulars) and which are at least in time and often in space as well (concrete). Nevertheless, as I indicated above, there is a point where the trope ontologist diverts from the classical nominalist. While the classical nominalist denies the ontological reality of properties, the trope ontologist takes properties for real by classifying them in a novel way, namely as concrete particulars, which is the only ontological category a nominalist accepts. Note however, that while tropes have concreteness and particularity in common with everyday objects like this keyboard, they differ from them in that their existence is strongly dependent on the existence of other entities, namely of other tropes. Other more common examples for strongly dependent entities are boundaries or states which are necessarily the boundaries or the states of something. Generally, tropes cannot exist by themselves

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but can only occur in clusters of tropes. A cluster of tropes which makes up an independently existing object is called a trope bundle. The tropes of a bundle are related to one another in a certain way called compresence relation.

13.2

Trope Bundles and Many-Particle Systems

Reading this subsection might come as a surprise since it is about particles and fields. It was to be expected that a revisionary ontology like trope ontology differs radically from the conventional particle or field interpretation of QFT. However, one has to realize that trope ontology does not claim that there are no particles and fields if the world is properly conceived. All that the trope ontologist claims is that particles and fields and all other objects in the usual everyday sense are not fundamental. It is possible to further analyse these ‘substances’ until one reaches tropes at the very bottom. This section deals with the outline of such an analysis in the domain of quantum physics and with some conceptual problems than can be solved by this approach. 13.2.1

‘Elementary Particles’

By the somewhat misleading term ‘identical particles’ physicists denote sets of elementary particles like electrons and photons whose permanent (or essential) properties are exactly the same. In the domain of QM rest mass, charge and spin are permanent properties. The introduction of spin is due to the fact that it turned out that it is necessary to ascribe a further internal degree of freedom to make sense of a certain behaviour of quantum mechanical particles in inhomogeneous electro-magnetic fields. The qualification internal is opposed to external degrees of freedom like the three degrees of freedom corresponding to the three dimensions of space. For a first approximation one can image spin to refer to the angular momentum of a spinning charged ball. Note, however, that conceptual problems are involved with a too figurative understanding of spin. For our purposes it suffices to just keep in mind that the spinning behaviour is essential for a quantum particle and labels some kind of intrinsic property. Specifying a certain combination of values (m, e, s) for these properties

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fixes a particular kind of elementary particle. For an electron this triplet is (m, e, s)electron = (0.000511 GeV/c2 , −1, 1/2) (13.1) where eV is the typical energy unit of high energy physics and the numerator c2 stems from Einstein’s equation E = mc2, so that GeV/c2 has the dimension of mass. Note that ‘mass’ refers to rest mass, i.e. the mass which is measured when a particle is at rest relative to the observer. Due to the special theory of relativity the effective mass of a particle diverges from its rest mass when the particle is in relative motion to the observer. In order to fix one particular specimen of a kind of elementary particle one has to specify more than its permanent properties, however. It seems obvious that one way to do this is by supplying its location in space and time. We can then say that we have, for instance, one individual electron here and another one there. Both have the same essential properties, since they are both electrons, but they have different locations and therefore they are two individual electrons. Of course, things are not quite that simple in quantum physics since generally a quantum object has no exact position. Instead, the quantum mechanical wave function gives us probabilities for finding the object in a given volume. One can refer to properties specified by the wave function as ‘relational properties’ of the quantum object since they tell us how this object relates to the continuum of space and time, or, depending on one’s taste in these matters, to other objects in the world. For an everyday object, like a house, such relational properties are the specifications of where it stands, how high it is, how deep it is etc. 13.2.2

Individuality of Quantum Objects

So far everything is fine. But things get more intricate in the case of two ‘identical particles,’ say electrons, which have been in mutual interaction in the past. Due to their interaction they get into the so-called quantum mechanical ‘entanglement,’ whose mathematical counterpart in QM is a superposition of wave functions. A collection of identical particles of the same kind which is in such an entangled state is called a many-particle system. Experimental results about the statistical behaviour of quantum me-

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chanical many-particle systems require a certain symmetrical structure of the wave function which describes the state of a many-particle system. With respect to their statistical behaviour in a many-particle system quantum mechanical particles fall into two groups. The first group of particles behaves according to Fermi-Dirac statistics and they are therefore called fermions. The second group of particles behaves according to EinsteinBose statistics and they are called bosons. Pauli’s famous theorem about the connection of spin and statistics tells us that all fermions have half integer spin and all bosons have integer spin. Examples for fermions are electrons, all the six types of quarks as well as protons which are composite particles. They all have spin 1/2. Examples for bosons are photons and gluons, each with spin 0. Common gloss has it that all ‘matter constituents’ are fermions and all ‘force carriers’ are bosons. The behaviour of fermions according to Fermi-Dirac statistics requires that the wave function of a many-particle system of fermions is antisymmetric. As an example, take the wave function of a system of two fermions (e.g. electrons),  1  (13.2) Ψ(x1 , x2 ) = √ ψα (x1 )ψβ (x2 ) − ψβ (x1 )ψα (x2 ) , 2 where ψα (x1 ) and ψβ (x2 ) are energy eigenfunctions of one-particle Hamiltonians and α and β represent sets of quantum numbers characterizing one-particle states. A wave function of an entangled system of fermions is a superposition of product wave functions, i.e. a sum of tensor products of one-particle states. The requirement that this wave function is antisymmetric means that it has to change its sign when the labels ‘1’ and ‘2’ are interchanged. Intuitively this change of labels refers to a swapping of two particles since the labels ‘1’ and ‘2’ stem from the two separate particles we started with before they became to be involved in an entangled compound state. Note that the antisymmetry requirement for many particle systems of fermions has the effect that the antisymmetric wave function automatically fulfills the well-known Pauli exclusion principle. When the two electrons described by equation (13.2) were to occupy the same state, the wave function of the compound state would vanish. This means that it is not possible to have a compound state with both electrons being in

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the same state.2 The explanations I gave in the last paragraph are standard explanations one can find in later chapters of any standard textbook on QM. Nevertheless, from an ontological point of view, the last paragraph contained some very misleading gloss. But although this is all well-known, it is very hard to avoid speaking and even thinking in somewhat misguided terms. The remainder of this section will show that an unsolved conceptual problem is the hotbed of this confusing situation.3 Since x1 and x2 have been introduced as variables of the ‘single particles’ 1 and 2, it is natural to ask what the states of these ‘single particles’ are. It turns out that it is impossible to give a satisfactory answer to this question if one holds on to the conception of individual particles. Each ‘single particle’ is in the same state as a part of the compound system, even though in the wave function of the compound system different one-particle wave functions appear. In the following I will spell out the reasoning which sustains this claim. The relevant considerations in this context have been worked out in the Quantum Theory of Measurement which was initiated by von Neumann. In the standard case, one studies the behaviour of a compound of two subsystems, namely a measurement apparatus and an object system to be measured. Since the Quantum Theory of Measurement, in contrast to Bohr’s approach, starts off by assuming that not only the object system but the measurement apparatus as well is to be described by quantum mechanics, similar considerations apply to any compound of two entangled quantum systems, just like the 2-electron-system from above. When two quantum systems, be it two electrons or one object system and one measurement apparatus, have been in interaction, they become to be entangled. In the case of the two electrons such an interaction could have been, e.g., their simultaneous emission from the same source. Formally, the resulting entanglement is described by a pure compound state which is a superposition of eigenstates of the subsystems. Since after the interaction 2

Note that all talk in the previous paragraph about the states of the single electrons meant pure states. However, subsystems of a compound are never in a pure state but rather in a mixed state. This will be a crucial point in the subsequent paragraphs. 3 See Saunders (2006) and Morganti (2009b) for the current philosophical discussion on quantum statistics.

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it is only the compound which is in a pure state, it is not obvious what the individual states of the two subsystems are. Taking the superposition seriously means that the subsystems cannot be in pure states as well. At this point the Quantum Theory of Measurement comes into play with the subsequent reasoning. The state of one subsystem of a compound system S1 + S2 in the pure state Ψ can be determined by means of the following requirement. For any observable A1 , the expectation value for measurements of the one subsystem, say S1 , in state W1 (using the more generally valid density matrix specification of states) must be identical to the expectation value for measurements of the observable A1 ⊗ 12 with respect to the compound system. ‘12 ’ denotes the identity operator in the Hilbert space of the second subsystem. Explicitly, this requirement reads (Ψ, A1 ⊗ 12 Ψ) = tr{A1 W1 },

(13.3)

where ‘tr’ denotes the trace operator, which for a given matrix maps to the sum of its diagonal elements. Note that W1 is a density operator acting on a Hilbert space and Ψ is a vector in a Hilbert space. These are two ways of describing states. However, whereas Ψ could be written in the form of a density operator, W1 could not be written in the form of a state vector since it is a mixed state. That W1 is a mixed state is a major result of the utilization of requirement 13.3. W1 is called a reduced state due to the reduction of the degrees of freedom of the second subsystem which is expressed by the requirement 13.3. W1 is explicitly given by W1 = tr2 P [Ψ].

(13.4)

I skip further details4 and state the relevant outcome for the case of the 2-electron system. It turns out that the reduced states for each of the two electrons is identical. With this result we have a serious threat for the individuality of the two electrons. Since they are both electrons they have the same permanent or essential properties. So far there is no problem. But as individuals we would expect the two electrons to be different in at least some relational time-dependent properties (position, momentum) which in quantum physics are given by the state. We saw, however, that the states of the two 4

See for instance Mittelstaedt (1998).

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electrons are identical. There is no property whatsoever that would allow us to make a distinction between these two electrons. It arises the question whether there are two individual electrons at all. This is a place where a principle that is known in philosophy as the Leibniz principle of the identity of indiscernibles seems to find a genuine application. When we suppose to start off with two things and there is no ‘legitimate’ property5 in which they are discernible, the Leibniz principle of the identity of indiscernibles says that there actually is just one thing. Using  Predicate Logic, the Leibniz principle reads ∀x1 ∀x2 ∀F F (x1 ) ⇔ F (x2) ⇒ x1 = x2 . Nevertheless, it seems impossible to agree to the conclusion that we are actually dealing with just one electron. We know that we started off with two electrons and we know that the compound system has, for instance, twice the charge of an electron and twice the mass, so that we cannot be talking about just one electron. On the other hand, in the face of the Leibniz principle it seems equally impossible that we have two electrons. We seem to have ended in a stalemate. Whichever alternative we choose we get into trouble.6 13.2.3

Dispositions and Tropes

The stalemate reached regarding the individuality of ‘identical particles’ as subsystems of many-particle systems can be overcome elegantly within a trope-ontological construal of particles. As I indicated in the introduction to this section, I am sympathetic with a variant of trope ontology advanced by P. Simons.7 In his ‘nuclear theory of tropes’, Simons combines the advantages of substratum and bundle theories while avoiding the respective disadvantages. According to Simons’ nuclear theory, an object is composed 5

An example for, as I call it, an illegitimate property is ‘being identical with S1 ’ which has the ad hoc function of undermining the Leibniz principle without actually saying anything about S1 . 6 St¨ockler (1999) contains a description and evaluation of this conceptual problem stressing the connection between the questioned individuality of quantum mechanical particles and the Leibniz principle of the identity of indiscernibles. 7 Most important for the present context is Simons (1994). Valuable additional information can be found in Simons (1987), Simons (1998a), Simons (1998b), and Simons (1999).

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of a tight bundle of essential tropes (‘nucleus’) and looser bundle of additional non-essential tropes. The nucleus accounts for the individual nature of a substance and therefore has a similar function as a substratum. This does not contradict the remarks on the role of essential and non-essential properties for the specification of elementary particles in the last section since in a trope-ontological approach properties are taken as particulars, which is marked by calling them tropes. For that reason a bundle of essential tropes (properties) fixes one particular object and not just a class of objects. Applying Simons’ nuclear theory of tropes allows for the following dissolution of problems with the individuality of ‘identical particles.’ When ‘identical particles’ form parts of a compound quantum system, they cease to be independent substances. Their nuclear trope bundles turn into one new nucleus and since they were both bundles they can easily restructure. Different kinds of ‘identical particles’ lead to different constraints for the formation of the new nucleus, namely to either symmetric or antisymmetric wave functions, as introduced in section 13.2. The advantage over a substratum theory consists in the fact that once the old nuclear bundles are broken up and form one new nuclear bundle, the question about the identification of the old substances in the new substance disappears. In a substratum theory, however, this question cannot be suppressed and leads to the described problems with the individuality of ‘identical particles,’ for the following reason. If the substratum is to fulfil its role as an individuating basis of an object it would be strange if it could pop in and out of existence as it is required for a satisfactory account of ‘identical particles’ as ‘parts’ of many-particle systems. One can say the same again with other words. Apparently, the investigation of quantum physics shows that quantum objects can quickly win and loose their status as independent concrete particulars, or short, of being substances. When two electrons interact and form one compound 2-electron system, two substances cease to exist, viz. the two initial electrons, while one new substance emerges, namely the 2-electron system. If one now performs a position measurement on this 2-electron system and detects an electron, the 2-electron system looses its status as a substance whereas the detected electron emerges as a new substance, and simultaneously with it there is a second electron as a substance too. Note that it has

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to be “a second electron” and not “the second electron” since it is not one of the original electrons that has been measured and regained its status as a substance. Now compare the classical substratum theory of substances and the trope bundle theory of substances as described above. Which of these ontological conceptions can explain the quick popping into and out of existence of substances most naturally? I believe that the trope bundle theory of substances rates far better than a substratum theory in this respect. Since substances are construed as bundles it is easily conceivable that these bundles can quickly turn into new bundles. Bundles are broken up and loose their status of substances. A different collection of tropes is bundled together and a new substance emerges. From the point of view of trope ontology these restructurings do not appear mysterious whereas they are inconceivable when substances are taken as fundamental entities which cannot be analysed any further. Dispositions Since Carnap’s Logical Construction of the World the definition of dispositional predicates has been a point of philosophical debate and the discussion is still going on. The starting point for the debate is the problem whether dispositions can be analysed within the language of extensional logic, first and second order logic. If dispositional predicates like ‘watersoluble’ are defined as material implications of the form x is water-soluble ≡ if x is put into water, it will dissolve we get a problem. Using this definition, everything would be water-soluble as long as it is not tested, because an implication is true whenever its antecedent is false. Later it turned out that definitions in terms of possibleworld semantics a` la D. Lewis are very successful. Besides non-probabilistic dispositions like ‘water-soluble’ there are probabilistic dispositions such as ‘being susceptible to a certain disease with a certain probability when living under certain conditions.’ One question about probabilistic dispositions is whether the probability is due to our ignorance of all details, so that it is inappropriate to assign such a probabilistic disposition as a real property to an object. Opposed to these merely epistemic probabilities are objective or real probabilities which can

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actually be ascribed to objects as their properties.8 Quantum physics is a case where, according to the standard view, one is dealing with objective probabilities since an ignorance interpretation of quantum probabilities can be excluded.9 So-called ‘non-objectification theorems’ show that the assumption of a merely epistemic understanding of quantum probabilities leads to conclusions which are in conflict with wellestablished experimental results.10 When in the following I use ‘prob’ as an abbreviation for ‘probability’ I am referring to these quantum probabilities which I assume to be attributable to single quantum objects. I understand quantum probabilities as objective probabilistic dispositions of quantum objects to display certain outcomes in the event of a measurement.11 All of this has of course a tentative flavour since two issues which are crucial in this context have not been settled yet. First, the notion of objective probabilities is notoriously troublesome.12 Second, the establishment and comparison of solutions to the problem of the quantum mechanical measurement process is an ongoing business. The results of future research might thus necessitate more or less extensive revisions to the ontological approach proposed in this section. 13.2.4

An Example

In order to give an explicit example of dispositional trope bundles in the context of quantum physics I will use the following notation. ‘[ · , · , · , ...]’ 8

Mumford (1998) advocates a radical and realist theory of dispositions, to the effect that in fact all properties are dispositional properties (as opposed to categorical properties). The spin of a subatomic particle is cited as one example of a dispositional property. The anthology Handfield (2009) contains various recent research articles and an introductory chapter by T. Handfield for non-specialists. 9 This is only true as long as we stick to the standard formalism of QM. In Bohm’s alternative version of QM, for instance, the situation is fundamentally different. In this approach, quantum probabilities are due to the ignorance of additional information, the so-called hidden variables. 10 See, e.g., subsection 4.3 (b) in Mittelstaedt (1998). 11 See Su´arez (2007) for an excellent analysis of four different attempts to solve quantum paradoxes by means of dispositional notions. Suarez himself argues in favour of what he calls the ‘selective-propensity interpretation’, according to which “a quantum system possesses a number of dispositional properties, among which are included those responsible for the values of position, momentum, spin and angular momentum.” 12 See chapter 4, part III. in Salmon et al. (1992).

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denotes the compresence relation which accounts for the tropes of a bundle being a bundle and not just an arbitrary collection of tropes. The ‘·’s stand for tropes. Their order is of no significance. I use ‘[ · , · , ...| · , · , ...]’ to make an explicit distinction between essential and non-essential tropes. Therefore the order does matter in this second bracket-notation. Tropes before the ‘|’ are essential tropes, tropes after the ‘|’ are non-essential tropes, i.e. ‘[e-tropes | n-tropes].’ Again, among essential tropes and nonessential tropes the order is of no significance. I will use a lower index at the unspecified compresence bracket - ‘[ · , · , · , ...]object ’ - to indicate which object these tropes make up. Doing so is no embarrassment for a trope ontologist since the claim of trope ontology is not that there are no objects in the usual sense (independent concrete particulars) but rather that such objects are not fundamental. They can be analysed in terms of tropes. In order to make my proposal clearer I will supply an explicit example of how a bundle of tropes for an ‘2-electron system’ could look like: (13.5) [e-tropes | n-tropes]‘2 el.-syst.’ = [m = 2mel , e = −2, s = 1 | prob(pos.), prob(spin), · , ...] where prob(pos.) and prob(spin) are just two examples for kinds of dispositional tropes with an informal notation. Note that I assumed the preparation of a particular spin correlation by choosing s = 1 for the compound system.13 I have set ‘2-electron system’ in quotes because, as I have argued in the preceding sections, one is not actually dealing with two objects, namely the two electrons, but rather with just one object, the one compound system. To make things more explicit, the infinitely many ‘prob(pos.)’-tropes, for instance, would be given by

Q (13.6) probΨ (V ) = dq1 dq2 |ψ(q1 , q2 )|2, V

IR3

where ψ is the spatial part of the ‘2-electron’-wave function Ψ, Q denotes the position observable and V a volume V which is a (Borel) subset of 13

Note further that there is no mass defect as in the case of an atom where the mass of a nucleus is less than the sum of the masses of its constituents in isolation. While the mass defect is due to the binding energy of the nucleons, the assumption for the above many-particle system is that the particles are no longer in interaction—although of course entangled due to their interaction in the past.

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IR3 , i.e. V ∈ B(IR3 ). The expression 13.6 is to be understood according to the standard interpretation, namely that it is the probability for detecting a particle if a suitable measurement is performed in volume V . The probability given by 13.6 comprises probabilities (N, V ), N ∈ {1, 2} probQ Ψ

(13.7)

for detecting N = 1 or N = 2 particles in the case of a position measurement. This means that the probabilities for N = 1 and N = 2 together have to sum up to unity for a disjoint partition of volumes. What I indicated with ‘prob(spin)’ and ‘ · , ...’ in the trope bundle 13.5 are the further terms for spin measurements, momentum measurements etc. One thing that ‘prob(spin)’ specifies are spin correlations in an EPRfashion. The trope bundle in equation (13.5) is thus to be understood as comprising an infinite number of dispositional tropes. The exposition in this section was primarily meant as a preparation of what follows in the next section, which contains the main ideas that I wish to defend. I have shown why a trope-ontological understanding of quantum physics is attractive even before considering QFT and in particular AQFT. In the next section it will become clear that trope ontology is even more compelling in the broader setting of (A)QFT.

13.3

The Trope Bundle Interpretation of AQFT

In this crucial section I will show what I consider to be the appropriate ontology for Algebraic Quantum Field Theory (AQFT), an axiomatic reformulation of QFT.14 I look at AQFT, first, because QFT is, as a candidate for the most fundamental scientific theory of nature, ontologically outstanding and, second, because I consider AQFT to be a particularly valuable formulation of QFT as far as ontological considerations are concerned. The main reason for my high esteem of AQFT in ontological matters is its conceptually perspicuous nature where careful thought is given to the question of which mathematical and physical entities come first and which are only derived, a concern that is very similar to the aim of ontological theo14

Kuhlmann (1999) already contains the basic idea, whereas the main argument as well the elaboration in the following are different.

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ries.These remarks suggest already what I mean by finding an ontology for a given scientific theory. It is finding an ontology where correspondences can be established between fundamental quantities and structural features of the scientific theory on the one side and basic entities of the ontological theory on the other side. 13.3.1

AQFT as a Model of Trope Ontology

The structure of AQFT and (the standard one-category version of) trope ontology closely resemble each other. Both theories deviate from traditional theories, namely QFT/substance ontology, by decisively putting algebras of observables/properties at the bottom.15 “Traditional” entities like particles and fields/substances and universals are seen as derivable or analyzable in terms of those basic entities, i.e. observable algebras/properties. The claim that observable algebras/properties are basic and not the usual entities in traditional accounts is bold and needs to be supported by convincing reconstructions of those traditionally acknowledged entities. Again, both in AQFT and in trope ontology much effort has gone into showing how (particles and fields)/(substances and universals) can be accounted for in terms of observable algebras/properties. Why the properties in AQFT should be rated as tropes and not as universals can be understood most effectively by considering the arguments in the general theory, i.e. trope ontology, just as the behaviour of a concrete torsion pendulum can be understood best by using the results in the general theory theory of the harmonic oscillator. In short, the point is that once properties are seen as the basic category of entities they can no longer be understood as universals, since universals are not suitable for individuation. In other words, properties need to be understood as particulars, i.e. tropes, if they are taken as the basis for all other entities. Naturally, it has to be shown first that the identification as a token of some general type is appropriate. The proof of a realizing model/abstract 15

Both theories assume one further kind of entity, namely states (in AQFT) or compresence relations (in trope ontology) respectively. In both cases the ontological status of these further entities is not as evident as for those entities that, for good reasons, are responsible for the naming of the respective theories.

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theory relationship, or of a token/type relationship, consists in showing that certain entities in the model play the same role or stand in the same relation to each other as do corresponding entities in the general theory. In the following I will exemplify the common structure of AQFT and trope ontology in one important aspect. 13.3.2

An Algebraic Argument for the Bundle Conception

In the following I will argue that a trope bundle theory is the most appropriate ontology for AQFT. I proceed in two steps. The first step of my argumentation consists in showing that a bundle conception of objects is the most natural understanding of the structure of AQFT. In the second step I argue that if one thinks that there are good reasons for adopting a bundle view, one is driven to understand properties as tropes. In other words, the gist of what I am saying is an argument in favour of a bundle conception of objects in opposition to a substratum approach. My ensuing plea for a trope theory is conditional on the first step of the argument in favour of a bundle view. The first step of my argument is the following: According to the bundle conception of objects it is wrong to think that properties can only belong to an object if they can rest on some underlying substratum.16 Schematically the substratum view looks like this: P1   

P2

P3

↑

↑ substratum

P4   

The most immediate candidate for a substratum ontology is the classi16

The substratum view is a particular kind of substance ontology. See chapter 3 in Loux (2002) for an up-to-date account. The substratum view is the main target of bundle theorists.

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cal theory of point particles.17 The general scheme from above could be realized as follows m 

p

E

↑

↑

q  



x(t)





where m is the mass, p the momentum, E the energy and q the electric charge of the particle. In the substratum view of point particles spatiotemporal localization accounts for the identity of a classical particle.18 The underlying view about the ontological status of space-time is substantivalism.19 The substratum view may fit to classical field theories as well. 20 In the case of an electric field, for example, one can defend the view that field values are attributed to space-time points (x, t) in the same way as properties are attributed to a substratum:21 x1 , t) E( 

↑  

17

x2 , t) E(

x3 , t) E( ↑ Space-time

x4 , t) E(   

Butterfield (2006) argues against the “doctrine that a physical theory’s fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there.” Also see Arntzenius and Hawthorne (2005). 18 A related line of thought can be found, for instance, in Strawson’s (1959) Individuals. 19 See Earman (1989). 20 I do not want to claim that classical field theories must be understood in terms of substrata. Wayne (2008), e.g., defends a trope-ontological view of classical fields. 21 I leave the problem of fields at a point aside since talk in terms of neighborhoods of points makes things more complicated without changing the argument I am presenting.

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At first glance it seems as if the central mapping in AQFT O → A(O) is quite similar to classical field theories. However, the situation is very different in AQFT, in a way that calls for a bundle conception. In contrast to classical field theories, one individual mapping O1 → A(O1) in AQFT does not contain any information about the physical system like the field strength in region O1 . Instead, the physical information is contained in the relation of all the mappings O1 → A(O1 ), O2 → A(O2 ), O3 → A(O3 ), ... or, in other words, the physical information rests in the net structure of algebras and not in the individual algebras.22 In particular, the physics of the system is not caught by an assignment of algebras to a disjoint partition of space-time into regions O1 , O2, O3 ... as one would expect if properties were assigned to some underlying substratum. The bundle view fits much better to AQFT. The bundle theorist holds that objects are made up by bundles of properties alone. The schematic representation of a property bundle in figure 13.1 is meant to highlight that the properties only stand in relation to one another without a substratum that holds them together. P2 P1 P3

P4

Figure 13.1: Schematic representation of a property bundle. The next step in my general argument in favour of a trope-ontological account of AQFT brings in tropes and is a purely philosophical step that is 22

For instance, it is specified how algebras evolve for growing diamond-shaped spacetime regions. Algebraic quantum field theorists favour considering diamonds because these are the most tractable regions of Minkowski space-time. In particular, since diamonds are globally hyperbolic space-time regions, they are guaranteed to be causally well-behaved, and moreover, to have a well-posed initial value problem.

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not specific to AQFT. So far I have argued that the philosophical notion of “things” as bundles of properties with no underlying substratum fits well with the physical theory of AQFT, where the net of observable algebras is the central structure that contains physical information. However, I have not said anything about the nature of the properties which are bundled together. In chapter 11 I argued that there are great advantages in understanding properties as tropes, i.e. as particulars and not as universals. Furthermore, in subsection 13.2.3 I presented an argument that trope theory can help to solve conceptual problems regarding quantum mechanical many-particle systems of identical particles. Now I wish to mention a third kind of argument in favour of tropes which is conditional on holding a bundle conception of objects. The point here is that if one wants to adopt a bundle view, then one has to do it with tropes, i.e. with particularized properties (and relations) and not with universals. The grounds for that claim is that the only other option, namely bundles of properties understood as universals runs afoul of the above-mentioned Leibniz principle.23 The bundle of universals view is one of the very few conceivable ontological options that actually seems to be out of the game. Although this third conditional argument in favour of tropes is a very important step in my current argumentation I will leave it to its mentioning since it is purely philosophical and has nothing to do with physics, not to speak of AQFT. 13.3.3

Representations and Properties/Tropes

So what could the trope bundles for AQFT be explicitly? After the very general line of argument in favour of a trope-ontological understanding of (A)QFT in terms of bundles of tropes (i.e. properties and relations) in the last subsection one would like to sort out those entities or structures in AQFT that can be identified with tropes, that allow for a distinction between essential and non-essential tropes and to give explicit examples of tropes. Regarding the distinction between essential and non-essential tropes one possibility is to say that the net O → A(O), O ∈ B(M ) encodes the essential ‘tropes’ and the state the non-essential ones.24 Note that tropes 23 24

For details see section 4.II in Armstrong (1989) or chapter 3 in Loux (2002). I considered this view for a while and Jeremy Butterfield proposed it as well in reaction

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comprise properties and relations, as long as they are thought of as particulars. Therefore it is no problem to have, e.g., the relation of the mappings O1 → A(O1) and O2 → A(O2) or of O2 → A(O2 ) and O3 → A(O3 ) among the tropes in this approach of a trope-ontological understanding of AQFT. Although I think that this line of thought is not wrong, but it is unsatisfactory for my taste. It puts all the burden of detail and explicitness on the shoulders of the physicist. I would like to see if it is possible to get at least one step further in order to reach a stage where a more explicit identification of entities or structures in AQFT with tropes can be given. Moreover, I would like to make further use of the elaborated corpus of AQFT which to me seems helpful in this context. The main idea of my proposal is to link up tropes with representations of algebras rather than with algebras (or their elements) themselves. At this stage it is helpful to consult section D.3 of the appendices which deals with the connection between representations of algebras and superselection sectors.25 There are a number of reasons that have driven me to this conclusion. My first reason is that in the variant of AQFT I am mainly considering, namely the abstract approach, the algebras of local observables are abstractly defined. The elements of these algebras are thus not explicitly given, they are only specified by certain requirements. What it means that the algebras are abstractly defined can be understood by looking at how commutation relations fix (groups and) algebras without dealing with any explicit elements of these sets. There is a very good justification for this highly abstract approach: It refrains from using concrete realizations that embody an implicit and unjustified choice among various available possibilities. Moreover, the abstract variant of AQFT yields the most immediate access to an understanding of superselections sectors which I consider to be very important for ontological questions. However, as far as physical understanding and ontological significance are concerned, the abstractness of the algebras in AQFT speaks, in my view, against a dito a talk I gave in Oxford in January 2003. However, in the sense of a final answer I find this view unsatisfactory for the reasons that will follow now. A related statement, although not in the context of trope ontology, can be found in Fredenhagen and Rehren (1998). 25 Also consult Earman (2008) for an account of superselection rules that is tailored for philosophers. Among other things Earman shows why the algebraic approach to quantum physics may explain the origin of superselection rules in a uniform and comprehensive manner.

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rect (representational) linkage between algebras and their elements on the one side and ontological entities on the other side. The algebraic formulation of QFT allows for deep insights into the underlying structure of QFT with respect to such issues as relativistic features, localizability as well as the general significance of and the relation between observables, states and quantum fields. However, the depth and generality have a price, namely a certain lack of physical explicitness on the level of algebras. In order to identify the mathematical counterparts of physical properties in AQFT one needs, I think, to look at the elaborated representation theory of algebras. Moreover, there is no specific physical information in individual algebras. Representations on the other hand are concrete and can have an immediate physical significance. Another reason against a direct linkage between tropes and elements of algebras has to do with the main difference between Segal’s approach and AQFT. One of the important early results of AQFT is the occurrence of inequivalent representations, which explains why superselection rules arise. This insight is one of the advantages of (abstract) AQFT over the C ∗ -algebra approach by Segal, who rated the occurrence of inequivalent representations as evidence for limiting all considerations to the level of the algebra. A third reason for linking up tropes with representations and not with algebras themselves is that the concept of the irreducibility of representations, it seems to me, is an appropriate mathematical requirement for ontological fundamentality or elementarity just as in Wigner’s analysis of the Poincar´e group. One might wonder now why one should take pains to go through the algebraic formulation of QFT when, in the end, one arrives at concrete realizations of algebras again, i.e. the initial starting point. Has it not been argued that in the standard formulation one makes implicit choices of particular representations, which are not equivalent because a field has an infinite number of degrees of freedom? And suddenly these very representations should yield the answer to the puzzles of ontology? Although it seems as if one has arrived at the very same point where one has started this is not true in one important sense in particular. The connection between the irreducibility of representations and ontological elementarity—my third reason for linking up tropes with representations—is not accessible on the level of representations without realizing them as representations of algebras.

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Moreover, different inequivalent irreducible representations correspond to different elementary systems—provided they have physical significance at all. Therefore the analysis of representations as representations of algebras allows for a classification of elementary physical systems, again in this sense like in the case of Wigner’s analysis of the Poincar´e group. There are a number of stages or forms of (ontological) redundancy with respect to representations of algebras in AQFT. The first two stages are reducibility of representations and unitary equivalence of (irreducible) representations. The physical irrelevance of some irreducible representations is the third stage of redundancy with respect to representations of algebras in AQFT. Because of a pivotal correspondence between inequivalent irreducible representations and superselection sectors (see section D.3 of the appendices) it is the theory of superselection sectors that matters for the following. It is the task of superselection theory to sort out those inequivalent irreducible representations with physical relevance. With respect to the division into essential and non-essential tropes I propose to make the following identification with entities in AQFT. I identify the essential tropes with the defining characteristics of one superselection sector. Since these defining characteristics are always conserved quantities (like various kinds of charges) one has very natural candidates for fundamental entities which specify the essential properties of a physical system. My pivotal point here is that these fundamental entities are properties (i.e. tropes) and not things or ‘substances’ like in most traditional ontologies. As far as the non-essential tropes are concerned I think one can identify them with the remaining characteristics of a vector state within a given superselection sector. By ‘remaining’ I mean those characteristics that go beyond specifying the membership to a particular superselection sector. Since the encoded physical information is often probabilistic, many tropes are dispositional and thus I call my proposal ‘Dispositional Trope Ontology’.26 26

Recently Morganti (2009c) has also advanced a trope-ontological account of QFT. As far as I can see his main motivation in favour of trope ontology is purely philosophical and the connection with physics shows how trope-ontological ideas can be applied in hard science. In contrast, I rather tried to give an argument that rests on the physics itself, namely the structure of AQFT. And since for me it is AQFT that supplies an independent and very strong reason for a trope-ontological view of fundamental physics I want to look at this same structure in order to find out what the basic tropes are. Morganti has instead

178 13.3.4

CHAPTER 13. DISPOSITIONAL TROPE ONTOLOGY Outlook on Potential Problems and Further Work

I will finish this section with some remarks about further work to be done and potential problems for my proposal. The first point that worries me a bit is that the essential properties are not linked to local algebras AO directly but to the quasilocal algebra A (via its irreducible representations). The ‘quasilocal’ algebra A is the smallest C ∗ -algebra containing all local algebras which means that A itself is not a local algebra but has a global flavour.In a sense it might be against the intitial spirit of AQFT to give an algebra with a global nature such a weight. However, this route has been taken in the development of AQFT itself, and in the end quantum physics is an inherently non-local theory anyway. Another point that is worth investigating is the relation of my proposal to the afore-mentioned view that the net O → A(O), O ∈ B(M ) encodes the essential properties (or ‘tropes’) and the state the non-essential ones. The relation of these two approaches is not obvious since the second approach is, as I have pointed out, very abstract and implicit, in comparison to the approach that I advocate. Maybe both approaches could be combined in a fruitful way. Moreover, I wish to mention that, first, the theory of superselection is very important for decoherence approaches to the measurement problem and, second, that there are alternative programs like Rob Clifton’s ‘beables’ for AQFT. Eventually, I want to speculate about a powerful but also very demanding option for putting my approach on an even firmer and more comprehensive formal basis.27 Both trope theory and the algebraic theory of superselection sectors suggest a formalization in terms of ‘category theory’, which was first introduced by S. Eilenberg and S. Mac Lane, in the 1940ies.28 Being a generalization of algebraic and topological concepts category thedirectly used the Standard Model of elementary particle physics for this identification. It is perfectly legitimate to do so, of course, but I think that it leaves the deeper reasons for the trope-ontological nature of the fundamental physical entities uncovered. 27 I want to underline, however, that this enterprise would certainly be very difficult to realize since it requires expertise in a number of different fields, each of which is highly intricate in itself already. 28 For many philosophers category theory is attractive because, among other things, as an abstract foundation for mathematics it is an alternative to set theory. See Marquis (2009) for an account of category theory from the philosophical perspective and a very comprehensive bibliography.

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179

ory attempts to capture axiomatically what is similar in various different classes of mathematical structures, e.g. group structures, by relating them to the structure-preserving functions between them. To my knowledge, in the case of trope theory there is just one advanced account of tropeontological ideas in the framework of category theory, more specifically the topological theory of sheaves, namely by Mormann (1995).29 Unfortunately, Mormann is primarily interested in how the resemblance of tropes can be understood by casting trope theory into the setting of sheaf theory. And I do not see how I could use his approach for the compresence of tropes in a bundle which I need for my analysis.30 In comparison, the formulation of the algebraic theory of superselection sectors in terms of category theory is more advanced.31 It would my hope that formulating both trope theory and the algebraic theory of superselection sectors in the common framework of category theory could make it possible to discern in a systematic way which elements of the scientific theory (i.e. AQFT) represent the tropes of the ontological interpretation of that theory. In my approach, which I outlined above, I think I got this identification right, but I went a long way without a map. 13.3.5

The Explanatory Power of the Trope Bundle Interpretation

As I point out in the appendix section D.3, the analysis of superselection sectors by Doplicher, Haag and Roberts uses an idealized notion of localization. In this subsection I will briefly sketch a more realistic approach to localization in AQFT. Furthermore, I will show why this approach adds to the arguments in favour of a trope-ontological understanding of (A)QFT. The so-called detector model establishes a contact of the abstract theory of AQFT with the practice of scattering experiments in High Energy Physics.32 From a philosophical point of view the most thought-provoking 29

Also consult Bacon (2008), section 10, for a brief exposition and discussion. As Bacon (2008) points out, it “is an oddity of the sheaf-theoretic approach, however, that it wet-blankets one of the most distinctive features of classical tropism: the construction of individuals as qualiton clusters.” 31 See Halvorson and M¨ uger (2007), for instance, and Sica (2006) for the broader perspective. 32 Compare section 8.2. The detector model is presented and discussed at length in 30

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concept in this model is that of almost localized operators/observables corresponding, in this context, to almost localized positions of the respective particles. In the detector model ‘almost localized in a certain region’ implies that all the rest of space-time still plays a role but only to a negligible extent. However, for the bearer of the unsharp property ‘almost localized,’ this has an important consequence for the competition between the substratum and the bundle view. It needs to be equally present in all of space-time because a property cannot be assigned in a part of space-time where the bearer of this property is not present. One can see this in the following way. The bearer, or ‘substratum’, of properties is by definition itself without any further properties. It gives only unity and individuality to objects. The substratum can begin and end to exist but since it is itself without any properties it is either fully there or not at all. This is the reason why the idea of the attribution of an unsharp position has such far-reaching consequences for the goal of describing particles within QFT. The bearer of the property ‘almost localized’ has to be uniformly spread out in the whole universe. This is effectively tantamount to a field ontology, the very opposite of trying to understand QFT in terms of particles. Thus in the course of trying to save the particle concept by introducing approximate localizability the problem comes in through the back door again. One way to handle this situation is to take the use of unsharp properties as a mathematical trick to smooth over the transition from theory to pretheoretical conceptions of nature and to dispense with ontological considerations altogether.33 Instead of that, I wish to propose an alternative ontology for unsharp properties. Obviously the very question of how an unsharp property can be attributed to the bearer of this property presupposes that properties always have to be attributed to some underlying substratum. In the light of trope ontology, however, this problem vanishes. An unsharp property, like any other property, does not need to be attributed to a substratum. A trope ontologist is therefore not forced to assume a particle to be equally present, via its substratum, in all of space-time. Objects understood as bundles of tropes are only as much present in a region sections II.4. and VI.1. of Haag (1996). Also, see Falkenburg (2007) who pleads strongly for taking experimental issues as highly important for interpretative concerns. 33 D. Buchholz advocated this view in private communication (July 1999).

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181

of space-time as the respective properties are present. One thus has a way to maintain the view that QFT speaks about particles—leaving the question of individuality aside. The price, however, is that one has to drop a basic assumption of the common account of particles, the assumption of a propertyless bearer of properties. As a piece of descriptive metaphysics, a trope-ontological account of the emergence of particles in QFT is more appropriate than a substratum account. First, because it makes far more sense of ‘almost localizability,’ and second, because it takes seriously that QFT effectively only has access to properties while the direct appearance of individual particles via particle labels in tensor products is purely conventional. Finally, I want to mention that there is a further logical alternative to understand an unsharp property, namely as a sharp property which is only attributed in an unsharp fashion.This alternative sounds compelling but turns out to be hard to grasp on second thought. One should think that something either has a property A or it does not and if it seems that it has the property A only approximately then what one should really be saying is that it actually has a property other than A, say A . In that case, A is only very similar to A and it is hard to see or to understand A without mentioning or thinking of A.

13.4

Summing Up

I hope to have successfully argued that my proposal of a dispositional trope theory is particularly appropriate as an ontology of QFT. I have pointed out that a dispositional trope ontology suggests itself when looking at that formulation which is most significant in an ontological sense, namely AQFT. Besides the immediate naturalness of a trope-ontological understanding of QFT in its algebraic formulation, two specific ontological problems could be solved. Both of these problems have the same structure. Although one should like to see a many-particle system as just one object or substance and although one would in general like to ascribe an unsharp localization to particles both manoeuvres are impossible as I have argued. I could show that the situation changes on a trope-ontological background. Now both wishes can be fulfilled due to a change in the ontological attitude.

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A more detailed account of how dispositional trope ontology can foster a more natural understanding of QFT will be given in the following conclusion to the whole thesis.

Part V

Concluding Remarks

Chapter 14 Physics and Philosophy I begin my conclusion with some remarks on the relation of physics and philosophy when ontological questions are under consideration. If only a fraction of my investigations have been convincing they should yield ample evidence for the theses I will lay out in this chapter. In short, I claim that neither philosophy alone nor physics alone are in a position to paint a coherent picture of the general structure of the physical world that takes all relevant knowledge into account which is available today. Note, however, that this claim is not meant to imply that physics and philosophy could not be pursued without taking notice of each other. All I want to argue for is that getting a comprehensive idea of the physical world on an upto-date level of discoveries necessitates a cooperation between physics and philosophy. Investigating the most general structures of what there is in the world two extreme ways can be chosen and often are chosen. One of them is deeply rooted in the tradition of philosophy from ancient and medieval times to modern rationalism and idealism until the twentieth century. Proponents of this way foster the idea that the structures of being qua being can be investigated by pure thinking in an a priori fashion. Some take a more modest stance and concede the necessity for at least some intuition or everyday experience. Nevertheless, defenders of the (quasi-) a priori tradition of ontology imagine their results to be immune against any specific scientific results. Opposed to this first way of investigating the most general structures of being is a second one. Proponents of this second way to ontology claim that only specific sciences, in particular natural sciences, are in the position to say anything about the basic entities there are and their irreducible 185

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characteristics. They contend that purely philosophical considerations on ontology are fruitlessly speculative and ill-founded and have no value in the light of ‘real scientific findings’. This pejorative stance towards ontology as a philosophical discipline has found emphatic support within some philosophical schools as well. Most notable are the British Empiricists in the seventeenth and eighteenth century and the Logical Positivists in the twentieth century. I believe that there is some truth to be found in both ways but that they can profit from each other and even need each other eventually. Using a similar statement by Kant I think that with respect to ontological questions ‘philosophy without sciences is empty and science without philosophy is blind’. I will try to give explanations for both sides of this claim. Why does philosophy depend on sciences when ontological matters are treated? I think that philosophy can get substantial results about our everyday ontological thinking and can uncover some hidden assumptions of our way to conceive of the world. However, when it comes to more fundamental questions about the ontological structure of the world apart from our possibly changing ways to think about it in everyday terms, it seems to me that philosophy alone comes to an end. All philosophy can do here, I believe, is to lay out a matrix of ontological options. In some cases it will be possible to exclude some of these options for internal reasons. In general, however, things are not so easy. I have the impression that most ontological conceptions have their merits with respect to those aspects which gave rise to their establishment while they have their weak points in other respects. In a situation where each conception has its successes but carries the burden of anomalies or unsolved problems as well an evaluation of different aspects becomes pivotal. How can physics profit from philosophical considerations about ontology? Probably the most important benefit for the physicist is not one that would help with his work. Nevertheless, I think ontological considerations can be helpful as heuristics when a theory is not completed yet as in the case of QFT due to the so far unsuccessful incorporation of gravitation.

Chapter 15 Summing Up 15.1

General Remarks

The investigation in this thesis can be devided into two main parts. The first part consists of general ontological considerations and their foundations in philosophy as well as quantum physics. The second part contains investigations of some either important or promising ontological approaches to QFT. The considered approaches have emerged partly in physics and partly in philosophy. Likewise, the arguments for their evaluation came in some cases from philosophy and in other cases from physics. Philosophical investigations about QFT form a relatively new area of philosophical research compared to similar studies in the philosophy of science and of course even more so with respect to general philosophy. Nevertheless, philosophical questions about QFT do have a tradition both in regard to methods and contents. As I have laid out in chapter 2 this two-fold anchorage in tradition applies to the history of general philosophy as well as to modern philosophy of science. With respect to the history of general philosophy the most prominent forerunner to questions about the ontology of QFT happens to be the same regarding methods and contents, namely the history of atomism. As regards contents the debate about atomism is the historical forerunner of ontological considerations about QFT since the atomistic point of view rests on the assumption that all natural phenomena can be reduced to a set of basic indivisible building blocks, namely the atoms. (Note that what we denote as atoms today are of course no candidates for the atoms of atomism.) The same program of reduction has always been at the core of QFT and naturally this holds for attempts of ontological analyses of QFT 187

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as well. As we saw in chapters 11 and 13 on trope ontology ‘reduction to a set of basic entities’ must be understood in a very wide sense since some revisionary ontologies like trope ontology strain the limits of this notion of reduction. Nevertheless, I rate trope ontology to be within the limits of the reduction program. As regards methods the development of atomism can be seen as a forerunner of ontological investigations about QFT for one general and one more specific reason. The general reason is that the development of atomism displays an interplay of physics and philosophy (to the extent that one can speak of two separate disciplines at all) which has gotten closer over its history. I have pointed out in chapter 2 that from ancient times to the middle ages the focus shifted from atomistic speculations as an aim in itself to the search of pragmatic explanations of natural phenomena on the basis of atomistic theories. The more specific reason why the debate about atomism is a methodological forerunner of the ontological analysis of QFT is that in both cases, atomism and QFT, the program of reduction to a set of basic building blocks was a very fruitful heuristic. It is one of the spectacular successes of QFT to correctly predict the existence of previously unknown particles on the basis of systematical reduction schemes. In my main study three general kinds of arguments (which can be either positive or negative) occured. The first kind of arguments are primarily philosophical. They refer to questions of consistency, simplicity, scope etc. The second kind of arguments are those which are mainly grounded in physical requirements, such as relativistic covariance or independence from the frame of reference. It turned out, however, that a third kind of arguments is predominant in the ontological analysis of QFT. These are arguments which cannot be construed as having separable philosophical and physical components. In these cases a physical argument can radically change its significance when different ontological approaches are considered. Arguments concerning non-localizability and unsharp properties are among the most important examples.

15.2. COMPARISON OF ONTOLOGIES FOR QFT

15.2

Comparison of Ontologies for QFT

15.2.1

Particles Versus Fields

189

I begin my final evaluation and comparison of the considered ontological approaches with the seemingly clearest result. Various arguments were marshalled in chapter 8 which put a heavy pressure on a particle interpretation of QFT. It was demonstrated in chapter 4 already that the formalism of QFT allows for the “creation and destruction of particles”. If particles are considered as the “substances of the world” (Armstrong) then the possibility of a creation and destruction of particles would spoil the whole conception. After all, as I have argued, the notion of substance is meant to ensure the very possibility of cognition in a changing world by reference to something stable and independent. Entities which simply pop into and out of existence do not fit this picture very well, to say the least. One could argue that the expression “creation and destruction of particles” is just a figurative way of talking about a certain piece of formalism rather than an ontologically significant statement. However, the same stance would then have to be valid with respect to N-particle states as well and this would imply that N-particle states have nothing to do with N particles. This would mean that degrading “creation and destruction of particles” to mere figurative talk is like throwing out the baby with the bathwater. Nothing is gained for the particle interpretation in this way. Some further problems for a particle interpretation of QFT were introduced and discussed in section 8.4 to the effect that the particle number is not an objective feature of a physical system. One of these problems is the possibility of a superposition of particle states with different numbers of particles. Although it first appears like an embarrassing fact for the particle interpretation it can probably be at least weakened by a proponent of this view. He can argue that the possibility of counter-intuitive superpositions (see Schr¨odinger’s cat) is rather an odd trait of quantum physics than a problem specifically for the particle interpretation. In the light of my arguments in section 13.3.1 I would back this general line of reaction by pointing out that quantum physics is inherently dispositional. Nevertheless, it is again not clear whether it would actually be a wise move by the particle ontologist to defend himself by reference to the dispositional character of the quantum world. As I have argued in chapter

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13 I think that not all ontological approaches can handle the dispositional nature of quantum theory equally well. I will come back to this point when discussing my own proposal of a dispositional trope ontology. I wish to add a general remark. The evaluation of arguments against the tenability of a particle interpretation of QFT displays a characteristic feature which we will encounter again in other discussions. There are some traits and problems of quantum theory which tend to pop up in disguise and notoriously cause trouble. Such issues are for instance the quantum measurement problem, the dispositional character of properties and the emergence of classical properties which are all closely related to one another. My point now is that reference to these problems must not in itself make up a good argument against a particular ontology. Since these problems are either yet unsolved in general or something we have simply not quite get accustomed to, no single ontology should get all the blame. However, the extent to which a general quantum feature is a problem can depend on the context, in particular on the ontology one has adopted. One and the same feature can be completely against the spirit of one ontology whereas it can be incorporated rather naturally into another ontological scheme. I think that the possibility of a superposition of particle states with different numbers of particles is such a feature which is against the spirit of a particle interpretation. I discussed some further results in section 8.4 which can all be seen as demonstrations for the non-objectivity of the particle number of a physical system. The best known of these results is the existence of vacuum fluctuations, i.e. local deviations from the global particle number zero. The second result was the Unruh effect which shows that the particle number is not independent of the state of the observer. The third of these results comes from the study of QFT in curved space-time and indicates that the existence of a particle number operator is a contingent property of the flat Minkowski spacetime. At least the first two apparent problems for a particle interpretation can be challenged (and were challenged, e.g. by P. Teller) in the same way as the above argument regarding the superposition of particle states with different numbers of particles. Naturally, the corresponding counter argumentation can again be used as well. Instead of elaborating on this discussion I will now leave the problems of the particle interpretation of QFT which are connected with the par-

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ticle number in favour of a different problem which might be particularly devastating for the idea of particles as basic entities, namely the problem of non-localizability. Over a little more than the last two decades a number of results have been produced which purport to demonstrate an unavoidable clash of causality and localizability when quantum theory is considered in a relativistic setting. Most notable of these are theorems by Hegerfeldt, Malament and Redhead. It is immediately clear why so much notice has been given to these results in ensuing discussions. Provided that the general line of interpretation given by the above-mentioned authors is correct, then particles could not be localized in any finite region of space-time, no matter how large it is. While it is obvious that the notion of a point particle is just a mathematical idealization, the smearedness of particles over all of space-time seems to stretch the idea of a particle beyond its limits. In section 8.3.2 I compared the theorems by Malament and Redhead in detail. It turned out that both theorems are equivalent although their appearances, their starting points, the formalism and mathematics used as well as the final conclusion drawn by the respective authors are quite different. Besides the in itself interesting fact of the equivalence of these results my comparative study effected a closer look at the respective assumptions and conclusions. Intuitive notions became explicit and could be given a second, more skeptical look. As with the evaluation of arguments concerning the objectivity of the particle number, the evaluation of arguments concerning the non-localizability of particles does not have as straightforward an impact on the tenability of a particle interpretation as it first appears. Locating the origin of non-localizability is more delicate than one might expect. I think that the conclusion that non-localizability speaks against the tenability of a particle interpretation is a legitimate way of interpretation. However, it is not the one and only legitimate way. There are some other alternatives left. I have considered other reactions to the problem of non-localizability which see its core not as a problem for a particle interpretation but rather as evidence that the concept of sharp localization itself is flawed. When discussing my own proposal for an ontology of QFT in subsection 15.2.3 I will argue for still another line of reaction to the problem of non-localizability. In short I will show that it depends on your ontological assumptions whether

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or not non-localizability is fatal for a particle interpretation. On the other side it was pointed out that the particle interpretation of QFT is an option which one is not easily willing to surrender. After all, QFT seems to rest on experiments with colliding particles. How can it be that the theory corresponding to these particle experiments tells us that there are no particles? Although this looks like a strange conclusion it is not absurd. It is perfectly possible that the impression that we have observed collisions of particles was deceiving. Like many other scientific observations the ‘observation’ of elementary particles is a highly interpretive business. Whereas a particle interpretation of QFT is confronted with various direct arguments against its tenability a field interpretation has a very different kind of trouble. It is not even clear what to argue against. As I pointed out the first immediate reason why QFT is often considered as a field theory is the occurrence of ‘quantum fields’ in the formalism of QFT. However, this does not lead to a viable conception of fields as the basic entities of QFT. ‘Quantum fields’ cannot be taken as these basic entities because they yield something physically real only together with the state vector. This problem cannot be cured by taking the state vector as a field as well because it is a fundamentally different kind of quantity. The other line of argumentation in favour of fields takes all arguments against a particle interpretation as evidence for a field interpretation which is tacitly taken to be the only alternative. However, this line of argument is even less apt to specify what these fields are. As I will explain later, I think there is a more promising way to handle arguments against a particle interpretation.1 15.2.2

Processes Versus Tropes

Formulating my final judgment when comparing process and trope ontology is an easy task. Process ontology might be apt to solve a number of conceptual problems which bar an ontological understanding of QFT. But I think that it is simply not necessary to go through such a radical revision of our ontological schemes to tackle these problems. I believe that with 1

In a similar manner Baker (2009) has recently argued that field interpretations are equally infected as particle interpretations so that one should rather think about the lesson taught by approaches that deal with algebras of observables.

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respect to QFT a trope-ontological account has the same problem-solving potential with much lower ‘revisionary expenses’. Whether or not internal philosophical considerations will speak in favour of process ontology in the end is a question I am not addressing here. So far at least, the arguments in that race have not yet come to an end. The idea of trope ontology is to analyse ordinary objects, like a chair or an electron, in terms of bundles of properties. However, and this is the main point, properties are taken as concrete particulars and not as universals, the last of which can be realized (or instantiated) many times. In order to express this difference to the standard view the trope ontologist denotes properties as tropes. Although tropes are now classified as concrete particulars there is an important difference to those objects which are commonly thought of first when speaking about concrete particulars. In contrast to these ordinary objects tropes cannot exist by themselves. Tropes are dependent concrete particulars which can only exist in bundles together with other tropes. Despite of the figurative talk of ‘bundles’ tropes should not be conceived as proper parts of a bundle. It would obviously lead to contradictions to think of a charge trope as a proper part of the bundle of tropes which makes up an electron. Instead of being in a part-whole relation the tropes of a particular bundle stand in the so-called compresence relation. There is no need to construe this relation as an additional entity besides the tropes of that bundle but rather as an internal relation which expresses how the tropes of a bundle necessitate each other. I have argued that I consider a trope-ontological account of QFT is superior to the other options. In order to justify that claim I have proposed a new ontological construal of QFT in terms of dispositional tropes. I will summarize the contents and merits of my proposal in the next subsection. Since a treatment of the notions of particles and fields is an integral constituent of my argumentation for a dispositional trope ontology, the summing-up of my proposal also contains my final evaluation of particle and field ontologies. The concluding subsection is therefore tantamount to a comparative judgment about the ontological options considered in this book.

194 15.2.3

CHAPTER 15. SUMMING UP The Merits of Dispositional Trope Ontology

Trope ontology is a revisionary approach which claims that the ontologically most fundamental entities are not those objects which are commonly taken to be fundamental. In trope ontology it is claimed that tropes are the most basic. On the one hand this claim should not be read as the assertion that e. g. electrons do not really exist since only tropes have true existence. All that trope ontology claims is that objects like electrons can be further analysed. On the other hand the idea of an analysis of ordinary objects in terms of bundles of tropes is different and in a sense more radical than the idea that it is possible to further divide electrons and quarks, e. g. into superstrings. The main idea behind trope ontology is the view that the standard conception of objects as such is misconceived. The stance of trope ontology is directed against a widespread and and mostly unrecognized construal of objects (or substances) into an invariable substratum on the one side and properties on the other side. As I have argued in a number of contexts it seems to me that it is this (mis-)conception of objects, to which particles and fields count, which is responsible for various problems which particle and field interpretations of QFT are confronted with. I will give two important examples. As I have set out the study of many-particle systems of quantum mechanical ‘identical particles’ revealed a serious problem for the idea of individual particles and therefore for a particle ontology in general. My point is now that these problems stem from the conception of a substance, here a particle, as somehow composed of an underlying substratum and its properties. One would like to say that once identical particles come together (via mutual interaction) to form a ‘many-particle’ or compound system there are no individual electrons any more. The is just one substance, namely the whole system. As I have argued I consider this to be mere rhetoric as long as one holds on to the substratum/properties-view. This gloss cannot bar the question where the ‘old’ substrata of the single electrons have gone. Within a trope-ontological scheme the same gloss of the ‘many-particle system’ being just one substance is more than rhetoric since it can be given a natural explanation. The bundles of the former single electrons have been ‘restructured’ to form one new trope bundle. In contrast to this explanation the corresponding explanation within the substratum view is

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far less convincing because the ‘comprehensive’ substrata of the former electrons are to rigid and ‘block-like’ to allow for such a restructuring. Now I come to the second example of an issue where I have argued that the problems have their origin in the misconception of an object as composed of a substratum plus its properties. I am talking about the apparent non-localizability of particles and the notion of unsharp properties as a way to handle these problems. I have set out that within a substratum framework the introduction of unsharp properties does not solve anything. The basic problem is just pushed to a different place. In order to ascribe an unsharp localization to an object, the bearer of this property, i. e. the substratum, has to be present in the whole universe. Since the substratum is so block-like nothing is gained. The substratum has to be assumed as equally present in the whole space. I have argued that a trope-ontological conception of objects sheds a very different light on unsharp localization. In a trope-ontological framework an unsharply localized object is just as much present in a certain region as the unsharp localization property tells. This would mean that the concept of unsharp localization unfolds its merits only when objects are construed as bundles of tropes but not when taken in the standard substratum view. The last point of my proposal for an ontological understanding of QFT to be mentioned again here concerns the nature of tropes which I take as the fundamental entities. I have set out that since QFT is an inherently probabilistic theory this salient feature has be reflected in its ontology as well in terms of dispositional tropes. This is one of the two contexts for which I have introduced Algebraic QFT (or short AQFT) which is a conceptually ‘clean’ axiomatic reformulation of QFT. I have shown that a dispositional trope ontology is the most natural ontology for AQFT which almost immediately suggests itself. In AQFT all the physical information about a quantum object is encoded in a certain nesting of algebras of observables. AQFT can be seen as a theory purely in terms of dispositions. And since there is nothing else than these dispositions I have argued that it is best to conceive of them as tropes. The punchline of my own proposal for an new ontology of QFT runs as follows. Both a particle and a field interpretation of QFT do have a certain legitimacy and neither can be refuted conclusively. However, I have argued that both particles and fields should not be taken as fundamental in an

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ontological sense. In the approach I propose dispositional tropes are the basic entities out of which everything is composed. Arguments against a particle and a field interpretation of QFT can thus at least be weakened although there is a price one has to pay: It is acceptable to speak and think of particles and fields provided one does not take them as being fundamental. Personally, I suppose that questions concerning the ontology of QFT are doomed to the same fate as questions concerning the quantum measurement problem: If our questions should ever come to an end the final solution, I believe, will neither be beautiful nor will it be accompanied by an alleviating eureka. I rather expect that we might in the end, at least partly, simply get accustomed to different ways of thinking which dissolve rather than solve our problems.

Part VI

Appendices

Abbreviations and Notation AQFT CCR GNS GRT prob SRT QFT QM

Algebraic Quantum Field Theory Canonical Commutation Relations Gelfand-Naimark-Segal General Relativity Theory Probability Special Relativity Theory Quantum Field Theory Quantum Mechanics

A, B, C R A, B, C, A1 , A2 B(H)

C ∗ -algebras Von Neumann (or W ∗ -algebra) Elements of C ∗ -algebras and of von Neumann (or W ∗ )-algebras Set of all bounded operators acting on Hilbert space H

199

Appendix A Special Relativity Theory: Some Notation and Required Results For the readers convenience I collect a few pieces of notation and some results which are important at different places of my investigation. Thus this appendix, in contrast to the later ones, is not even approximately self-contained.1 Using the metric tensor gµν   1 0 0 0  0 −1 0 0   gµν =  (A.1)  0 0 −1 0  0 0 0 −1 one can relate xµ and xµ according to xµ = gµν xν

(A.2)

where Einstein’s summation convention applies. The same relation holds for any pair of a contravariant and the corresponding covariant vector. On a passive interpretation the metric tensor merely represents a change between two coordinate systems which does not change the Lorentz distance between two points. ∂ 1∂ , −∇) (A.3) ∂µ ≡ ≡( ∂xµ c ∂t is the contravariant derivative and ∂ 1∂ ∂µ ≡ µ ≡ ( , ∇) (A.4) ∂x c ∂t 1

For a more comprehensive story as well as more details see, for instance, Goldstein (1980) and Rindler (2006).

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is the covariant derivative. In relativity theory the non-relativistic expressions for energy and momentum are no longer conserved separately because the non-relativistic mass (not the rest mass) changes with the velocity. Instead, there is a conservation law for the relativistic energy-momentum 4-vector E E (A.5) pµ = ( , p), pµ = ( , −p) c c which is called ‘4-momentum’ or ‘world momentum’ as well. The formal expression of the relativistic conservation law is that the modulus of pµ is constant, so that there is an invariant world scalar m2 c2 = pµ pµ = E 2 /c2 − p · p

(A.6)

which immediately leads to the relativistic relation between the total energy E, the momentum p and the rest mass m (the mass of the system at rest) which reads (A.7) E 2 = p2 c2 + m2 c4 where c is the velocity of light. Equation (A.7) is the relativistic counterpart of the non-relativistic equation E = p2 /2m (or E = 1/2mv2 for constant mass). Einstein’s famous relation ∆E = ∆mc2 (or E = mc2 ) follows by looking at the change ∆E of the energy which results from a change ∆m of the rest mass, i.e. when ones takes ∆p = 0. The additional term m2c4 (the square of the rest energy) in equation (A.7) is the main difference between the relativistic and the non-relativistic equation.

Appendix B Ontologically Oriented Survey of Quantum Mechanics In this and the next two appendices I will deal with some salient features of quantum mechanics (QM), quantum field theory (QFT) and the algebraic approach to quantum field theory (AQFT), an axiomatic reformulation of QFT. Instead of aiming at completeness I will emphasize those issues which are of particular significance for ontological considerations. Moreover, I will introduce certain pieces of formalism which prepare the ground for investigations in some later chapters.1 One important aspect which has governed the choice of material in this and the next two chapters is the transition from one theory or formulation to another, e. g. the transition from QM to QFT, from the standard formulation of QFT to axiomatic reformulations or from a Hilbert space formalism to an algebraic one. Since I think that a study of these transitions reveals various ontologically significant features I will review such transitions in some detail and I will investigate a number of questions. Which were the (historical) reasons to look for another theory and were these reasons justified from a current point of view? How did one try to overcome the difficulties or weaknesses of the older theory? How successful are the new theories in these respects? I will ask these questions in particular regarding those transitions where new categories of formal entities are posited or where they emerge. In order not to be misunderstood I wish to emphasize that I do not mean omega particles or top quarks when I 1

Literature with general introductions to QM, QFT and AQFT can be found in my suggestions for further reading on page 271 at the end of glossary D.3.

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speak of ‘new categories of formal entities’. I rather think of categories such as rays in a Hilbert space, elements of operator algebras, smeared fields and superselection sectors. It is this general perspective which will be under scrutiny and not so much the rich corpus of the elaborated theory. For this reason the weights in this and the next two chapters will be different from those in most of the standard literature about QFT. The more comprehensive glossary D.3 on Physics (and Mathematics) is partly meant to balance this emphasis by introducing some further standard issues. In chapters 5 and 6 I will argue in some detail why I consider AQFT, an algebraic reformulation of QFT, to have a salient importance for the ontological investigation of QFT. I will not spend much time on those parts of QFT which have primarily been introduced in order to ease calculations, such as Feynman diagrams, the use of functional integrals and various quantization techniques to name just a few. Instead of trying to cover all the potentially important issues I have decided to concentrate on those aspects of QFT and in particular the algebraic formulation (i.e. AQFT) which I consider to have the highest impact on ontological considerations and where I have new ideas about what this impact is. The reason why I hold AQFT in high esteem regarding ontological matters will be one of the main issues in chapter 6. Since I expect a very diverse readership there will sometimes be a certain tension between quite advanced topics and elementary explanations. I cannot see a better way to proceed and therefore I can only ask for the patience of those readers who are already familiar with the physics and mathematics discussed in these passages. In order not to disturb the flow of argumentation too much, I have interspersed various footnotes in this and the next two chapters which are only meant for the reader with no or very little physics and mathematics background (or as a reminder). These footnotes are printed in italics and can be skipped by the more advanced reader in these matters without loss of information.

B.1. THE HILBERT SPACE FORMALISM

B.1 B.1.1

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The Hilbert Space Formalism States and Observables

Much of the formal and interpretive setting which was successfully introduced for non-relativistic standard quantum mechanics is part of the foundations of standard QFT as well. Unfortunately, this applies no less to most of the conceptual problems of QM. In the present work I assume a certain acquaintance with the basics of standard QM as well as with the connected problems. Nevertheless, for the reader’s convenience and in order to introduce some notation I will give a brief summary of the most important definitions, notions and equations as well as of some of the standard interpretations. The emphasis will be on the general quantum setting which has either retained its validity in QFT or which is the immediate basis for modifications. The two basic kinds of formal entities in standard QM are states and observables. All of the following formulae will deal with either or both of these kinds. Since von Neumann’s systematic and mathematically rigorous formulation of QM2 it is common to associate with each given physical system (e. g. one particle) a separable Hilbert space H as the state space, i.e. the set of all possible states of this system. (Complex) vectors ψ, or more accurately rays with ψ = 1 are interpreted asrepresenting   iα possible pure iα states of the given physical system (note that e ψ  = e  ψ = ψ, α ∈ IR). Besides pure states there is another category, namely mixed states, which play, besides their use in quantum statistical mechanics, a very important role in connection with many-particle systems and quantum measurements. The definition of a mixed state will be introduced later. Hermitian (also called self-adjoint) operators3 on a Hilbert space H are taken to represent the observables with respect to a physical system 2

Von Neumann’s pioneering monograph von Neumann (1932) is the basis for all further examinations of the foundations of quantum physics. The definition of separability and of a Hilbert space as well as further connected details can be found in the ‘Physics (and Mathematics) Glossary’. 3 Metaphorically speaking an operator “can be thought of as an animal that eats vectors and spits out other vectors” (p. 251 in Earman (1992)). For a concise definition and connected notions see the glossary. Note that footnotes in italics are meant for readers with only little background knowledge in mathematics and physics. For that reason I appeal to intuition rater than to rigour.

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associated with that Hilbert space. Eigenvalues of a Hermitian operator are interpreted as possible measurement outcomes, i.e. those values the observable can show in an appropriate measurement of the physical system. The standard argument for associating observables with Hermitian operators is that according to a theorem of linear algebra eigenvalues of Hermitian operators are always real which is usually taken to be prerequisite for their interpretation as measurement values. The second important part of the mentioned mathematical theorem states that in the case of a Hermitian operator the eigenvectors belonging to different eigenvalues are orthogonal to each other. This second feature of Hermitian operators has a convenient consequence in the choice of (a complete set of) eigenvectors as basis vectors in the decomposition of a given state since the scalar product of two orthogonal vectors vanishes. As far as the interpretation is concerned states which are represented by orthogonal vectors in Hilbert space are one special case in the probability interpretation (which I will deal with in the next section). There, the vanishing scalar product of two orthogonal states will correspond to a certain vanishing probability. The above-mentioned eigenvalues of an operator are the solutions of its so-called eigenvalue equations. In order to make the described setting more explicit and in order to introduce some necessary notation, consider the eigenvalue equation4 A|ψi  = ai |ψi  (B.1) of a Hermitian operator A on the Hilbert space H which is the starting point for actual calculations.5 For simplicity I assume that A is discrete 4

With the help of an eigenvalue equation one can determine those vectors which are transformed in a particular way: when an operator acts on them the result is simply a multiple of the initial vector. Eigenvalue equations for operators are used in all parts of physics in order to find out salient states of a given system whose characteristics are in some aspects represented by operators. The dynamics of rigid bodies rotating about an axis might be helpful for getting an intuitive idea of the physical significance of eigenvalue equations. In an appropriate basis the eigenvectors of the inertia tensor are the principal axes of inertia and the corresponding eigenvalues are the principal moments of inertia. In the case of a box of matches, e. g., the principal axes of inertia coincide with the three main symmetry axes. 5 The so-called ‘bra-ket’-notation, e. g. of state vectors like |ψi , goes back to Dirac and proved to be very useful. The bracket from which it is derived is the direct product v1 |v2  of two vectors v1 and v2 which gives a scalar. One example for the usefullness of

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and nondegenerate. Physically, the numbers ai (the eigenvalues of A) are interpreted as possible measurement outcomes which can be read off the pointer of a measurement apparatus that can measure the observable A with respect to the examined physical system. To each eigenvalue ai corresponds an eigenvector (or eigenstate) |ψi . The appropriate ontological interpretation of this setting seems to be quite straightforward. A physical system in state |ψi  has the value ai of property A (e. g. spin). And reversely it is quite common to say that |ψi  is the state of the system after a measurement with outcome ai . Nevertheless, it will become clear in section B.2.3 why this gloss is somewhat problematic. To be sure one can say that |ψi  can be interpreted as that state of a system which would with certainty yield the measurement outcome ai if A were measured. The described connection is sometimes called the ‘eigenvector-eigenvalue link’ (or postulate). B.1.2

Probability Interpretation

So far the situation is comparatively unproblematic. Each eigenstate of a certain observable is—in some sense—linked with the corresponding eigenvalue. Consider one such eigenstate, say |ψk , with the corresponding eigenvalue ak . Ontologically, it might be possible under these circumstances to say that the system has the property Pak . Unfortunately, even leaving any interpretive difficulties at this stage aside, this is only a special case. In general, a physical system will not be in an eigenstate of the observable in which one is interested. And in this case it is in no way clear what to say about the value of that observable. According to the statistical interpretation, which is the minimal consensus among physicists, the expression probA(ψ, ai ) = | ψi |ψ |2

(B.2)

can be understood as the probability to get the value ai if a measurement of A is performed on a physical system in state ψ. This piece of interpretation is sufficient for all practical purposes and the basis for the slogan “Shut up and calculate!”. If one is not willing to shut up—or if practithe ‘bra-ket’-notation is that an eigenvector belonging to an eigenvalue, say n, can very conveniently be written as |n. Mathematically, the split-up of a direct product into the left side v1 | and the right side |v2  involves the notion of the dual space of a vector space.

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cal purposes are not one’s concern anyway—things do in fact get quite intricate. Ontologically, nothing is said about the value of observable A in state ψ, i.e. before the measurement. Moreover, even after the measurement it is possible that nothing can be said since, e. g., in the case of a measurement of a photon, the photon can vanish. But even in the more common case that the physical system does survive the measurement a straightforward understanding of the situation is obstructed by the socalled quantum measurement problem (see section B.2.3). Thus it is open at this stage whether any property can be ascribed to the physical system with respect to observable A. For continuous observables, i.e. observables with a continuous spectrum of eigenvalues, the situation is even harder to handle and has in fact plagued quantum theorists. Unfortunately, two of the most important observables, position and momentum, are in fact continuous observables. For the position observable one has the following probability interpretation which goes back to Max Born and yields the standard interpretation of the well-known wave function ψ(x) as well. For the one-dimensional case, which I choose for simplicity, the integral

(B.3) probQ(ψ, X) = dx |ψ(x)|2 X

gives the probability that an appropriate measurement apparatus responds positively, i.e. clicks for instance, if a quantum mechanical particle in the (normalized) state ψ(x) is measured in the interval X ⊂ IR.6 Based on the statistical interpretation it is often advantageous to use the expectation (or average) value (B.4) Aψ = ψ|A|ψ for measurements of an observable A with respect to a large ensemble of 6

In order to link the case of the continuous position observable to the above-mentioned expressions for discrete observables one can give the following catchy but problematic explanation. Writing Q|x = x|x as the eigenvalue equation for the position observable Q, the vectors |x would be the eigenstates of the position observable Q with the corresponding eigenvalues x. Comparing with the above-mentioned discrete case, x|ψ = ψ(x) is in close analogy to ψi |ψ in expression (B.2) and thus gives the stated probability when its modulus, i.e. |ψ(x)|, is squared. One reason why this gloss is problematic is that the vectors |x understood as ‘δ-functions’ are not elements of any Hilbert space which is assumed as the state space. Moreover, they are not even well-defined functions at all. In the physics glossary I deal with the notion of ‘δ-functions’ more closely.

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systems which are identically prepared in the pure state ψ. So far I have only dealt with pure states, namely ψ and the ψi . The difference between these states is that the ψi are eigenstates with respect to the observable A while ψ is an arbitrary pure state—possibly one of those eigenstates ψi of A but in general some other pure state. Apart from pure states there is, as I have already mentioned above, a more general concept of states, namely that of mixed or statistical states. Therefore, formulae for pure states do arise as special cases of any formulae dealing with mixed states. Mathematically, a mixed state can be represented by a weighted sum of pure states |ψi 
wi P|ψi  (B.5) ρ= i

with the probability weights wi and the projection operator P|ψi  ≡ |ψi ψi |.

(B.6)

One can understand such a mixed or statistical state as a situation where a physical system can be found in one of the states |ψi  with the corresponding probability wi . A mixed state must not be confused with a superposition which is a pure state. Since the so-called density operator ρ comprises pure states as well—if all but one wi are zero—one can use a more generally valid alternative formula for expectation values, namely Aρ = tr(ρA)

(B.7)

where ‘tr’ stands for the trace of an operator which is defined as the sum of its diagonal elements. If ρ represents a pure state then equation (B.7) reduces to equation (B.4). Mixed states play a central role in what is well-known as the entanglement of states, a notion that was introduced by Schr¨odinger in connection with his famous cat scenario. If a compound system is in a pure state then the subsystems are generally in mixed states. From an interpretive point of view the most important example is the compound of a physical system and the measuring apparatus if a measurement interaction has occurred. If a physical system is in a pure state but not in an eigenstate of the observable to be measured, i.e. in the standard case, the act of measurement will inevitably bring the system into a mixed state. And in this

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state the question arises how to understand the probabilities in formula (B.5). The two main types are objective and subjective interpretations. In the last case, probabilities are taken to express our ignorance of the real situation where no probabilities are needed. In the case of objective interpretations the probabilities are somehow ascribed to something real in the measured object. A very prominent proposal for how to understand probabilities objectively is the so-called propensity view which says that probabilities are real irreducible tendencies to display a certain definite properties upon measurement. Obviously the stance one adopts towards measurements play an important role for the ensuing ontological picture.7 Again, I postpone a further discussion of possible answers to the separate section about the measurement problem in section B.2.3. The ascription of properties to physical systems is one the central ontological problems which is a challenge for any interpretation of quantum physics. So far I dealt with this issue only with respect to one kind of property, or one observable respectively.8 Connected but even more troublesome problems arise in the case of two observables if they have the genuine quantum feature of being incompatible. This situation was particularly stressed by Bohr and can be made more precise by using the very important definition of a commutator [A, B] ≡ AB − BA.

(B.8)

of two observables A and B. The physical interpretation is the following. If the commutator [A, B] vanishes, then the order in which the two observables are measured is irrelevant and A and B are said to be compatible observables. In general, however, the commutator of two observables does not vanish and this fact is one of the most salient non-classical features 7

Still another distinction that is sometimes made and which goes back to Heisenberg is the one between a mixed state (Gemisch) and a mixture of states (Gemenge) after a measurement interaction. In contrast to a mixed state it is unproblematic for a mixture of states, which can be written as {(wi , P|ψi
) : i = 1, 2, ...}, to give a classical ignorance interpretation—provided the ψi are taken to be orthogonal. The mixed state which is determined by this mixture of states is the one given in (B.5) again. See section 2.5 in Busch et al. (1991) for details. 8 The relation of the philosophical as well as the everyday notion of properties on the one side and the physical notion of observables on the other side is a very controversial matter. In particular, the attitude one has depends crucially on the position regarding the measurement problem. I will present my own point of view in chapter 13.

B.1. THE HILBERT SPACE FORMALISM

211

of quantum physics. For this reason quantum operators have been called q-numbers in contrast to classical numbers (c-numbers) which commute always. The most important case of non-commuting observables is expressed by the canonical commutation relations, i.e. the commutation relations for the canonical variables, e.g. position and momentum coordinates, which are given by [ˆ pi , qˆj ] = −i
δij (B.9) [ˆ pi , pˆj ] = [ˆ qi , qˆj ] = 0 where i, j = x, y, z and δij is the so-called Kronecker delta which is 1 if i = j and 0 otherwise. The interesting case, which displays the quantum character, is the one where the commutator [ˆ pi , qˆj ] does not vanish as, e.g., for the pair pˆx and qˆx , so that pˆx qˆx = qˆx pˆx . Thus, in contrast to the classical case, the order of terms in the product does make a difference which has the above-mentioned interpretation. Mathematically speaking, pˆi and qˆj are operators (see glossary D.3) which is sometimes indicated by hats on top of these variables. It is not possible to ascribe sharp values for noncommuting observables to a system simultaneously. In the case of position and momentum Heisenberg’s uncertainty relations specify to which degree simultaneous measurements are possible. 9 Quantum systems with more than one ‘particle’, so-called many-particle systems, require a non-trivial extension of quantum mechanics.10 Moreover, the analysis of these systems reveals a number of strikingly nonclassical phenomena. The mathematical description of many-particle systems uses the notion of the tensor product which can be applied to state spaces as well as to states in these spaces.11 For two particles whose state spaces are the Hilbert spaces H1 and H2 the tensor product H1 ⊗ H2 . 9

(B.10)

Probably the most extensive research on this issue has been done by Paul Busch beginning with his publication Busch (1982) and many others afterwards. In section 13.3.5 I will present an argument which rests on the notion of unsharp properties in connection of my proposal for a trope-ontological interpretation of QFT. 10 As it is common practice I talk about ‘particles’ although it should be clear that the legitimacy of this talk is part of the very question of my study. 11 Mathematical properties tensor products can be found in many textbooks on QM. For a relatively elementary but mathematically concise introduction to the notion of the tensor product see, e.g., section 1.6 in Redhead (1987) or section 16.5 in Hannabus (1997). A more advanced discussion can be found in section 1.2 of Busch et al. (1991).

APPENDIX B. QUANTUM MECHANICS

212

is the state space of the two-particle system. If {φi } is a basis of H1 and {ψi } a basis of H2 then one possible two-particle state is φj ⊗ψk . In general, however, a two-particle state will be a linear superposition of such states. Many-particle systems are one bridge from QM to QFT since fields arise as a limiting case of many-particle systems when the particle number goes to infinity. B.1.3

Dynamics

A central part of any comprehensive physical theory is concerned with the question how the physical systems under study evolve in time in a given setting. The quantum mechanical equation of motion is the famous Schr¨odinger equation ∂ (B.11) i
ψ(r, t) = Hψ(r, t) ∂t where for instance   1 2
2 2 ˆ + V = Ekin + Epot ∇ +V = p H=− (B.12) 2m 2m is the Hamiltonian operator, short Hamiltonian, of a one-particle system without spin.12 The Hamiltonian is the generator of time-translations which means the following. Due to equation (B.11) the time-evolution of states is described by the transformation |ψ → |ψ
 = U |ψ

(B.13)

i

where U is the unitary13 operator U = e−
Ht . The dynamics which is governed by the Schr¨odinger equation is deterministic and in itself unproblematic. Problems arise only with the question how the emergence of measurement results, which is a dynamical physical process too, is related to the only dynamical law one has in QM, namely the one expressed by the Schr¨odinger equation. In section B.2.3 I will explicate why the measurement process seems to be governed by a different kind of dynamics. Since the present chapter is intended as a preparation for the later investigation of QFT a remark is needed concerning different ways how the dynamics can be represented in the formalism. In the usual representation 12 13

The Laplacian operator ∇2 is defined in the physics glossary. An operator U is called ‘unitary’ if U U † = U † U = 1 or, equivalently, if U −1 = U † .

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213

of QM, the so-called Schr¨odinger picture, states evolve with time while observables are time-independent. One alternative representation is the Heisenberg picture in which states are time-independent while the observables evolve in time according to the transformation |O → |O
 = U OU † .

(B.14)

A useful third representation is the interaction picture (see section 4.5). Although the choice of the picture is only a matter of representation and not of content, it does play a role, of course, which subject matter is to be described. In the context of QM the Schr¨odinger picture is most useful and widespread, in the case of QFT (without interaction) it is the Heisenberg picture and for the description of interaction processes in QFT the interaction picture is custom-made, so to say.

B.2

Problems for an Ontology of QM

Up to this point I have introduced the formal setting of QM and the standard interpretations, i.e. how the mathematical formula can be linked with physical phenomena and entities. I have sometimes pointed out that the interpretation of certain parts of the formalism is controversial. In this section I will concentrate on interpretative problems which are central for an ontological analysis of QM. The three problems I will deal with have a direct bearing on the question which kinds of entities QM is describing. Although QM is not the subject matter of my further study this is a very important step because I have selected those problems which maintain their relevance when looking at QFT. The legacy of QM for ontological considerations about QFT is mostly a negative one. Most of the notorious obstacles for an ontological understanding of QM are equally troublesome in QFT. In section 6.1 I will reflect upon the question whether it is appropriate to start ontological investigations about QFT before corresponding matters with respect to QM are settled. For now, I put that concern aside. Problems concerning the individuation and reidentifiability of particles, are the most fundamental ones for the present context, especially when considered together. They are concerned with the distinguishability of particles in its synchronous and in its diachronous aspect respectively. Both aspects have notoriously caused trouble for the idea of individual traceable

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particles. These problems already arise in QM and lose nothing of their importance in QFT. B.2.1

The Problem of Individuation

Individuality is a central so-called category feature in various conceptions of things like tables or particles.14 In the context of QM a problem of individuation occurs for systems with many quantum particles. The starting point for the statistics of such systems is the fact that the Maxwell-Boltzmann statistics (i.e. the energy distribution law) which is valid in classical statistical mechanics leads to false predictions for systems of ‘identical’ quantum mechanical particles. In QM, particles are called ‘identical’ when they have all their permanent properties (e.g. rest mass, charge, spin) in common. A set of permanent properties fixes a class of particles (e.g. electrons) rather than a particular particle.15 It turned out that one gets the experimentally correct statistics when the possible micro states which lead to the same macro state are counted differently for systems of identical particles: Micro states which differ merely by the ‘exchange of two particles’ must be counted as just one state. This fact is referred to as ‘non-occurrence of degeneracy of exchange’16 or ‘indistinguishability of identical particles.’ What are the consequences of the indistinguishability of identical particles for our main issue, the basic entities of QFT? The emerging problems become clearer after a short look at the symmetrisation postulate which follows from the indistinguishability of identical particles, given some additional assumptions.17 Depending on the spin of the respective particles the wavefunction of a many-particle system has to be symmetric or antisymmetric under the ‘exchange of two particles,’ or, to be more careful, under the exchange of two particle labels.18 ‘Non-symmetric’ wavefunc14

See section 7.3 for a detailed treatment. Cf. chapter viii in Mittelstaedt (1986). 16 In German: “Nichtauftreten von Austauschentartung”. 17 The logical connection of the indistinguishability of identical particles and the symmetrisation postulate is discussed in detail in St¨ockler (1988), p. 12. 18 Pauli’s well-known ‘exclusion principle’ is thereby fullfilled automatically: The wavefunction of two fermions in the same single state, i.e. with the same quantum numbers, vanishes as can be seen in equation (B.15) on p. 215. In other words, there is no compound state where two fermions have the same quantum numbers. 15

B.2. PROBLEMS FOR AN ONTOLOGY OF QM

215

tions (which are neither symmetric nor antisymmetric) are excluded. The symmetrisation postulate is only necessary19 in the Schr¨odinger many-particle formalism which, despite the problems with indistinguishability, uses labels that were originally meant to refer to individual particles of the overall system. Because of the symmetrisation postulate, however, not all wavefunctions of the ‘overall’ or ‘compound’ system, which could be constructed by the standard way of forming the tensor product of one-particle states, are allowed any more. Within the Schr¨odinger manyparticle formalism non-symmetric wavefunctions can be formed but get excluded. The formalism is therefore richer than the experimental reality which it is designed to encompass.20 This fact could be taken as an indication that the theory is built upon inadmissible assumptions which lead to a piece of structure that has to be excluded artificially. A closer look at a symmetrisized wavefunction of a compound system hints at a reason for these difficulties: An anti-symmetric wavefunction of a system of fermions (e.g. electrons) is a superposition of product wavefunctions, i.e. a sum of tensor products of one-particle states. A sufficient example is the wavefunction of a system of two identical fermions,  1  (B.15) Ψ(x1 , x2 ) = √ ψα (x1 )ψβ (x2 ) − ψβ (x1 )ψα (x2 ) , 2 where ψα (x1 ) and ψβ (x2 ) are energy eigenfunctions of one-particle Hamiltonians and α and β represent sets of quantum numbers characterizing one-particle states. Since x1 and x2 are variables of the “single particles” 1 and 2 it is natural to ask what the states of these “single particles” are. It turns out that it is impossible to give a satisfactory answer to this question if one holds on to the conception of individual particles. Each “single particle” is in the same state as a part of the compound system even though in the wavefunction of the compound system different one-particle wavefunctions are used. Talking of a “particle exchange” has obviously become problematic. The usage of labels for individual particles in the usual sense might lead one astray. One thus has an ontological problem since on the one hand one can successfully use labels which seem to number something 19

The symmetrisation postulate is true but trivial in the so-called ‘occupation number representation’ which I am going to discuss later. 20 Especially M. Redhead worked on the so-called “surplus structure” of scientific theories: Redhead (1975), Redhead (1980).

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but on the other hand one is not dealing with particles in the usual sense any more for which the labels were introduced originally.21 In section 13.2 I will elaborate on this issue in more detail. B.2.2

The Problem of Reidentifiability

The problem of individuation exerts its full force only in connection with the second one, the problem of reidentifiability: If in certain classes of microparticles we cannot distinguish individual particles by permanent properties why do we not simply look where they are and keep track of their location while time elapses? The following argumentation shows that even this way is obstructed. On first sight the claim that we cannot follow a particle in space-time is astonishing since we seem to have exactly these looked-for tracks of particles in cloud-chamber photographs, showing, for example, charged particles on curved trajectories. A closer look reveals, however, that these ‘particle tracks’ have very little in common with sharp trajectories of classical physics. On the micro level we have smeared tube-like objects. Each of these tube-like trajectories is the result of a vast amount of unsharp quantum mechanical position measurements22 in close succession. The degree of unsharpness or “smearedness” is even far bigger than the theoretical minimum which is given by Heisenberg’s uncertainty relation for position and momentum. With a particle track being the result of many successive measurements, the immediate suggestion is that one and the same particle gave rise to the track because numerous measurements were performed on this particle. Unfortunately we have difficulties with this assumption in QM: Even if the time interval between two quantum mechanical position measurements that contribute to one particle track is extremely small we cannot be sure to have measured the same particle. The reason for this is the fact that even in the case of a sharp position measurement, after an arbitrarily small time interval, there is a non-vanishing probability that the measured 21

Problems with particle labels are discussed extensively in Teller’s book Teller (1995) on some philosophical problems of free QFT, with main emphasis on QED. 22 In our context a measurement is every interaction of a quantum object and a macroscopic system with a definite result, e.g. a dot on a photographic plate.

B.2. PROBLEMS FOR AN ONTOLOGY OF QM

217

object can be detected infinitely far away from the first point of detection.23 Between the results of two quantum mechanical measurements of a continuous observable - like position - there is in principle no deterministic connection.24 The problems discussed above partly reflect a general difference between classical and quantum mechanics. In classical mechanics, when position and momentum of a free particle are given, both are fixed for all later times; we have a so-called ‘path law.’ In quantum mechanics, however, we do not have a deterministic law of this kind. The first reason is that a quantum object cannot have a sharp position and a sharp momentum at the same time. The second reason is more fundamental: There is, in general, no deterministic connection between single (or groups of) measurement outcomes. All we have in quantum mechanics is a law for the evolution of the statistics of measurement outcomes: The statistics is given by the ‘state function’ and the deterministic law according to which it evolves in time is the Schr¨odinger equation. One knows, therefore, how the statistics of possible measurements are connected, but one does not know in general how single measurements are connected. This means that a particle track cannot be interpreted as a succession of connected measurements of one object. The possibility to identify an object by tracing it through spacetime is excluded. Problems concerning the individuation and reidentifiability of quantum mechanical particles cause severe difficulties for the conception of individual quantum objects. Because of the problem of individuation it is impossible to distinguish individual quantum objects which are specified by the same permanent properties. The immediate proposal for a solution to this obstacle is ruled out by the problem of reidentifiability: Quantum objects cannot be identified as individual entities by localizing and tracing them in space-time. 23

See for example Hegerfeldt and Ruijsenaars (1980). The situation is different with discrete observables, i.e. where the measurement outcomes are discrete numbers. In this case ‘repeatable measurements’ are possible if we are dealing with ‘state preparation measurements.’ After the measurement, the measured object can be found in that eigenstate which corresponds to the measured eigenvalue. A repeated measurement leads with certainty to the same measurement outcome (this is the defining property of a ‘repeatable measurement’). For a proof of the impossibility of repeatable measurements of continuous observables see the classical paper Ozawa (1984). 24

218 B.2.3

APPENDIX B. QUANTUM MECHANICS The Measurement Problem

There are two fundamentally different kinds of attempts to solve the measurement problem—apart from dismissing it as a purely metaphysical problem with no empirical consequences. The first kind of attempt are collapse theories (e.g. GRW), the second kind are no-collapse theories (e.g. many worlds, Bohm).25 Collapse theories, which are more orthodox than nocollapse theories, try to find, e.g., a dynamics other than the unitary time evolution given by the Schr¨odinger equation in order to account for the apparent collapse of the wave function during a quantum measurement. In the GRW (Ghirardi-Rimini-Weber) approach, for instance, a non-linear time evolution is sought which could describe the transition to definite measurement outcomes in the general case where the quantum object is not in an eigenstate of the measured observable. No-collapse theorists, on the other hand, argue that no collapse is needed in order to make sense of the supposedly mysterious measurement process. In Everettian many worlds theories, for instance, quantum measuremens are explained by a branching of the universe into “many worlds”. Schr¨odinger’s cat never comes into the paradoxical situation where it is undecided whether she is dead or alive, rather the dead version of her can be found in one branch of the universe and the alive version in another branch. We, as observers, never observe this as a paradoxical situation since one version of us too goes into one of these branches and another version of us goes into the other branch and these two versions of us have no access whatsoever to each other.26 One argument that is often put forward against an Everettian approach is that it seems to be impossible to make sense of the probabilities for different measurement outcomes.27 Assume, e.g., that there is a probability of 1/3 for the cat to end up dead and a probability of 2/3 for it to stay alive. 25

A recent overview using the distinction between collapse and no-collapse theories can be found in Dickson (1998). 26 See, e.g., Wallace (2001) for a more detailed account of the Everettian fate of Schr¨odinger’s cat. 27 Barrett (1999) is a very instructive and comprehensive exposition of problems faced by those approaches which follow Everett’s relative-state formulation of QM, i.e. the many worlds interpretation for instance. An impressively short and persuasive analysis of the stalemate reached in the interpretation of QM can be found in Peres and Zurek (1982) where only the most general features of different approaches are used.

B.2. PROBLEMS FOR AN ONTOLOGY OF QM

219

What do these probabilities mean if both cases actually become realized in different branches? It does not seem to make much sense to say that there is a probability of 2/3 to end up in one branch when another version of you ends up in another branch at the same time. No matter what the (non-zero) probabilities are you can be sure that each of the possibilities will be realized. The Everettian can respond in two ways. One is to point at the fact that the very concept of probability is not well understood so that the Everettian is no worse off than the proponents of any other theory in that respect. Another, more compelling, reaction is to make sense of probabilities for branching universes on the basis of decision theory.28

28

See Deutsch (1999) and Wallace (2002).

Appendix C Advanced Foundational Topics in QFT C.1

Gauge Invariance

Some theories are distinguished by the mathematical property of gauge invariance which means that transformations, so-called gauge transformations, of certain terms do not change the observable quantities. The requirement of gauge invariance has the mathematical advantage that it provides an elegant way to introduce terms for interacting fields. Moreover, requiring gauge invariance plays an important role for the selection of theories. The prime example of an intrinsically gauge invariant theory is the theory of the electromagnetic field. It is well-known from the classical theory that Maxwell’s equations can be stated in terms of the vector potential A and the scalar potential φ or in terms of the 4-vector potential Aµ = (φ, A). The link to the electric field E(x, t) and the magnetic field B(x, t) is given by ∂A − ∇φ (C.1) B = ∇ × A, E = − ∂t or covariantly (C.2) F µν = ∂ µ Aν − ∂ ν Aµ where F µν is the electromagnetic field tensor. The important point in the present context is that given the identification (C.1), or (C.2), there remains a certain flexibility or freedom in the choice of A and φ, or Aµ . In order to see that, consider the so-called gauge transformations ∂χ (C.3) A → A − ∇χ, φ → φ + ∂t or covariantly (C.4) Aµ → Aµ + ∂ µ χ 221

222 APPENDIX C. ADVANCED FOUNDATIONAL TOPICS IN QFT where χ is a scalar function (of space and time or of space-time) which can be chosen arbitrarily. Inserting the transformed potential(s) into equation(s) (C.1), or (C.2), one can see that the electric field E and the magnetic field B, or covariantly the electromagnetic field tensor F µν , are not effected by a gauge transformation of the potential(s). Since only the electric field E and the magnetic field B, and quantities constructed from them, are observable, whereas the vector potential itself is not, nothing physical seems to be changed by a gauge transformation because it leaves E and B unaltered. Note that gauge invariance is a kind of symmetry that does not come about by space-time transformations. In order to link the notion of gauge invariance to the Lagrangian formulation of QFT one needs a more general form of gauge transformations which operates on the field operator φ and which is supplied by φ → e−iΛ φ, φ → eiΛ φ

(C.5)

where Λ is an arbitrary real constant. Equations (C.5) describe a global gauge transformation in contrast to a local gauge transformation φ(x) → e−iα(x) φ(x)

(C.6)

which can vary with x. The requirement of invariance under a local gauge transformation is essential for finding the equations describing fundamental interaction. Take for example the Lagrangian for a free electron. The requirement that the Lagrangian should be locally invariant under the same type of transformation can only be fulfilled by introducing additional terms. The form of these terms is determined by the symmetry requirement, which results in the introduction of the electromagnetic field. In a sense, the electromagnetic field is a consequence of the local symmetry of the Lagrangian for the electron. This procedure can be generalized to more complex transformations (for example referring to mixing the components of field operators) and new interactions. By requiring local gauge invariance additional fields can be introduced. These additional fields describe the interaction between the original fields. The gauge principle provides a general schema for introducing interaction by constructing gauge field theories. To this end one starts with a Lagrangian for a matter field and derives the interaction by introducing exactly those fields that make the Lagrangian invariant under

C.2. EFFECTIVE FIELD THEORIES AND RENORMALIZATION 223 a relevant local gauge transformation. It seems that all fundamental forces can be described by such local gauge field theories. Gauge symmetry plays a crucial role in determining the dynamics of the theory since the nature of gauge transformation determines the possible interaction. The structure of these transformations are characterized by special mathematical groups: U(1) for QED, SU(2) ⊗ U(1) for electroweak interaction, SU(3) for strong interaction. The relations between these groups are exploited in programs for the unification of the fundamental types of interaction. There is also a strong analogy to general relativity where a local gauge symmetry is associated with the gravitational field. From a more technical point of view gauge symmetries are important tools in proofs of renormalizability. The upshot is that the fulfillment of gauge invariance has an importance for the selection of theories which makes gauge invariance a player in the same league as Lorentz invariance. Since gauge invariance plays a pivotal role in the discovery of quantum field theories it is a paradigm case for how a rich mathematical structure can help in the construction of theories. General introductions to gauge theories can be found in Cao (1997) and Schweber (1994). Auyang emphasizes the general conceptual significance of invariance principles in her book Auyang (1995) while Redhead (2002a) as well as Martin (2002) focus specifically on gauge symmetries. Lyre (2004b) is a study of the significance of gauge theories for the debate on structural realism, with a related paper Lyre (2004a) in English. The ontological significance of gauge potentials is discussed in particular with respect to the Aharanov-Bohm effect, e. g. in Healey (2001).

C.2

Effective Field Theories and Renormalization

In the 1970s a program emerged in which the theories of the standard model of elementary particle physics are considered as effective field theories (EFTs) which have a common quantum field theoretical framework. EFTs describe relevant phenomena only in a certain domain since the Lagrangian contains only those terms that describe particles which are relevant for the respective range of energy. EFTs are inherently approximative and change with the range of energy considered. EFTs are only applicable on a certain energy scale, i. e. they only describe phenomena in a certain

224 APPENDIX C. ADVANCED FOUNDATIONAL TOPICS IN QFT range of energy. Influences from higher energy processes contribute to average values but they cannot be described in detail. This procedure has no severe consequences since the details of low-energy theories are largely decoupled from higher energy processes. Both domains are only connected by altered coupling constants and the renormalization group describes how the coupling constants depend on the energy. The main idea of EFTs is that theories, i. e. in particular the Langrangians, depend on the energy of the phenomena which are analysed. The physics changes by switching to a different energy scale, e. g. new particles can be created if a certain energy threshold is exceeded. The dependence of theories from the energy scale distinguishes QFT from, e. g., Newton’s theory of gravitation where the same law applies to an apple as well as to the moon. Nevertheless, laws from different energy scales are not completely independent from each other. A central aspect of considerations about this dependence are the consequences of higher energy processes on the low-energy scale. On this background a new attitude towards renormalization developed in the 1970s which revitalizes earlier ideas that divergences result from neglecting unknown processes of higher energies. Low-energy behaviour is thus affected by higher energy processes. Since higher energies correspond to smaller distances this dependence is to be expected from an atomistic point of view. According to the reductionistic program the dynamics of constituents on the microlevel should determine processes on the macrolevel, i. e. here the low-energy processes. However, as, for instance hydrodynamics shows, in practice theories from different levels are not quite as closely connected because a law which is applicable on the macrolevel can be largely independent from microlevel details. For this reason analogies with statistical mechanics play an important role in the discussion about EFTs. The basic idea of this new story about renormalization is that the influences of higher energy processes are localisable in a few structural properties which can be captured by an adjustment of parameters. “In this picture, the presence of infinities in quantum field theory is neither a disaster, nor an asset. It is simply a reminder of a practical limitation—we do not know what happens at distances much smaller than those we can look at directly.” (Georgi (1989), p. 456) This new attitude supports the view that renormalization is the appropriate answer to the

C.3. STRING THEORY

225

change of fundamental interactions when the QFT is applied to processes on different energy scales. The price one has to pay is that EFTs are only valid in a limited domain and should be considered as approximations to better theories on higher energy scales. This prompts the important question whether there is a last fundamental theory in this tower of EFTs which supersede each other with rising energies. Some people conjecture that this deeper theory could be a string theory, i. e. a theory which is not a field theory any more. Or should one ultimately expect from physics theories that they are only valid as approximations and in a limited domain?

C.3

String Theory

Up to now string theory is the most promising candidate for bridging the gap between QFT and general relativity theory, thus supplying a unified theory of all natural forces, including gravitation.1 The basic idea of string theory is not to take particles as fundamental objects but strings which are very small but extended in one dimension. This assumption has the pivotal consequence that strings interact on an extended distance and not at a point. This difference between string theory and standard QFT is essential because it is the reason why string theory also encompasses the gravitational force which cannot be treated in the framework of QFT. Gravitation is so hard to be reconciled with QFT because the typical length scale of the gravitational force is very small, namely at Planck scale, so that the quantum field theoretical assumption of point-like interaction leads to untractable infinities. To put in another way, gravitation becomes significant (in particular in comparison to strong interaction) exactly where QFT is most severely endangered by infinite quantities. The extended interaction of strings brings it about that such infinities can be avoided. In contrast to the entities in standard quantum physics strings are not characterized by quantum numbers but only by their geometrical and dynamical properties. Nevertheless, “macroscopically” strings look like quantum particles with quantum numbers. A basic geometrical distinction is the one be1

Two of the standard introductory monographs to string theory are Polchinski (2000) and Kaku (1999). A very successful popular introduction is Greene (1999). An interactive website with a nice elementary introduction is ’Stringtheory.com’.

226 APPENDIX C. ADVANCED FOUNDATIONAL TOPICS IN QFT tween open strings, i. e. strings with two ends, and closed strings which are like bracelets. The central dynamical property of strings is their mode of excitation, i. e. how they vibrate. Reservations about string theory are mostly due to the lack of testability since it seems that there are no empirical consequences which could be tested by the methods which are, at least up to now, available to us. The reason for this “problem” is that the length scale of strings is in the average the same as the one of quantum gravity, namely the Planck length of approximately 10−33 centimeters which lies far beyond the accessibility of feasible particle experiments. But there are also other peculiar features of string theory which might be hard to swallow. One of them is the fact that string theory implies that space-time has 10, 11 or even 26 dimensions. In order to explain the appearance of only four space-time dimensions string theory assumes that the other dimensions are somehow folded away or “compactified” so that they are no longer visible. An intuitive idea can be gained by thinking of a macaroni which is a tube, i. e. a two-dimensional piece of pasta rolled together, but which looks from the distance like a one-dimensional string. Despite of the problems of string theory, physicists do not abandon this project, partly because there seem to be no better candidates for a reconciliation of quantum physics and general relativity theory with the possible exception of the so-called “loop quantum gravity”. Correspondingly, string theory has also received some attention within the philosophy of physics community in recent years.2

2

One philosophical investigation of string theory is Weingard (2001) in Callender and Huggett (2001), an anthology with further related articles. Another more recent study is Dawid (2007). Dawid argues that string theory has significant consequences for the philosophical debate about realism, namely that it speaks against the plausibility of antirealistic positions.

Appendix D Assumptions and Results of AQFT D.1

Assumptions of AQFT

The basic elements in the algebraic formulation of QFT are C ∗-algebras which are associated with bounded openregions O in Minkowski spacetime M . One thus has a mapping O → A(O) from spacetime regions to algebras.1 The basic structure upon which the assumptions or conditions of AQFT are imposed is the following:2 • Local observables: self-adjoint elements in local (non-commutative) von Neumann-algebras A(Oi ), which are subsets of B(H), i.e. the set of all boundedoperators acting on Hilbertspace H. • Physical states: Positive, linear, normalized functionals ω on A which map elements A of local algebras to real numbers, i. e. ω : A → ω(A), where ω(A) is the expectation value of A in state ω. Relativistic Axioms (i) Locality: 1

For the following it will be useful to refer to different spacetime regions as O1 , O2 ..., i.e. by means of an index. 2 Apart from the works cited in chapter 5 see Redhead and Wagner (1998) and Buchholz (1994) for brief but helpful expositions of the assumptions in AQFT.

227

228

APPENDIX D. ASSUMPTIONS AND RESULTS OF AQFT If O1 ⊂ O2
, then A(O1) ⊂ [A(O2)]
, where O
denotes the set of all points which are spacelike separated from all points in O and [A(O)]
, which is defined as the set of all operators (in A) which commute with all operators in A(O). The assumption of locality (or ‘spacelike commutativity’ or ‘Einstein causality’ or ‘microcausality’) requires that observations in spacetime regions which are causally separated (in the sense of special relativity theory) must be statistically independent.

(ii) Covariance: The Poincar´e group P+↑ is represented by automorphisms on the net of local algebras, i.e. αg [A(O)] = A[g(O)]. The map g → αg (A) is assumed to be strongly continuous for any A ∈ A, where A is the global algebra generated by all local algebras A(O). Covariance: A transformation (active or passive) of all spacetimecoordinates (according to the formula of special relativity theory) must not change the physics. Specifically for local algebras: It makes no difference whether we take a local algebra w.r.t. a spacetime region which is translated in Minkowski space or whether we take the local algebra w.r.t. the original spacetime region and transform it by the corresponding automorphism on the net of local algebras. (iii) Diamond: A(O) = A[D(O)], D(O): ‘causal shadow’ of O. General Physical Assumptions The next set of assumptions are ‘general physical assumptions’ in the sense that they can be motivated and understood by themselves on purely physical grounds (but not related to relativity theory). We will see in the next paragraph why this is not the case with all of our assumptions. (iv) Isotony:

D.1. ASSUMPTIONS OF AQFT

229

If O1 ⊆ O2 , then A(O1 ) ⊆ A(O2 ). Isotony: The set of local observables grows with the considered spacetime region, i.e. the more room we have (in spacetime) the more measurements are possible. This gives rise to the so-called net structure of local algebras. The net structure of local algebras, in turn, contains the physical information which distinguishes one quantum field theory from another (as the map from spacetime regions to local algebras depends on the considered quantum field theory). As a consequence of the isotony condition the set-theoretic union of all local algebras has *-structure.3 (v) Spectrum: The spectrum of the translation subgroup of P is contained in the closed forward lightcone. A more familiar and physically more intuitive formulation is to postulate that the spectra of the energy operator H = pˆ0 (Hamiltonian) and the mass operator m = (ˆ p2)1/2 are nonnegative.4 The spectrum condition requires the energy of a physical system to be bounded from below, i.e. that there must be a lowest possible energy, in order to exclude a perpetuum mobile. (vi) Vacuum: ∃ unique state Ω, invariant under all Poincar´e transformations, i.e. U (g)Ω = Ω ∀g ∈ P. It is not necessary in all approaches to postulate the existence of a vacuum state. The most famous example where this postulate appears 3

See Buchholz (2000), p. 5, and Haag and Kastler (1964), p. 849, for a further discussion of the isotony condition. 4 From a mathematical point of view the spectrum condition makes it possible to use various theorems from the theory of analytic functions. A famous example of this connection, although not within AQFT, are Hegerfeldt’s articles on the incompatibility of causality and particle localization, e. g. Hegerfeldt (1974), Hegerfeldt and Ruijsenaars (1980) Hegerfeldt (1985) and Hegerfeldt (1998). Hegerfeldt’s main assumption is merely the positivity of the energy. Starting with this assumption Hegerfeldt derives his results, from a mathematical point of view, primarily by the use of the theory of analytic functions.

230

APPENDIX D. ASSUMPTIONS AND RESULTS OF AQFT is the Wightman theory. In other approaches certain stability conditions are sufficient. The best way to handle the situation seems to be a question of ongoing research.5

Technical Assumptions The last set of assumptions can be called ‘technical assumptions’ since they are closely connected to ‘technical’ (i.e. relating to the mathematical formalism) requirements which turn out to be necessary but which cannot be given a satisfactory justification in physical terms. These assumptions are physical assumptions only in the sense that are needed to get a formalism which is physically meaningful. The separate significance of these assumptions apart from features of the formalism as a whole is not yet fully understood. Hence it is possible that there will be some changes in the future to understood and justify the foundations of AQFT better. Weak additivity is essentially the assumption that the spacetime continuum is homogeneous, so that no subquantal phenomena like minimal length are present. The irreducibility condition is a global condition requiring that the global algebra R is irreducible which means that it is a factor of type I∞ . Physically this condition states that the considered system has no superselection rules, i.e. the system can be described within one single coherent superselection sector.6 ‘Without Loss of Generality’-Assumptions In order to make the derivation of theorems easier one often imposes further conditions on the set of bounded spacetime regions. Such conditions are meant to render the study of the inherent structure of AQFT more lucid without diminishing the generality of the derived results. Obviously the choice of spacetime regions to be studied should be such that all possible spacetime regions can be covered or approximated and that this choice determines the whole net of local algebras. A common choice is to consider only the so-called double cones which are a subset of the diamonds. 5

Private communication with D. Buchholz. The existence of invariant subspaces of the global algebra is the very essence of superselection. Superselection rules are due to certain conserved physical quantities, like total electric charge. The eigenspaces of the corresponding operators are called superselection sectors. 6

D.2. REPRESENTATIONS AND STATES

D.2

231

Representations and States

In this appendix I will introduce the crucial notions of representations, states and superselection sectors.7 Let A be a C ∗ -algebra, H a Hilbert space, B(H) the set of all bounded operators acting on Hilbertspace H and π a map π : A → B(H). The pair (H, π) is called a representation of A if the map π is a  -homomorphism8. The basic idea is very similar to the corresponding issue of representations of a group. In both cases something is called a representation if it preserves the structure of the group/algebra which it represents in a different space. Preserving the structure refers in particular to the composition of actions in the respective sets. The Hilbert space H is called the representation space. It is common practice not to mention H and to call π alone a representation. A representation of a C ∗-algebra is thus a map from this algebra to another set which maintains the algebraic structure. Note that B(H), the set of all bounded operators on H, is itself a C ∗-algebra. For this reason it is sometimes distinguished between abstract C ∗ -algebras and those C ∗ -algebras which are concretely realized by bounded operators on a Hilbert space. A physically relevant example for a representation is generated by the assignment of a bounded self-adjoint operator to each (possibly unbounded) operator on the Hilbert space L2 of square-integrable functions. A representation is called faithful if π is isometric, i. e. if and only if π(A) = 0 entails A = 0 where A ∈ A.9 The following definition is of particular importance for ontological ques7

The more general mathematical setting is presented in the authoritative introduction Bratteli and Robinson (1979). A very nice and refreshingly irreverent introduction to various notions and issues that are dealt with in this section can be found in Ruetsche (2003). Some parts of this section are inspired by her compact and lucid way to present a very technical subject matter. 8 The map π is called a  -homomorphism if it holds that π(cA + dB) = cπ(A) + dπ(B), π(AB) = π(A)π(B) and π(A ) = π(A) where A, B ∈ A and c, d ∈ C, the set of complex numbers. 9 An equivalent but more technical definition of faithfulness involves the notion of the kernel of an algebra A which is defined as the set ker π = {A ∈ A; π(A) = 0}. Based on this notion a representation is called faithful if and only if ker π = {0}.

232

APPENDIX D. ASSUMPTIONS AND RESULTS OF AQFT

tions that will be discussed in later chapters. A representation is irreducible if the representation space H has no nontrivial closed invariant subspaces, meaning that only the trivial spaces {0} and H are closed linear subspaces which are invariant under the action of B. In other words, if a representation is irreducible then any subspace of H other than {0} and the whole space H will be changed if the elements of B operate on them. The notion of irreducible representations—be it of groups or algebras—is so important for ontological (and computational) issues because it allows for a decomposition of a given representation into irreducibles, or correspondingly, it allows for a decomposition of the representation space into a direct sum of invariant subspaces. These invariant subspaces can then be seen as referring to simpler systems.10 Let π1 and π2 be two representations of A in Hilbert spaces H1 and H2 respectively. π1 and π2 are said to be unitarily equivalent if there is a unitary operator U : H1 → H2 such that11 U π1 = π2 U ∀ A ∈ A. The unitary equivalence of two representations π1 and π2 is denoted by π1  π2 . A central fact about representations of a C ∗ -algebra is that they depend on the state of the system that is described by the algebra. This will be the topic of the next paragraphs. In 1947 Irving Segal introduced a novel notion of a state and proved that there is an important correspondence between these so-called algebraic states, as defined below, and representations of a C ∗ -algebra.12 According to Segal’s algebraic definition a state is a normed positive linear functionalover a C ∗-algebra, i. e. a linear map ω : A → C 10

(D.1)

It is intuitively helpful to understand the reduction of a representation to a host of simpler irreducible representations as a generalization of the decomposition of a periodic function by means of Fourier analysis. In the case of Fourier analysis one is dealing with irreducible representations of the planar rotation group. It is very easy to see that the representations occurring in the Fourier analysis are irreducible since the subspaces which are spanned by the basis vectors are 1-dimensional. 11 It might be helpful to note that multiplication with U ∗ “from the right” and using the defining property of a unitary operator, viz. U U ∗ = 1, yields the alternative equation U π 1 U ∗ = π2 . 12 See Segal (1947a).

D.2. REPRESENTATIONS AND STATES

233

for which ω(A∗ A) ≥ 0 for all A ∈ A (positive) and ||ω|| = 1 (normed) where C is the set of complex numbers. In view of this definition ω(A) can be interpreted as the expectation value of the observable A. In relation to standard QM, where observables act on states, the algebraic notion of a state is a dual formulation since here states act on observables as is obvious from the map D.1.13 The general correspondence between algebraic states over and representations of a C ∗-algebra is of great importance for the algebraic approach to QFT. The first side of this correspondence leads from representations to states. To be more accurate a given representation (Hπ , π) provides us with a whole supply of algebraic states in the following way. For any nonzero vector Φ ∈ Hπ one gets an algebraic state, i. e. one that fulfills all the requirements14 that are stated in the above definition, by setting ωΦ (A) = (Φ, π(A)Φ), A ∈ A.

(D.2)

This means that, for one given representation, all vectors in the representation space Hπ give rise to algebraic states via formula D.2. Note that the subscript ‘Φ ’ which is attached to ‘ω’ indicates that this entity ω which is proven to be an algebraic state depends on the given vector Φ. For this reason ωΦ (A) is called a vector state of the representation. So the first result is that for a given representation π any vector Φ in the representation space Hπ yields an algebraic state, which is of a particular kind, namely a vector state.15 In fact, any algebraic state is a vector state if one only works with an appropriate representation. This possibility is one result of the following construction. The second direction in the asserted correspondence between algebraic states and representations leads from algebraic states to representations. 13

Note that the algebraic states over a C ∗ -algebra A are elements of the dual space of this algebra which is the set of continuous, linear functionals over A. The continuity of algebraic states follows from the requirement of positivity according to proposition 2.3.11 in Bratteli and Robinson (1979). 14 See, e. g., section 2.3.2. in Bratteli and Robinson (1979) for a proof. 15 In a similar way, for a given representation, any density matrix ρ on the representation space Hπ gives rise to a so-called normal state of the representation by setting ωρ (A) = T r(ρπ(A)), A ∈ A.

234

APPENDIX D. ASSUMPTIONS AND RESULTS OF AQFT

To this end one has to construct from a given algebraic state ω a representation (Hω , πω ) and a unit vector Φω so that ω(A) = (Φω , πω (A)Φω ).

(D.3)

Note that in contrast to equation D.2 now there are subscripts ‘ω ’ attached to the ‘Φ’ and ‘π(A)’ since for the other direction of the correspondence the representation π and the vector Φ depend on the given algebraic state ω. The unit vector Φ thus represents the algebraic state ω in the representation space Hω . The actual construction of a representation of a C ∗-algebra for a given algebraic state was achieved by Segal in a famous paper Segal (1947a) where he generalized a representation theorem by the mathematicians Gel’fand and Naimark from 1943.16 The result of this so-called GNS-construction is that for any given algebraic state over a C ∗algebra A there is a cyclicrepresentation (Hω , πω , Φω ) such that ω is given by the construct from equation D.3 for all A ∈ A. Moreover, the given algebraic state ω uniquely specifies the triple (Hω , πω , Φω ) up to unitary equivalence. Eventually, it should be noted that a state over a C ∗ -algebra is pure if and only if the corresponding representation is irreducible. An example for a GNS-construction is Wightman’s reconstruction of a quantum field when the vacuum expectation values of products of field operators are given. Concluding this passage on representations of and algebraic states over ∗ C -algebras I wish to emphasize that the notion of a representation of a C ∗-algebra is the primary one whereas the notion of an algebraic state is introduced in particular in order to construct representations.

D.3

Superselection Sectors

The discovery of superselection rules dates back to a collaboration by Wick, Wightman and Wigner in 1952.17 Superselection rules forbid the interference of states that belong to different subsets into which the state space H 16

From a mathematical point of view the GNS-construction theorem rest on the same idea as one of the most famous theorems of functional analysis, viz. the Hahn-Banach theorem. 17 See Wick et al. (1952). A good exposition of the theory of superselection rules and its significance for AQFT is presented in chapter 2 of Horuzhy (1990).

D.3. SUPERSELECTION SECTORS

235

of a quantum field theory decomposes. Thus one has a decomposition H = ⊕ Hi

(D.4)

with orthogonal subspaces Hi , which are called superselection sectors, where i labels the quantum number of the superselected (or conserved) quantity, e.g. the electric charge. This means that there is a zero probability for a transition between states of different superselection sectors. Moreover, a superposition of states from different superselection sectors cannot be a pure state. Each superselection sector is the eigenspace of a conserved quantity which gives rise to a superselection rule. The phenomenon was first recognized by Wick, Wightman and Wigner with respect to the example of fermion and boson states, i.e. states with half odd integer or integer spin respectively. Further examples are charge and baryon number so that states with different charge and baryon number cannot be superposed.18 Haag and Kastler (1964) generalized these results by linking them to the existence of inequivalent irreducible representations of A, the algebra of quasilocal observables.19 Whereas unitarily inequivalent (irreducible) representations of the algebra belong to different superselection sectors, unitarily equivalent (irreducible) representations are from the same sector. This means that a superselection sector can be defined as a unitary equivalence class of an (irreducible) representation, sometimes denoted by [π] where π is one arbitrary representation from the equivalence class. It is thus the occurrence of inequivalent representations which explains why superselection rules arise. This insight is one of the advantages of (abstract) AQFT over the C ∗ -algebra approach by Segal who rated the occurrence of inequivalent representations as evidence for limiting all considerations to the level of the algebra. One of the basic obstacles in the resulting theory of superselection sectors is that there is a multitude of representations not all of which are physically relevant. Since the Hilbert space in standard QFT is a direct sum of all physically relevant representation spaces (i. e. sectors) it is a main target of the theory of superselection sectors to select the relevant representations in order to link up with the standard formulation. The task 18 19

See chapter 1 in Streater and Wightman (1964). See their pioneering work Haag and Kastler (1964)

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APPENDIX D. ASSUMPTIONS AND RESULTS OF AQFT

of selecting physically relevant superselection sectors has been addressed in the famous analysis by Doplicher, Haag and Roberts (1969-1974) which is an illuminating example. The aim of the analysis by Doplicher, Haag and Roberts (DHR) is the identification of the set of states with vanishing matter density at infinity for a massive theory.20 In other words DHR try to pick out only locally generated superselection sectors. More precisely, this locality of the chosen representations/sectors is defined relative to the vacuum representation π0 in the sense that those representations are selected whose restriction to observables measurable in the complement of any diamond-shaped region in Minkowski spacetime is unitarily equivalent to π0 . In mathematical terms the DHR selection criterion reads as follows: π|A(O
)  π0 |A(O
) , O ∈ K for a sufficiently large region O. In this formula  means unitary equivalence as defined on page 232, O
is the causal, i. e. spacelike, complement of O (O
is thus not bounded). Furthermore K is the set of all open bounded double cones21 (“diamonds”) in Minkowski spacetime and A(O
) is the C ∗-algebra generated by all A(Oi ), Oi ∈ K, Oi ⊂ O
. Finally, π|A(O) is the representation π of the quasilocal algebra A restricted to observables measurable in O. Note that the DHR selection criterion is not dealing with any particular region O. The idea is to select all those states where matter is localized in some region no matter where this region is or how large it is (as long as it is finite). One can say that the DHR criterion selects those representations which represent localized excitations of the vacuum. And different localized excitations are distinguished, via superselection rules, by the kinds of charges that are localized, e.g. baryon number. Generally the DHR criterion selects representations (or superselection sectors) with charges localized within some region O. However, one example for an exception is the electric charge which is not in the scope of the DHR analysis. The reason is that an electric charge is, by Gauss’ law, observable via the flux of the electric field through a sphere containing O. 20

A good account of the DHR analysis can be found in Roberts (1990). An open double cone is the intersection of the interior of a forward and the interior of a backward lightcone w.r.t. to two time-like related points. 21

D.3. SUPERSELECTION SECTORS

237

Thus for any O there is flux that is measurable in O
since the sphere may, according to Gauss’ law, have an arbitrarily large radius. Therefore the DHR criterion is too restrictive. While it is appropriate for charges with only short range effects, such as baryon number, it excludes electric charges and topological charges. A refined treatment was given by Fredenhagen and Buchholz (1982). Note that the DHR selection criterion, although it selects certain sectors, is not on a par with superselection rules, e. g., for charges. Rather, the DHR selection is a preselection, namely of physically relevant sectors, before the proper superselection analysis among these preselected sectors begins. While the idealized notion of localization expressed by the DHR criterion thus turns out to be too strong the general ideas of the DHR analysis of superselection sectors are crucial for the algebraic theory of superselection because they connect localization properties with statistical features of charged states. As Roberts stresses in his highly useful “Lectures on Algebraic Quantum Field Theory” one of the great triumphs of AQFT is the understanding of the general structure of superselection sectors.22 On the basis of the theory of operator algebras it is possible to understand the origin of three very important aspects of relativistic quantum field theory, first, the composition of charges, second, the classification of particle statistics and, third, charge conjugation. It will turn out that, mathematically, the correlate of the composition of charges is the tensor product of representations, the mathematical correlates of charge quantum numbers (describing particle statistics) are the labels of inequivalent irreducible representations (or sectors), and charge conjugation can be found in the mathematical procedure of going over to the complex conjugate representation. In the context of algebras a mapping ρ : A → B from one algebra to another algebra is called a morphism if ρ(αA + βB) = αρ(A) + βρ(B) and ρ(AB) = ρ(A)ρ(B) for all A, B ∈ A and α, β ∈ C. This means that in 22

Although these issues are obviously by no means easy Roberts (1990) is a comparatively accessible introduction. See Buchholz (2000) for a short and slightly more up to date introduction. Chapter 9 in Baumg¨artel and Wollenberg (1992) is concise and mathematically detailed and very helpful if the general line of argumentation is already clear. From a physical point of view proposition 9.2.6 and remark 9.2.7, proposition 9.4.3 and remark 9.4.5 as well as proposition 9.6.5 and remark 9.6.6 are particularly important. Also, as always, see Haag (1996), where chapter IV matters for the present context.

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the case of a morphism it makes no difference whether one first carries out certain actions on elements in A and then maps the result to B or whether one first maps these elements from A to B and then carries out the same actions in the algebra B. For C ∗ -algebras one needs the natural supplementary requirement π(A∗) = π(A)∗ for the adjoint operation or involution to define a C ∗ -morphism. It is common to suppress the C ∗ if one is only dealing with C ∗ -algebras and to just talk about morphisms. A morphism ρ is a special kind of representation. The crucial aspect of working in terms of morphisms is the possibility of the composition of a sequence of morphisms. This defining characteristic of morphisms implies that, with the help of a morphism ρ, one can immediately construct for any given representation π another representation π ˜ by setting π ˜ = π ◦ ρ. A morphism ρ is called localized in region O if it is an endomorphism, i. e. a map from R into R, that acts trivially in the causal complement of O, i. e. ρ(A) = A for all A ∈ R(O
) which is equivalent to saying that ρ|A(O
) = id. Note that for computational reasons the spacetime regions O are usually taken to be diamond-shaped as described above. A morphism ρ which is localized in O is called transportable if there is, with respect to every region arising from a Poincar´e transformation of O, a morphism that is equivalent to ρ. Here the notion of unitary equivalence of representations carries over in a straightforward fashion. Physically relevant or “admissible” representations23 can be described by localized transportable endomorphisms since it can be shown24 that a representation π is admissible if and only if it is unitarily equivalent to π0 ◦ ρ for an appropriate localized and transportable endomorphism ρ. This insight makes it possible to classify the kinds of superselection sectors by an analysis of the localization behaviour of localized morphism. And moreover, this is carried out by studying the properties of the algebra A itself. From the above-mentioned three important aspects of relativistic quantum field theory—composition of charges, classification of particle statistics and charge conjugation—I will now highlight the issue of statistics in order 23

This is the term used in Baumg¨artel and Wollenberg (1992), defined in subsection 9.2.1. 24 See Proposition 9.2.6 and its instructive proof in Baumg¨artel and Wollenberg (1992).

D.3. SUPERSELECTION SECTORS

239

to clarify the general connection between the mathematical formalism of AQFT on the one side and physical as well as ontological issues on the other side. The statistics of superselection sectors is determined by the permutation properties of sequences of morphisms. Physically, a sequence of morphisms corresponds to a composition of charges which explains why the permutation properties lead to the statistics. For the permutation group of n elements, Pn , a representation in terms of unitary operators is (n) given by the ρ (Pn). (n) The analysis of the representation ρ (.) depends only on ρˆ, the equivalence class of ρ. Let ρ be admissible and let Φ be a corresponding left inverse Φ(ρ ) = λ1 where a left inverse Φ is defined as a positive bounded linear mapping from the quasilocal algebra A into L(H0 ), the set of bounded operators from H0 into H0 (the Hilbert space of a vacuum representation), which has the properties stated in definition 9.5.2 of Baumg¨artel and Wollenberg (1992). Since ρ is admissible if it is irreducible the parameter λ can be used to classify the irreducible equivalence classes ρˆ. Different values of the parameter λ thus yield a first classification of superselection sectors of A, i. e. the equivalence classes of irreducible admissible representations. In physical terms λ classifies the statistics of the sectors. For λ = 1 one has the usual Bose statistics and for λ = −1 the usual Fermi statistics. Furthermore there are so-called strange statistics where the values λ = ± d1 , d > 1 correspond to para-Bose (for +) and paraFermi (for -) statistics of the order d. The case λ = 0 is called infinite statistics and is not physical.25 These results confirm the so-called BoseFermi alternative, i. e. that every particle must be either a boson or a fermion, and the generalization of this alternative to para statistics.26 In subsection 13.3.3 these results have a direct ontological bearing.

25

The expression ‘parastatistics’ was coined by Green in 1953 bringing together earlier considerations by Gentile (1940) and Wigner (1950) that Fermi-Dirac statistics on the one side and Bose-Einstein statistics on the other side may not exhaust all the options. 26 Note that representations with d > 1 are no longer 1-dimensional as they are in the usual Bose and Fermi cases.

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Physics (and Mathematics) Glossary Adjoint Operator  Involution. Algebra A vector space X with a coefficient field IK is called an algebra if a multiplication is defined which associates the product xy to each pair x, y ∈ IK and the multiplication satisfies the following laws: (A1) x(yz) = (xy)z (associative law) (A2) x(y + z) = xy + xz, (x + y)z = xz + yz, (distributive law) (A3) λ(xy) = (λx)y = x(λy), where x, y, z ∈ X and λ ∈ IK. Note that on a vector space only the addition of vectors and scalar multiplication, i. e. the multiplication of a scalar and a vector, are defined. If in addition it holds for the multiplication that (A4) xy = yx

(commutative law)

then the algebra is called an commutative (or abelian) algebra. An example for a commutative algebra is the real function algebra of all functions f : D → IR (with an arbitrary domain D) provided that this set of functions is a vector space which if the functions f and g are amoung itss elements contains f · g as well. A second example with a high significance for the axiomatic development of quantum physics is supplied by the (canonical) commutation relations which generate a so-called commutator algebra. A particularly famous case is the Lie algebra which is fixed by the commutation relations of the six generators of the (homogeneous) Lorentz group (for boosts in three directions and for rotations about three axes). Antiparticle The theory (i. e. QFT) predicts and experiments have confirmed that to each particle with non-zero charge there is a corresponding antiparticle. The only difference between a particle and 259

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a corresponding antiparticle is that their respective charges have opposite signs. Otherwise they have exactly the same properties. The most famous example of a particle-antiparticle pair are the electron and the positron. The spectacular first observation of a positron in a cloud chamber photograph of cosmic rays is due to C. Anderson in 1931. The existence of antiparticles was first predicted on the basis of the Dirac equation. While the Dirac equation was intitially meant to describe only electrons it turned out to exibit a duplication of solutions the second half of which were finally interpreted as describing antiparticles. As Weinberg points out (p. 199) in Weinberg (1995) “the reason for antiparticles” is that the existence of e. g. an annihilation field for one type of particle necessitates a creation field for the corresponding antiparticle so that conserved quantum numbers like charge can remain constant. The occurence of antiparticles is usually taken to be one of the inherently relativistic features of QFT with no classical counterpart. Banach Algebra (and Normed Algebra) A Banach space A is called a Banach algebra if it is an algebra and for the product defined on this algebra the norm has the property ||AB|| ≤ ||A|| ||B||

∀A ∈ A.

Note that while completeness with respect to the norm is required for Banach spaces and for Banach algebras this is not the case for normed spaces and normed algebras. Banach Space A normed space is called a Banach space if is complete with respect to its norm. Banach ∗ -Algebra A Banach algebra is called a Banach ∗ -algebra if it is equipped with an involution A → A∗ such that ||A∗ || = ||A||

∀A ∈ A.

Borel Set B(H) The set B(H) of all bounded operators acting on the Hilbertspace H is a Borel set. Boundedness of Operators An operator A : X → Y is said to be

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261

bounded below if there exists a number k such that (φ, Aφ) ≥ k

∀A ∈ X, ||φ|| = 1,

i. e. for all unit vectors φ in the domain of A. The set of all bounded operators acting on the Hilbertspace H is mostly denoted by B(H). Sometimes this set is denoted by L(H) due to the fact that continuity and boundedness happen to coincide for linear operators where L(H) is initially introduced as the set of all linear and continuous operators on H. Note that in the general case one talks about B(X, Y ) and L(X, Y ) respectively. It is convenient to work with bounded (instead of unbounded) operators because they can be added and multiplied without restriction. Canonical Commutation Relations (CCR) [pi , qj ] = −i
δij [pi , pj ] = [qi , qj ] = 0

(D.5)

where i, j = x, y, z if we assume the three-dimensional configuration space of standard QM. The dimension of the configuration space equals the number of degrees of freedom. The famous Stone-von Neumann uniqueness theorem shows the fundamental significance of the CCRs leading to the abstract Hilbert space formulation of QM. C ∗ -Algebra A Banach algebra is called a C ∗ -algebra if it has an involution such that ||A∗ A|| = ||A||2 ∀A ∈ A. Note that this so-called “norm property” has more consequences than one expects. One important consequence for reflections about AQFT is that it can be shown quite easily using the norm property that B(H), the algebra of all bounded operators on a Hilbert space, is a C ∗ algebra. Moreover, any selfadjoint subalgebra of B(H) which is closed w. r. t. the uniform topolgy is a C ∗ -algebra as well. The letter ‘C’ in ‘C ∗ -algebra’ indicates that one is dealing with a complex vector space. Following the custom in the physics literature I denote C ∗-algebras with A, B, C and elements of C ∗-algebras with A, B, C. The notion of a C ∗ -algebra is due to Gelfand. It can be shown that a C ∗ -algebra is isomorphic to a norm-closed selfadjoint algebra of bounded operators on a Hilbert space.

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Classification of von Neumann Algebras Von Neumann has classified von Neumann (or W ∗ )-algebras into three different types with further subdivisions. While type I algebras are the relevant ones for standard QM, type III algebras are characteristic for QFT. Completeness (and Metrical Space) A metrical space is complete if any Cauchy sequence converges in this space, i. e. the limit itself belongs to the metrical space. A metrical space is a pair (X, d) where X is a set and d is a metric on this set, i. e. a mapping d : X × X → IR, (x, y) →  d(x, y) which fulfills certain requirements (Definitheit, symmetry and triangle inequality). d(x, y) is called the distance between the points x and y. An example for a metrical space is IR with the metric d(x, y) := |x − y|, x, y ∈ IR where| · | denotes the map on the absolute value. D‘Alembert operator The d‘Alembert operator is defined as follows 1 ∂2 ∂2 ∂2 ∂2 1 ∂2 2 ≡ −∇ + 2 2 = − 2 − 2 − 2 + 2 2 . c ∂t ∂x ∂y ∂z c ∂t 2

(D.6)

Degrees of Freedom The dimension of the configuration space equals the number of degrees of freedom. In order to give an example, the number of degrees of freedom of one spinless particle is three. The recognition of spin adds one further degree of freedom. Dirac “δ-Function” In various areas of physics it is common to use the so-called “δ-function” which has been introduced by Paul Dirac in the twenties. Dirac postulated a “function”  ∞ δ(x) which vanishes for all x ∈ IR \ {0} and has the property that −∞ δ(x) dx = 1. Although it is a very convenient tool in order to describe (idealized) point masses or point charges the “δ-function” cannot be a function in the usual sense since it is a basic mathematical fact that the integral over a function is zero if the function differs from zero only for a finite number of arguments as the “δ-function” does. In order to cope with this situation without dispensing with the main idea of the “δ-function” altogether

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L. S. Schwartz (since about 1945) and other mathematicians have built a sound (and sophisticated) mathematical ground which is known as distribution theory and forms a part of functional analysis. Pivotal for distribution theory is the notion of test functions which play a role in axiomatic formulations of QFT as well. One important application is Wightman’s smearing of fields with test functions in order to avoid difficulties with quantum fields at a point in space-time. Dirac Equation The Dirac equation is the result of the second attempt to find a relativistic wave equation after the first attempt, leading to the Klein-Gordon equation, had been considered by Dirac and others as a failure. The Dirac equation (iγ µ ∂µ − m) ψ = 0

(D.7)

is an equation for a spinor wave function ψ with four components. The γ µ are the Dirac matrices which fulfill anticommutation relations [γ µ , γ ν ]+ ≡ γ µ γ ν + γ ν γ µ = 2g µν and thus specify an algebra. g µν is the metric tensor (see appendix A). In the “standard representation” the Dirac matrices are explicitly given by 4 × 4 matrices     0 1 0 −σ j 0 j γ = , γ = 1 0 σj 0 where the σ j , j = 1, 2, 3 (or x, y, z) are the Pauli spin matrices       0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = . 1 0 i 0 0 −1 The covariant derivative ∂µ is defined in appendix A. According to common gloss the spin follows automatically from the Dirac equation supposedly showing that spin is an inherently relativistic feature while in non-relativistic QM it has to be added on the basis of experimental data. However, although this argument sounds convincing and and is accordingly widespread it is unfortunately wrong. Spin can be derived in a non-relativistic setting as well. Feynman’s path integral formalism The central notion in Feynman’s path integral formulation is that of a functional integral, instead of

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BIBLIOGRAPHY using operators as the standard Hilbert space formulation. In the case of quantum mechanics the path integral formulation for the amplitude of the transition probability from an initial state |xi ti  to a final state xf tf | is given by

xf tf |xi ti  = D[x(t)]eiS[x(t)] (D.8) where S is the action functional and D[x(t)] is the analogue of dx in the usual integral. The integral in equation D.8 is a functional integral— indicated by the use of square brackets—since functions (the paths x(t)) are mapped onto numbers (the transition amplitudes xf tf |xi ti ) where the integration runs over all functions x(t). In more physical terms, the integral in D.8 is taken over all possible trajectories or paths x(t) between the initial state and the final state. It is clear of course that this gloss must not be understood literally. The main reason for this caution is not that it is inconceivable that all paths are travelled upon simultaneously but rather that sharp trajectories are ruled out in quantum mechanics, at least in its standard version (see sections B.2.2, 8.2 and 10.3). This situation is different if one assumes a revisionary hidden variable approach, as the followers of David Bohm do, where particles do travel on trajectories. In the path integral formulation of QFT the paths x(t) are replaced by field configurations φ(x, t), over all of which the integration is taken so that they all contribute to a certain degree. As Weinberg stresses in Weinberg (1995), “although path integration is by far the best way of rapidly deriving Feynman rules from a given Lagrangian, it rather obscures quantum mechanical reasons underlying these calculations” (p. xxii). It thus seems that the path integral formalism is primarily a useful mathematical device. As far as ontological considerations are concerned I think that path integrals are of little relevance even though they are very important for the working physicist. The path integral formalism is introduced, e. g., in Ryder (1996) starting from a fairly elementary level. See in particular the very accessible introduction to functional calculus, namely section 5.4 on differentiation and section 5.5. on integration.

Functional Integral In order to quantise gauge theories it is very convenient to use functional integrals which is the reason why functional

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methods are quite popular in modern treatments of QFT. Operator and Functional. Hilbert Space (and Pre-Hilbert Space) John von Neumann coined the notion of the abstract Hilbert space which has a very high importance for the foundations of quantum theory. A Banach space equipped with an inner product which is suitable to the norm is called a Hilbert space. Alternatively one can define a Hilbert space as a pre-Hilbert space which is complete where a pre-Hilbert space is a vector space equipped with an inner product. In particular in the context of QM the best-known examples of infinite-dimensional Hilbert spaces are the space L2 of square-integrable functions and the space l2 of squaresummable sequences. The fact that L2 and l2 are isomorphic Hilbert spaces (Fischer-Riesz theorem) is the mathematical counterpart of the discovery that the wave mechanics approach to QM by Schr¨odinger (using L2 ) and Heisenberg’s matrix mechanics (using l2) are equivalent representations of the same underlying structure, viz. of a Hilbert space. 

-Homomorphism In this work the definition of a -homomorphism is needed in the context of representations of C ∗ -algebras. For this reason the definition of a  -homomorphism will be specified for that particular context. Let A be a C ∗ -algebra, H a Hilbert space and B(H) the set of all bounded operators acting on Hilbertspace H. The map π : A → B(H) is called a  -homomorphism if it has the following properties: (H1) π(cA + dB) = cπ(A) + dπ(B) (linearity), (H2) π(AB) = π(A)π(B), (H3) π(A ) = π(A) where A, B ∈ A and c, d ∈ C, the set of complex numbers.

Involution (or Adjoint Operation) The involution (or adjoint) of an operator is the generalization of the conjugate complex of a complex number for operators. Isotony Property Set of observables increases with the size of the localization region. See Buchholz (2000), p. 4.

BIBLIOGRAPHY

266 Klein-Gordon Equation The Klein-Gordon equation ( + m2 )φ = 0

is the result of first attempt to construct a relativistic version of the Schr¨odinger equation. After an initial rejection it is known today that the Klein-Gordon equation is valid for a massive particle with spin zero, e. g. a meson (except the ρ-meson) which mediates the interaction between hadrons. Thus the Klein-Gordon equation is not valid for electrons as originally intended. KMS Condition A given state is an equilibrium state with a certain temperature if it satisfies the KMS condition, where the abbreviation is after the work of Kubo (1957), Martin and Schwinger (1959). Laplacian operator The Laplacian operator ∇2 is defined as follows ∂2 ∂2 ∂2 ∇ ≡ 2+ 2+ 2 ∂x ∂y ∂z 2

where ∇ alone is called Nabla operator. Local Commutativity ‘Local commutativity’ is here used synonymously to the term ‘locality’. Both denote the commutation of operators/ observables which refer to space-time regions which are spacelike separated from one another. The postulation of ‘local commutativity’ or ‘locality’ respectively is the central condition of AQFT. It states the statistical independence of measurements in spacelike related regions of space-time. See appendix D. Natural Units Going over to the so-called natural units by setting c =
= 1 means that all velocities are multiples of the basic unit c and expressions for action are multiples of the basic unit
. The fundamental dimensions in the system of natural units, abbreviated as n.u., are mass, action and velocity while in c.g.s units (centimetre, gram, second) the fundamental dimensions are mass, length and time. The calculational effect of the transition from c.g.s. units to natural units is a considerable simplification of the formulae in relativistic QFT. Note that another effect of the transition to natural units is that various quantities have the same dimension now, e. g. mass, momentum, energy and wave number all have the natural dimension mass.

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More details about how to handle the n.u. system can be found, e. g., in Mandl and Shaw (1993), section 6.1. Norm (and Normed Space) Let X be a vector space and IK a set of numbers. A mapping || · || : X → IR is called a norm if it has the following properties: (N1) ||0|| = 0, ||x|| > 0 for x = 0, (N2) ||λx|| = |λ| ||x||, (N3) ||x + y|| ≤ ||x|| + ||y|| (triangle inequality) where x, y ∈ X and λ ∈ IK. A linear space (or vector space) which is equipped with a norm || · || is called a normed space. Note that a norm refers to single elements of a vector space while a metric refers to (the distance between) two elements of a vector space. Nevertheless, a given norm on X generates a metric d(x, y) := ||x − y|| which is called the canonical metric or uniform topology or norm topology on X. The completeness required for Banach spaces is always with respect to this norm. A well-known example for a norm is the euclidean norm ||x|| :=

√

< x, x >, x ∈ X

where < ·, · > is the inner product on a euclidean vector space X. N-particle state An N-particle state (where N is a positive integer) is the eigenstate of the number operator corresponding to its eigenvalue N. The conventional but problematic gloss is that an N-particle state is a state where N particles are present. See chapter 4. Operator and Functional (and Related Notions) The characteristic feature of operators which distinguishes them from ordinary numbers, which are scalar quantities, lies in the fact that two operators in a product can generally not be switched without changing the product. When Heisenberg recognized this peculiar behaviour of “quantum numbers” he did not know that there is a whole field in mathematics which is

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268

concerned with such entities, namely the theory of operators within functional analysis. Today functional analysis is an important prerequisite for any physicist working on the foundations of quantum physics. According to the more precise mathematical definition a continuous linear operator is a continuous linear map T : X → Y between two normed spaces. If the elements of Y are scalars then T is called a functional instead of an operator. In contrast to conventional func f (x)) a functional, say F , can map e. g. a tions (e. g. f: IR → IR, x → whole function (and not just one or more arguments) to a scalar. The norm of a functional F is defined as ||F || = sup{|F (A)|; A ∈ X, ||A|| = 1}. Note that in |F (A)| one can employ the usual absolute value | · | since F (A) is a scalar. The norm || · || in ||A|| is the operator norm. A linear functional ω over a C ∗ -algebra A is said to be positive if ω(AA) ≥ 0

∀A ∈ A.

Operator Norm Let A be an operator A: X → Y . The operator norm of A is defined as ||A|| = sup{||Aφ||; φ ∈ X, ||φ|| = 1}. Note that the norm || · || in ||Aφ|| and ||φ|| denotes a different norm which is defined on the normed spaces X and Y whose elements are vectors (and not operators). Pure state A pure quantum state is a maximal set of information or properties respectively which can be ascribed to a quantum object. It is a characteristic feature of a quantum mechanical state that it can never contain answers to all possible experimental questions. A state is pure if it cannot be written as a linear combination (with positive coefficients) of two other states. Quantum Chromo Dynamics (QCD) The theory of the strong interaction, i. e. of the force which holds nucleons, the building blocks of an atom’s nucleus, together is called Quantum Chromo Dynamics (QCD). The term ‘chromo’ (greek: colour) is due to the purely mnemotechnical association of particular colours with the fundamental

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types of ‘colour charge’ of QCD which is the property of elementary particles that accounts for their sensitivity to strong interaction. Only quarks and gluons carry colour charge and thus experience strong interaction while gluons mediate the strong interaction. The notion of colour charge is analogous to that of electrical charge which is responsible for the sensitivity of electrically charged particles, e. g. electrons and protons, to eletromagnetic interaction which is mediated by photons. One peculiarity about colour-charged particles which could have ontological implications is the so-called quark confinement according to which quarks and gluons are always confined into hadrons, i. e. composite particles, with an overall neutral colour charge. Quantum Field Theory The most meaningful short characterization of quantum field theory (QFT) can be given by pointing at the reasons which have motivated its construction. Historically as well as systematically the most pressing need was a reconciliation of quantum mechanics and (special) relativity theory which has finally led to QFT. However, although this explanation is quite common the possibility to formulate a non-relativistic QFT shows that the explanation is somewhat misleading. As the term quantum field theory indicates the field aspect is vital (which can of course be connected with relativity theory most naturally). The introduction of fields leads to various mathematical and conceptual complications as compared to QM since a field is by definition a system with an infinite number of degrees of freedom. From a more intuitively physical point of view QFT is meant to give a unified description of particles and forces. Any particle gives rise to a corresponding quantum field and to each kind of force (which is mediated by a field) one associates particle-like force carriers like photons for the electromagnetic interaction or gluons for the strong interaction. Dirac’s quantization of the electromagnetic radiation field in 1927 is commonly seen as the first systematic step towards QFT or even the inception of QFT. The date shows that the formation of QFT began already before QM had reached its mature form. Quantum Mechanics Unless stated otherwise I use the term quantum mechanics for non-relativistic quantum physics of at most a finite

270

BIBLIOGRAPHY number of particles in the standard Hilbert space formulation, which has been introduced by von Neumann.

Quantum Physics I use the term quantum physics, primarily as opposed to classical physics, each time when it is not important to distinguish between standard quantum mechanics and quantum field theory, relativistic and non-relativistic quantum theories or between free theories and theories incuding interactions. To put it another way, I use ‘quantum physics’ as a generic term for all kinds of quantum theories. Relativistic Quantum Mechanics It is often distinguished betweeen relativistic quantum mechanics and quantum field theory. This distinction might be somewhat confusing in the light of the common proposition that QFT results from the attempt to reconcile quantum mechanics with special relativity theory. The solution to this puzzle is that there are two different historical roads to QFT. On the first road one starts from ordinary (non-relativistic) QM and tries to find relativistically invariant versions of the basic quantum mechanical formulae. On the second road one starts with a relativistic theory (viz. the classical theory of the electromagnetic field) and one then tries to introduce quantum concepts by quantizing the classical field theory (Canonical Quantization). In the middle of the first road stands relativistic quantum mechanics with e. g. the famous KleinGordon equation and the Dirac equation. “Second quantization” of these one-particle equations of relativistic quantum mechanics then supplies the bridge to quantum field theory. Separable (Hilbert space) A Hilbert space is called separable is there is a denumerably infinite sequence of vectors which are dense with respect to the Hilbert space. Stone’s Theorem Stone’s theorem shows the correspondence between representations of the Lie group and infinitesimal representations of the corresponding Lie algebra. Type III Algebra The local algebras in algebraic quantum field theory are of type III according to von Neummann’s classification scheme. Physically this means that locally any state looks like a mixed state.

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Vacuum state The vacuum state is the eigenstate of the energy operator with the lowest eigenvalue, i. e. the lowest possible energy. Note that it is characteristic of quantum physics that the lowest possible energy of a quantum system is not zero. The label ‘vacuum state’ stems from the fact that the vacuum state is also the eigenstate of the number operator corresponding to the eigenvalue 0. This fact explains the common but problematic gloss that the vacuum state is the state with no particles present. See section 8.4 for some surprising problems of this gloss. Von Neumann algebra (or W ∗ -algebra) The characteristic feature of a von Neumann algebra (or W ∗ -algebra) is that it is closed in the weak operator topology, hence the letter ‘W ’ in ‘W ∗ -algebra’. The ‘∗’ indicates that the algebra is stable under taking adjoints, i. e. it is a ∗-algebra. Note that every von Neumann (or W ∗ )-algebra is a C ∗ -algebra. Unlike C ∗ -algebras von Neumann (or W ∗ )-algebras do always contain projections. Sometimes von Neumann algebras are distinguished from W ∗-algebras similarly as in the distinction between concrete and abstract C ∗ algebras. W ∗-algebras are thus the abstract counterparts of von Neumann algebras. Weak additivity Spacetime is homogeneous, no phenomena like minimal length exist. Horuzhy, p. 13. Wedge Image of the set {|x0 | < x1 } under a Poincar´e transformation. Monographs Suggested for Further Reading. Further details about standard QFT can be found in Mandl and Shaw (1993), Ryder (1996), Weinberg (1995) and Weinberg (1996). For my taste the accounts of these three authors all have their merits and drawbacks. Amoung these Mandl and Shaw present the least advanced account of QFT but it has the advantage of great clarity and is very well thought through from a didactical popint of view. Ryder is often very helpful as far as mathematical and calculational aspects are concerned. Moreover, Ryder is a good source for the standard textbook gloss which is usually spelled out more than in many other works. However, in conceptual matters Ryder is not always reliable

272

BIBLIOGRAPHY

with respect to a higher standard than textbook gloss. Finally, Weinberg’s books clearly present the state-of-the-art as far as standard QFT is concerned. Weinberg purveys various valuable insights to the reader who is not repelled from his often Moreover, Weinberg often presents issues in an unusual order reflecting his views about what comes first and what is derived which sometimes diverts considerably from the more standard presentations. For comprehensive accounts of AQFT (and other axiomatic approaches to QFT) see Araki (1999), Haag (1996) and Horuzhy (1990) as well as the classics Streater and Wightman (1964), Jost (1965) and Bogolubov et al. (1975). There are some highly recommended papers with more current surveys of AQFT, e. g. by Buchholz and Fredenhagen, as cited in chapter 4 and appendix D. See Bratteli and Robinson (1979), Reed and Simon (1975) and Reed and Simon (1980) for definitions and theorems concerning functional analysis, in particular the mathematical theory of operator algebras. Further useful information about the history of QFT are in Cao (1997) and Schweber (1994). Mittelstaedt (1998) deals with the conceptual foundations of quantum physics.

Philosophy Glossary Concrete Concrete objects have at least a temporal location, as the summer of 1969, and mostly have a spatial location as well. Dependent Something is dependent when it cannot exist of itself. For more details see section 7.4 and for the distinction of different senses of (in-)dependence see Simons (1998a), p.236. Nominalism Denies the existence of universals. Particular See ‘universal’. Substance The expression ‘substance’ as used in this study indicates an independent concrete particular. This matches Aristotle’s (primary) original use of ‘substance’ in the Categories for which a particular horse or a particular human being are prime examples. It deviates, however, from Aristotle’s later use in the Metaphysics where substance or ousia is identified with the individual form of a concrete particular. Substance understood in this way only makes up a concrete particular together with matter which is not part of the substance in this second sense of substance. In the later philosophical tradition a third usage of substance was very important, namely substance as the ‘factor of particularity’ (Locke) in things as opposed to their properties. This third usage of ‘substance’ is synonymous to substratum. For more details see chapters 3, 7 and 11 in this thesis as well as Hoffmann and Rosenkrantz (1997), Loux (2002), chapter 3 and Simons (1998a). Substratum In various ontological approaches a substratum is assumed which is responsible for the individuality of a concrete particular. Since one gets to the substratum of an object by abstracting from all the object’s properties the substratum is sometimes called bare particular (Gustav Bergmann). If Aristotle’s later identification of a substance with the form of an individual is taken to mean a universal form then 273

274

BIBLIOGRAPHY prime matter would take the role of the individuating substratum. This interpretation of Aristotle is controversial, however. See section 7.2 for a different interpretation. Substratum theorists commonly take the individuating power of the substratum to be irreducible. For details see Armstrong (1989), p. 60 f, Loux (2002), chapter 3 and Simons (1998a), p. 237.

Trope A trope is an individual property instance or a dependent concrete particular. This is the same terminology as in Simons (1994). K. Cambell characterizes tropes as abstract particulars where ‘abstract’ (as opposed to concrete) is his usage means ‘(capable) incapable of independent existence’ Campbell (1990). Universal A universal (opposite: particular) can be multiply instantiated or exemplified.

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