Gauge Theories: a Case Study of how Mathematics Relates to the World A thesis presented by Antigoni Nounou to University of London in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Philosophy London School of Economics and Political Sciences University of London London April 2002 O 2002 by Antigoni Nounou All rights reserved. UMI Number: U615236 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI U615236 Published by ProQuest LLC 2014. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 « O F ^ POLITICAL Abstract The goal of this thesis is to investigate the relation between mathematics and physics and the role this relation plays in what physics does best, that is in scientific explanations. The case of gauge theories, which are highly mathematical, is used as an extended case study of how mathematics relates to physics and to the world and these relations are examined from both a historical and a philosophical perspective. Gauge theories originated from an idea of Weyl which turned out to be wrong, or in other words, empirically inadequate. That original idea underwent a dramatic metamorphosis that turned the awkward caterpillar into a beautiful butterfly called gauge theories, which were very successful and dominated theoretical physics dur­ ing the second half of the twentieth century. The only leftover from Weyl’s faux pas was the very name of the theories and the question how it is possible for something as wrong as his original idea to result in a theory so relevant to the world. We argue that it is thanks to a very dynamic and dialectic relation between mathematicians and physicists, both theoretical and experimental, that the resulting theory turned out to be so successful. From a more philosophical perspective, we take the view that the relation between mathematics and physics has a structuralist character, in general, and we recognize that what we call ambiguity of representation of the third type lies at the heart of gauge theories. Our claim is that it is precisely this type of ambiguity of representation and the non-physical entities that it inevitably introduces which ex- Abstract iii plain the physical facts. However, the non-physical entities should be attributed a non-causal status in order to provide valid and legitimate scientific explanations. The fibre bundles formulation of gauge theories is considered to be their unique formulation that allows for this shift and the Aharonov-Bohm effect which is ex­ amined within the fibre bundle context provides a narrower yet very fruitful case study. Acknowledgments I would like to thank the Greek State Scholarship Foundation for their gener­ ous support over the first three and a half years of this Ph.D. and the LSE Depart­ ment of Philosophy, Logic and Scientific Method for the Popper scholarship and for the financial support they offered me for a year. Also, I would like to thank all the people who helped me in completing this thesis. Craig Calender Nancy Cartwright, Jordi Cat, Carl Hoefer, Jeff Ketland and Stathis Psillos contributed, one way or an­ other, to the opening of my philosophical horizons. From Imperial College, Chris Isham and Kelly Stelle patiently explained to me again and again difficult technical issues and guided me through gauge theories, gravity and fibre bundles. I am most grateful to Michael Redhead, D.K. and my mother because without their support on all levels this thesis would have never been completed. Finally, I am indebted to my two examiners, Dr. Chang and Professor Kilmister for their valuable comments that helped me improve the thesis. Contents Introduction.................................................................................................. 1 1 Some History.............................................................................................6 1.1 The Quest for the holy Grail of a Unified Theory ....................................................7 1.2 The Weyl-Einstein D ebate.......................................................................................14 1.3 The Metamorphosis of Weyl’s Idea .........................................................................19 1.4 Swimming Against the Phenomenological Tide1 ..................................................28 1.5 A very Brief History of Fibre Bundles ................................................................... 32 1.5.1 From Sphere Spaces to Sphere Bundles to Fibre Bundles ..........................35 1.6 The Aftermath..........................................................................................................38 2 Mathematical Representations of Physics .............................................41 2.1 The Mathematical and the Physical........................................................................ 42 2.1.1 Raising the Issu es ......................................................................................... 42 2.1.2 The Question of Choice: Which Mathematical Representation and W hy?.............................................................................................................. 46 2.2 Field’s Id e a ...............................................................................................................47 2.2.1 Science Without Numbers: a Defence of Nominalism ............................. 48 1 The title of this section is borrowed from a phrase that can be found in O’Raifeartaigh’s The Dawning o f Gauge Theory , p.7. O’Raifeartaigh’s book is highly recommended as a wonderful resource for more precise and complete historical detail. For a standard physics introduction to this material reference may be made to Aitchison & Hey’s Gauge Theories in Particle Physics. V Contents vi 2.2.2 In What Ways ’Utility of Mathematical Entities’ is Different from ’Utility of Theoretical Entities’ ................................................................................ 51 2.2.3 Illustration of Why Mathematical Entities are Useful: Arithmetic, Geometry and Distance ..................................................................................54 2.2.4 Nominalism and the Structure of Physical Space ..................................... 55 2.2.5 A nominalistic Treatment of Newtonian Gravitational Theory ................ 57 2.2.6 Criticism of Field’s programme by Malament ...........................................59 2.2.7 Criticism of Field’s programme by Shapiro .............................................. 62 2.3 Structuralism 2.4 Michael Redhead’s Surplus Structure .................................................................... 66 2.4.1 Symmetries...................................................................................................70 2.4.2 Surplus Structure and Gauges .....................................................................72 2.4.3 Comparing Field & Redhead .......................................................................76 3 Formulations of Gauge Symmetries ..................................................... 80 3.1 Ambiguity of Representation of the Second Type and the Third Type: More Canonical Variables/Degrees of Freedom than the Ones Needed? .............80 3.2 Gauge Symmetries and Constrained Hamiltonian Systems or Structures ............84 3.2.1 The Free Electromagnetic F ield ................................................................. 93 3.3 Symmetries, Conserved Quantities and Interactions ............................................94 3.3.1 Noether’s First Theorem and Conservation Law s ..................................... 95 3.3.2 Noether’s Second and Third Theorems and Interactions ...........................99 3.3.3 Symmetry, Ambiguity of Representation and Indeterminism ................ 103 3.4 Local Symmetries Giving Rise to Interactions .....................................................105 3.4.1 Spacetime, Matter, Interactions and Numbers .............................. 108 Contents vii 3.4.2 Yang-Mills Theories: the Weak and the Strong .......................................116 3.5 Constrained Hamiltonian Systems or Fibre Bundles? ........................................ 122 3.5.1 Explaining Fibre Bundles ..........................................................................124 3.5.2 Science With Numbers, but not Necessarily With Coordinates ..............139 4 Scientific Explanation: Four Ways to the Aharonov-Bohm Effect.. 142 4.1 Scientific Explanation ........................................................................................... 143 4.1.1 Holistic vs C ausal...................................................................................... 146 4.2 Abstraction, Approximation and Idealization: the Laws of Physics do not Lie, it’s Just that the Mappings are Not-All-Inclusive and Non-Exact ...........................................................................156 4.2.1 Galileo and the Problem of Accidents .......................................................156 4.2.2 Models and Analogies in Science ..............................................................160