Playing with pretty mathematics: Dirac and Mathematical Beauty (2)

The last post in this series explored the question of whether Dirac was “Eddingtonian” in the sense Olivier Darrigol argues for. Did Dirac follow Eddington’s metaphysics and methodological principles? From an analyses of Darrigol’s evidence and a 1931 paper from Dirac, the conclusion was overall “no,” particularly about the metaphysics. The case is stronger when just considering Dirac’s methodological views. This post goes into more detail about Dirac’s methodology and his aesthetics. It takes a closer look at Dirac’s famous 1928 paper that introduced his relativistic theory of the electron to the world.

According to the contemporary report of Charles G. Darwin, referring to Dirac’s 1928 theory, “Dirac’s guiding principle [was] that the ‘Hamiltonian equation’ must be linear” (Darwin 1928). Darwin was referring to the fundamental equations of Dirac’s approach: the Hamiltonian of classical mechanics that represented the total energy of a system of particles (or fields). This was the formalism upon which Dirac’s “general transformation” theory of quantum mechanics was based. (Dirac invented this theory in 1926.) And certainly the linearity of this equation mattered to Dirac. Dirac wrote the Hamiltonian for a general system in this theory as

(H − W)ψ = 0.

But was Darwin right to focus on linearity as the core of Dirac’s method?

Let’s take a look at what Dirac actually said about his methodology in this famous paper. According to Dirac’s published papers, the reason the the Hamiltonian must be linear is that linearity preserved a sense of causality: “so that the wave function at any time determines the wave function at any later time.” Dirac continued with a second reason. “The wave equation of the relativity theory must also be linear in W if the general interpretation is to be possible.” So causality and consistency with Dirac’s general transformation theory were the reasons undergirding the formal requirement of linearity.

We can further inquire as to why causality and consistency with the general theory were important. For this we may stay within a close reading of Dirac’s 1928 introduction of his theory. Dirac started the paper with the observation that present theories of quantum mechanics—which included the presupposition that the electron was a point-like object—were not empirically adequate. They could not account for “duplexity” phenomena that Goudsmit and Uhlenbeck had attributed to the electron having spin angular momentum in 1925 and more broadly with a postscript by Bohr in 1926.

But there were challenges to the picture of a spinning electron, known since Lorentz: given known limits on the radius of the electron and the required angular momentum, each electron would have to spin faster than the speed of light. “The question remains as to why Nature should have chosen this particular model for the electron instead of being satisfied with the point-charge” (Dirac 1928). Dirac continued with a statement about the incompleteness of the existence theories. If there were some incompleteness, and it could be rectified, if the theory could be made whole, perhaps one would not need recourse to the spinning electron model. One could continue with the point charge picture.

One would like to find some incompleteness in the previous methods of applying quantum mechanics to the point-charge electron such that, when removed, the whole of the duplexity phenomena follow without arbitrary assumptions. In the present paper it is shown that this is the case, the incompleteness of the previous theories lying in their disagreement with relativity, or, alternatetively [sic], with the general transformation theory of quantum mechanics. (Dirac 1928)

Here Dirac emphasized the importance of consistency, of seeing physical theory as part of a whole.

We can see here the emergence of an aesthetics of physics: proper theories should fit together. The theory of the electron should be relativistic, should fit with the rest of quantum mechanics, and should fit with Bohr’s correspondence principle.

Simplicity also came into play: “It appears that the simplest Hamiltonian for a point-charge electron satisfying the requirements of both relativity and the general transformation theory leads to an explanation of all duplexity phenomena without further assumption” (Dirac 1928)

Here we can see Dirac’s developing aesthetics of science: physics should be a whole, where different theories fit together according to rules such as Bohr’s correspondence principle.* Relativity, atomic theory, and the general transformation theory of quantum mechanics should be united. Nature should be simple, too. The electron was the simplest type of object—a point charge. Thus Dirac’s aesthetics extended to epistemological views on the nature of theories and ontological views on the constituents of nature. But in his famous 1928 paper, we do not yet see the emergence of a fully-articulated principle of mathematical beauty.

In the next post in the series I’ll take a closer look at Dirac’s concept of mathematical beauty as it was developed in the late 1930s, and put forth Kragh’s views on the subject.

Thanks for reading!

Aaron

*Bohr’s correspondence principle, for Dirac, was the principle that in the limit of large quantum numbers, a quantized theory should give the same results as a classical theory.

 

 

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2 Responses to Playing with pretty mathematics: Dirac and Mathematical Beauty (2)

  1. Graham Farmelo says:

    Much enjoyed this, and Part 1, too! Graham Farmelo

    • Thanks Graham! I’m a little limited because I don’t want to go through the whole publishing permissions process from the archive just for blog posts. This will eventually be part of my dissertation though.

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