Sommerfeld’s Miracle: The Ultimate Challenge to Scientific Realism

Peter Vickers

A ‘no miracles’ argument is still prevalent in the scientific realism debate, even if a lot has changed since Hilary Putnam’s formulation of it, and even if the word ‘miracle’ is generally avoided. For example, realists think that if the most central ‘working’ parts of a scientific theory were not even approximately true (for any serious theory of ‘approximate truth’), then it would be incredibly unlikely (‘miraculous’) for that theory to deliver successful novel predictions with ‘perfect’ quantitative accuracy (e.g. to several significant figures). It would be like perfectly predicting the time and position of the next solar eclipse based on a completely false (not even approximately true) model of how the sun, moon, and earth interact. Here it is appropriate to talk in terms of ‘counterexamples’ to scientific realism: any historical case where a scientific theory delivered ‘perfect’ predictions but where the central working parts of the theory are now thought to be radically false would be a very serious thorn in the side of nearly every contemporary scientific realist position.

Fig 2

Arnold Sommerfeld (left) and Paul Dirac (right)

How many such ‘counterexamples’ are there in the history of science? Are there any at all? For more than twenty years, the main historical challenges to scientific realism were those on Larry Laudan’s famous ‘list’. Nearly all of these examples are now quite widely considered (by realists at least!) to be somewhat less dramatic for realism then they perhaps initially appeared: either the success achieved by the false theory is only quite impressive, or else the confirmation resulting from that success can reasonably be seen to extend to only certain ‘working’ parts of the theory, which are indeed approximately true. So why weren’t there tougher historical challenges on Laudan’s list, cases where the success was hugely impressive and where the most central parts of the theory were not even approximately true? Is it because such cases don’t exist, because the scientific realist is basically right in her conviction that there are no ‘miracles’ in science?

I know of one such ‘miracle’. Even physicists, with no knowledge of the scientific realism debate (including Heisenberg) have been inclined to call it a ‘miracle’. It is widely referred to in the physics literature as a remarkable coincidence (‘perhaps the most remarkable numerical coincidence in the history of physics’, according to Ralph Kronig writing in 1960) and one physicist has even called it a ‘cosmic joke’. Without any intention, these remarks directly challenge the central ‘no miracles’ or ‘no cosmic coincidences’ argument for scientific realism.

Fig 1

Figure 1. Part of the hydrogen emission spectrum: only light of very specific frequencies is emitted from hydrogen. The fine structure is not visible here.

The case concerns Arnold Sommerfeld’s 1916 derivation of the ‘fine structure formula’, which predicts the frequencies of the spectral lines of hydrogen. The theory leading to the formula built on Bohr’s much simpler model of the hydrogen atom, which had been able to account for the ‘gross’ structure of the hydrogen emission spectrum (specific frequencies of light that are emitted from hydrogen; see Figure 1). But it turns out that when one looks at this gross structure through a high resolution spectroscope, each line is actually not a single line, but two or more lines grouped very closely together. This is the ‘fine structure’, a phenomenon Bohr’s model was incapable of explaining.

Between 1913–16, Sommerfeld developed Bohr’s model, introducing elliptical orbits, a second quantum number, and relativistic corrections. These were all well-motivated de-idealizations, and indeed Bohr himself had attempted (without success) to de-idealize in this way. The de-idealizations led to the fine structure formula, which, it turns out, perfectly predicts the fine structure of the hydrogen spectral lines. That is, it gives exactly the same formula as the fully relativistic Dirac quantum mechanics of 1928, and it gives better predictions for this phenomenon than basic (1925) Schrödinger–Heisenberg quantum mechanics. Since the success is so dramatic, the realist would dearly like to explain the success in terms of the (approximate) truth of the central working parts of Sommerfeld’s theory (dismissing the radically false parts of the Sommerfeld theory as ‘idle wheels’). However, Sommerfeld reached his formula by considering the velocity variations for an electron in an elliptical orbit, and the variation of electron mass with velocity as dictated by special relativity. There seems to be a clear dependence relation here between the final formula and (what are now taken to be) radically false assumptions. As Griffiths notes in his popular textbook Introduction to Quantum Mechanics: ‘It’s not even clear what velocity means in QM’.

Despite this drama, the realist is not without options. First, consider the suggestion that a realist commitment to Sommerfeld’s theory was in fact not warranted, for two separate reasons: (i) although Sommerfeld’s success vis-à-vis the fine structure formula was astonishing, the theory also had some significant failures (e.g. spectral line intensities), and a realist inclined towards Bayesian updating might insist that after iterating one’s degree of belief in the theory given all available evidence, one’s final degree of belief might not be very high; (ii) the scientific landscape was changing rapidly at that time (1916–25), and perhaps a cautious realist should not make a realist commitment to any theory, however successful, until a few years have passed and the dust has settled.

However, concerning (i), usually a doxastic commitment is thought to be warranted in the face of (‘normal’) anomalies so long as the successes are good enough. And on (ii), Sommerfeld’s success came in 1916, and the theory vis-à-vis the hydrogen atom still looked like the ‘complete truth’ to mainstream physicists nine years later in 1925 (as detailed by Helge Kragh in his excellent paper, ‘The fine structure of hydrogen and the gross structure of the physics community, 1916-26’). They certainly thought they had been cautious enough, and the realist might be wary of insisting that scientists made a big mistake in their epistemic judgements.

What if the realist accepts that a realist commitment is necessary? Can she hold out hope that a certain sort of doxastic commitment, to certain features of Sommerfeld’s theory, can in fact be sustained? It is hard to see how a standard selective realist move could work, given the radical discontinuities between the two theories. Lawrence Biedenharn’s paper, ‘The “Sommerfeld Puzzle” Revisited and Resolved’, sounds very promising for the realist(!), but it is far from clear that the ‘resolution’ in this paper is what the realist is after. For example, Biedenharn certainly does not demonstrate that Sommerfeld’s derivation can succeed without reference to continuous worldline orbital trajectories (something totally absent from Dirac’s theory), nor does he demonstrate how Dirac’s derivation can succeed without reference to electron spin (something totally absent from Sommerfeld’s theory). And these are perhaps the two most important things on the realist’s to-do list here.

Apart from Biedenharn, there is one other prominent attempt in the physics literature to resolve the puzzle. Stefan Keppeler (‘Die “alte” Quantentheorie, Spin präzession und geometrische Phasen) makes use of a semi-classical approach to Dirac quantum mechanics, and uses this to derive the fine structure formula. He then makes a comparison with Sommerfeld’s derivation, and concludes that Sommerfeld succeeded because two missing terms happen to cancel each other out. Thus Keppeler favourably quotes Yourgrau and Mandelstam, who in 1951 had already written, ‘Sommerfeld’s explanation was successful because the neglect of wave mechanics and the neglect of spin by chance cancel each other in the case of the hydrogen atom’. If this is right, it wouldn’t be very satisfying for the realist: on Keppeler’s explanation, the success remains ‘miraculous’, since it is miraculous that the two errors exactly cancel each other. Even the most conscientious selective realist certainly would go wrong in her doxastic commitments, since Sommerfeld’s classical approach is definitely not ‘idle’ for Keppeler. It’s just somehow ‘pure luck’ that Sommerfeld’s approach gives the right results given that his mechanics is wrong and that he neglects the effect of electron spin.

That isn’t the end of the story, however, since Biedenharn’s analysis indicates a possible mistake in Keppeler’s reasoning. He insists that ‘Sommerfeld’s success is not at all a matter of blind luck’ and writes: ‘It is now clear that the argument of Mandelstam and Yourgrau (that Sommerfeld succeeded “because the neglect of wave mechanics and the neglect of spin effects by chance cancel each other”) cannot be correct’. The reason is that Biedenharn takes his analysis to have shown that ‘wave mechanics per se makes no change in the answer’, in which case this can’t be one of the two errors that cancel each other out.

Thus a hope remains for the realist that the right application of selective realism might yet explain the puzzle in realist terms. Biedenharn’s claim that there is ‘the closest possible correspondence between the two (suitably formulated) calculations’ and a ‘most remarkable and detailed correspondence between the Sommerfeld procedure and the quantal solution’ suggests, perhaps, that some sort of structural realism might fit this case rather well. That is, if the realist restricts her doxastic commitment in this case to ‘high level structure’, it might be possible to be a realist here without committing to ‘radically false’ claims about the underlying nature of hydrogen. Of course, the standard (causal) explanation of the fine structure within modern quantum mechanics makes essential use of electron spin, but this doesn’t rule out an alternative, non-causal (structural) explanation of the fine structure where spin plays no role.

Whatever the case, it is hard to get away from the idea that there is something miraculous about Sommerfeld’s success. For example, supposing the basic structure of Sommerfeld’s theory (or at least the working part of Sommerfeld’s theory) is indeed identical to some underlying structural features of Dirac quantum mechanics, it would still be a miracle that Sommerfeld managed to hit upon that structure, given the purely classical, concrete physical ideas that dictated the content of his theory. But in fact this isn’t a problem for the scientific realist; it just goes to show that the ‘no miracles argument’ is a misnomer, since miracles of a kind are perfectly compatible with realism. What the realist needs is a principled, non-ad-hoc restriction on the scope of confirmation of a theory achieving significant predictive success. So long as the realist does not end up doxastically committing to something that is radically false, she is in the clear. And, notwithstanding appearances to the contrary, that does remain possible vis-à-vis the Sommerfeld miracle. The realist is surely bruised, but not beaten. And if the realist can indeed turn this case around, that would be quite something.

Peter Vickers
Durham University