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CRC Press eBooks

Dirac

Jonathan Allday

Published Sep 21, 2022• 0 citations
Publisher

Abstract

Between 1925 and 1933, Paul Adrian Maurice Dirac made a series of crucial contributions to the development of quantum mechanics and set much of the agenda for fundamental research through the following decades. Any one of his insights would have been sufficient to ensure his lasting recognition in physics, but together they amount to the work of a genius. In 1925 Dirac published a paper building on Heisenberg’s outline of a mathematical structure for quantum theory. Dirac’s next two papers came out in 1927 and developed the theory to describe the interaction between electromagnetic waves and atoms. This work set the foundations of quantum field theory and developed the technique called ‘second quantization’. Between 1932 and 1933, Dirac wrote three more papers on quantum field theory, managing to cast it in a fashion that made its consistency with relativity apparent.

DOI
10.1201/9781003225997-24

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Between 1925 and 1933, Paul Adrian Maurice Dirac made a series of crucial contributions to the development of quantum mechanics and set much of the agenda for fundamental research through the following decades. Any one of his insights would have been sufficient to ensure his lasting recognition in physics, but together they amount to the work of a genius. In 1925 Dirac published a paper building on Heisenberg’s outline of a mathematical structure for quantum theory. Dirac’s next two papers came out in 1927 and developed the theory to describe the interaction between electromagnetic waves and atoms. This work set the foundations of quantum field theory and developed the technique called ‘second quantization’. Between 1932 and 1933, Dirac wrote three more papers on quantum field theory, managing to cast it in a fashion that made its consistency with relativity apparent.
Abstract Following the path by which humanity learned quantum mechanics can lead to an improved understanding of the theory and the origins of its perceived limitations. Our goal is to retrace the development by investigating primary sources, including original published papers and letters, with attention to their timing and influence. Quantum mechanics began in 1900 with the introduction of Planck’s constant, which lead to the “old” quantum theory and Bohr’s model of the atom. With the introduction of matter waves in 1924, a second more intense period began. By 1928 the complete fundamental structure of quantum mechanics was established with the introduction of the Schrödinger equation, the proof of its equivalence to Heisenberg’s matrix mechanics, and the introduction of the relativistic Dirac equation. The theory retains the same essential form today, fueling the technological revolution that began in the last century. We culminate with an introduction to quantum information and computing. This completely new physical scientific paradigm was developed largely in Germany during a period of enormous social, economic, and political upheaval—in a hostile intellectual environment—that parallels the overturning of classical physics. It is striking that all the problems debated today were immediately recognized by the founders of the theory and instructive to see how they dealt with the various issues. We emphasize the centrality of the often neglected second quantization form of the theory for questions of interpretation and give a detailed examination of von Neumann’s widely misunderstood hidden variable theorem.
The year 2025 marked the centennial of quantum mechanics, inaugurated by Heisenberg's matrix formulation and the foundational contributions of Pauli, Schrodinger, and Dirac. Concurrently, 2026 marks the centennial of the Klein - Gordon equation, the second-order relativistic wave equation from which both the Schrodinger and Dirac equations were derived. This article supplements the recent review published in J.Phys. A: Math.Theor.,58 (2025) 053001 by providing a more detailed examination of the formative period 1925 - 1928, with particular attention to contributions that have received insufficient recognition in the standard narrative. We reconstruct Kramers' independent derivation of the Dirac equation - obtained essentially simultaneously with Dirac's own result yet unpublished for seven years - and discuss its relation to Van der Waerden's group-theoretical approach. The role of Charles Galton Darwin in elucidating the physical content of the Dirac equation is also highlighted. In addition, we present two modern derivations not catalogued in the earlier review: one based on Operational Dynamical Modeling, which deduces the Dirac equation from relativistic Ehrenfest relations and the canonical commutation algebra, and one rooted in the Madelung hydrodynamic formulation. Three broad periods of quantum theory development -- foundational, consolidation, and the modern era of quantum information -- are briefly surveyed.
Göttingen and Copenhagen were the undoubted capitals of the new discipline of quantum mechanics. The expertise in experimental and mathematical physics and in pure mathematics made Göttingen the epicentre of the revolution which was taking place in the mathematical physics of quanta. Whilst this was to remain the case for the next few years, other actors soon appeared on the scene who were to contribute to Born's ‘tangle of interconnected alleys’. What was truly remarkable was how quickly the different approaches to the problems of quantum theory were developed and the rapid assimilation of all of them into a coherent and self-consistent theory of quantum mechanics. Whilst the theory itself was completed relatively quickly, the understanding of its physical content was to take many more years. The new players on the scene included Paul Dirac at Cambridge, Erwin Schrödinger in Vienna and Norbert Wiener at the Massachusetts Institute of Technology. Each of them brought quite new approaches to the development of quantum theory – their innovations were to supersede the matrix mechanics of Born, Heisenberg and Jordan, but there can be no doubt that the success of that theory indicated clearly the route ahead. They were however to involve the introduction of new mathematical techniques into the description of quantum phenomena. Dirac's approach to quantum mechanics Paul Dirac was trained as an electrical engineer at Bristol University, but he had a very strong mathematical bent. He was a solitary character who was notoriously quiet and self-effacing. He simply worked things out on his own.
At the beginning of the twentieth century, it became clear that microscopic particles do not obey Newton's laws of classical mechanics. The idea of light quanta was at odds with the established wisdom that light is a wave phenomenon, of course, but no other explanation of the photoelectric effect seemed possible. The next step in the development of quantum mechanics was taken by Niels Bohr, who introduced postulates about the discreteness of the energy of the bound electrons in an atom similar to the discreteness of the energy of light quanta. Modern quantum mechanics was founded by Werner Heisenberg, Max Born, Erwin Schroedinger, Paul Adrian Maurice Dirac, Wolfgang Pauli, Enrico Fermi, and others. In quantum mechanics, it is in general fundamentally impossible to determine the exact outcome of an experiment; one can only determine the probability of any given outcome.
I conclude the first part of this book by carrying the story up to 1928, to the lightning flash in Eddington's mind produced by Dirac's paper on the wave equation of the electron. Setting out the context and Eddington's thinking about it is a very different matter for quantum mechanics and relativity. Both theories are commonly expressed in austere mathematical language. In the case of relativity the basic ideas behind the mathematics are now well understood and can be set out with little complication. These ideas were mostly already clear to Eddington. But for quantum mechanics the mathematics, yoked to the wealth of experimental results, drove the development of the theory at a breakneck speed, mostly without anyone pausing for deeper understanding. It is indeed only in the last decade that it has become generally accepted that quantum mechanics still lacks any coherent interpretation. To explain Eddington's context it is therefore necessary to say something about the mathematical formalism. It is true, as will become clear, that Eddington was a little more sceptical than most, but by and large the mathematics drove him as it did those directly working in the theory. I begin, then, with a description of that part of the early (pre-1925) history of quantum mechanics that was in Eddington's mind in 1923. Then I explain what happened in 1925–6 and Eddington's reaction to it. The chapter concludes with Dirac's relativistic wave equation of 1928 and Eddington's further reaction to that. The old quantum theory What is now called the old quantum theory arose at the turn of the century. Two related experimental contradictions with classical physics played important roles in this.