Sir Arthur Eddington and the Foundations of Modern Physics Ian T. Durham Submitted for the degree of PhD 1 December 2004 University of St. Andrews School of Mathematics & Statistics St. Andrews, Fife, Scotland 1 Dedicated to Alyson Nate & Sadie for living through it all and loving me for being you Mom & Dad my heroes Larry & Alice Sharon for constant love and support for everything said and unsaid Maggie for making 13 a lucky number Gram D. Gram S. for always being interested for strength and good food Steve & Alice for making Texas worth visiting 2 Contents Preface … 4 Eddington’s Life and Worldview … 10 A Philosophical Analysis of Eddington’s Work … 23 The Roaring Twenties: Dawn of the New Quantum Theory … 52 Probability Leads to Uncertainty … 85 Filling in the Gaps … 116 Uniqueness … 151 Exclusion … 185 Numerical Considerations and Applications … 211 Clarity of Perception … 232 Appendix A: The Zoo Puzzle … 268 Appendix B: The Burying Ground at St. Giles … 274 Appendix C: A Dialogue Concerning the Nature of Exclusion and its Relation to Force … 278 References … 283 3 I Preface Albert Einstein’s theory of general relativity is perhaps the most significant development in the history of modern cosmology. It turned the entire field of cosmology into a quantitative science. In it, Einstein described gravity as being a consequence of the geometry of the universe. Though this precise point is still unsettled, it is undeniable that dimensionality plays a role in modern physics and in gravity itself. Following quickly on the heels of Einstein’s discovery, physicists attempted to link gravity to the only other fundamental force of nature known at that time: electromagnetism. Both Hermann Weyl (1885-1955) in 1918 and Arthur Stanley Eddington (1888-1944) in 1921 developed field theories that in essence were early attempts at unification employing the new concept of the geometrisation of physics. Also in 1921 the German theoretical physicist Theodor Kaluza (1885–1954) attempted this by extending Einstein’s field equations to five dimensions (Kaluza 1921). Essentially he postulated a five dimensional Riemannian space by adding to the four known dimensions a fifth one where particles always followed closed paths. Both electromagnetism and relativity were contained within this grand scheme but it did not contain any of the relatively young quantum theory leaving most physicists to realize it bore no resemblance to reality. The Swedish theoretical physicist Oskar Klein (1894–1977) added the quantum aspect to Kaluza’s theory in 1926 (Klein 1926) and similar subsequent theories have been loosely grouped into the category of Kaluza-Klein Theories. In Klein’s theory the fifth dimension was unobservable whereas Kaluza’s was macroscopic in size. This unobservable dimension’s physical reality was akin to a quantity that was conjugate to the electrical charge. In this way Klein also sought to explain Planck’s quantum of action. The lack of sufficient mechanisms for testing such an idea and finding a practical application for the theory kept Kaluza-Klein theories largely out of the mainstream until their revival in the 1970s. This did not stop many scientists from studying unification, however. Einstein essentially devoted the final thirty years of his life to it while Eddington devoted the last fifteen. 4 Unification today is widely regarded as the Holy Grail of physics. Physicists have successfully unified the strong, weak, and electromagnetic forces with special relativity under the guise of quantum field theory, but any definitive link to gravity or general relativity remains elusive. String theory is currently the mainstream theory of choice for this but remains unproven. Unifying gravity and quantum theory then must be at the heart of this quest, and theories of quantum gravity have been at the forefront of research in physics for nearly forty years. But attempts at such a unification actually date to at least 1928, when Paul Adrien Maurice Dirac (1902-1984) derived his relativistic equation for the electron (Dirac 1928a and 1928b). Eddington, disappointed that Dirac’s equation did not appear in tensor form,1 sought to reformulate Dirac’s work in 1929-1930 to put quantum theory into the language of relativity, i.e. tensor calculus (Eddington 1929). This led to the development of several theories of cosmology in the 1930s developed primarily by Eddington, Dirac, and Edward Arthur Milne (1896-1950). Several unification theories that did not directly address cosmological questions were also developed at this time. Eddington’s work rested on the premise that quantum mechanics and relativity could be united under a common framework both in the formalism and the philosophy. He began by analyzing uncertainty and became convinced that its introduction into physics heralded such a monumental change that every physicist needed to consider its philosophical implications in their work. He clearly opposed the Einstein-Podolsky- Rosen (EPR) interpretation saying that any scientist who accepted the idea of hidden variables as an explanation of indeterminacy “wants shaking up and waking” (Eddington 1935b, p. 84). He saw this fundamental indeterminacy as the foundation on which to build a unified theory of physics. Though his work on uncertainty is clearly debatable in its validity it actually foreshadowed some later developments in physics, including the need for a quantum- mechanical standard of length. This led him to the next major component of his work: an analysis of The Pauli Exclusion Principle. He develops exclusion into a richer framework that serves, in combination with uncertainty, as the basis of later versions of 1 Charles Galton Darwin (1887-1962) was the first to note that Dirac’s equation was not in tensor form; see C.G. Darwin, “The Wave Equation of the Electron,” Proceedings of the Royal Society [A] 118 (1928), 654- 680. 5 his complete theory, which held that physical events depend solely on dimensionless ratios. Later, this idea was taken up by Dirac in proposing his Large Numbers Hypothesis (Dirac 1937). Eddington’s work hinted at some of the underlying principles of modern theories including some aspects of grand unified theories (GUTs) and string theory. In fact, as this monograph discusses, Fundamental Theory, as it was posthumously titled, is a very early attempt at quantum field theory that quite remarkably predicts future advances in that field. It’s greatest relevance to modern science is in its unique interpretation of the foundational aspects of modern physics and its philosophical implications for the underlying structure of the physical world. In fact Eddington’s work has seen somewhat of a renaissance in recent years and has been studied in greater detail by a growing list of scholars. I first discovered this aspect of Eddington’s work when reading a brief account of his cosmology in Helge Kragh’s Quantum Generations. I had known of Eddington from my work in astronomy for his many mainstream accomplishments, but this brief encounter with his unorthodox worldview turned my research from work on general problems in cosmology to addressing truly foundational problems in modern physics. The results of my initial foray into his work on uncertainty, that reveal a deep distrust of standard measurement techniques and a worldview incorporating uncertainty into the very fabric of space-time, led naturally to his extension of the Exclusion Principle. One of the many amazing insights that continued to fuel my work was the fact that Eddington modifies the interpretation of this fundamental principle and extracts results from the new interpretation that point to a deeper philosophical meaning behind exclusion. This presented me with several fundamental questions about the nature of exclusion: could it be more than a relatively straightforward quantum phenomenon; could it reside in that fundamental area inhabited by the conservation laws, the forces of nature, and the uncertainty principle, and, if it does, what does this mean for modern physics? My conclusions in this endeavour have led to several extended pieces of research in fundamental physics that, in itself, emphasizes the surprising relevance of his work despite its chequered past. 6 Examining these questions is not only important for a complete understanding of exclusion and Eddington’s unorthodox worldview, but they are also at the heart of the relationship between science and philosophy. When analyzed in full compliment with his work on uncertainty, the whole of his thinking begins to unravel itself. To say that Eddington went from being one of the subjects of my dissertation to being the only subject of my dissertation does not do proper justice to his influence on me. Delving into the deep questions of uncertainty and exclusion, particularly in the context of unification and the nature of the universe itself, his work has led me into many new uncharted areas and has helped to focus my general research interests onto more fundamental and foundational questions. But aside from my personal interest in the subject, Eddington’s philosophical and even some of his mathematical work is often overlooked by modern scholars. Bohm, Fred Hoyle (1915 - 2001), and Hermann Bondi (b. 1919) are well- known despite their controversial theories making up a large portion of their body of work, while Eddington, whose diverse work included the first observational verification of general relativity and the nearly single-handed creation of the field of stellar structure, tends to be overlooked and even marginalized.2 It was this historical treatment that contributed to my focus solely on Eddington. My research concentrated primarily on what comprises the first six chapters of Fundamental Theory and is often referred to as his statistical theory. These six chapters focus their efforts on reinterpreting and applying Heisenberg’s Uncertainty Principle and Pauli’s Exclusion Principle. They form the philosophical and interpretive basis of his entire program of research. I analyzed each in detail both philosophically and mathematically in search of any morsel of truth or potential application to modern physics.