Nounou, Antigoni
(2002)
Gauge theories: A case study of how mathematics relates to the world.
PhD thesis, London School of Economics and Political Science.
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 during 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-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 examined within the fibre bundle context provides a narrower yet very fruitful case study.
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