Last month, I was at Foundations 2018 in Utrecht. It is one of the biggest conferences on the foundations of physics, bringing together physicists, philosophers, and historians of science. A talk I found particularly interesting was that of Alexander Blum, from the Max Planck Institute for the History of Science, entitled Heisenberg’s 1958 Weltformel & the roots of post-empirical physics. Let me briefly summarize Blum’s fascinating story.

In 1958, Werner Heisenberg put forward a new theory of matter that, according to his peers (and to every physicist today) could not possibly be correct, failing to reproduce most known microscopic phenomena. Yet he firmly believed in it, worked on it restlessly (at least for a while), and presented it to the public as a major breakthrough. How was such an embarrassment possible given that Heisenberg was one of the brightest physicists of the time? One could try to find the answer in Heisenberg’s personal shortcomings, in his psychology, in his age, perhaps even in his deluded attempt at making a comeback after the sad episode of his work on the nazi Uranprojekt during World War II. Blum’s point is that the answer lies, rather, in the very peculiar nature of modern physical theories, where mathematical constraints strongly guide theory building.

Heisenberg’s theorizing was allowed by the strong constraints that quantum field theory (QFT) puts on consistency. His goal was to find the ultimate theory not with the help of empirical results (like those coming from early colliders), but from pure theory, with one principle in addition to those of QFT. His idea was to ask for radical monism: deep down, there has to be just one fundamental featureless particle. It has to be spin 1/2 so that integer spin particles can be effectively obtained as bound states. The only non-trivial option is then to add a non-renormalizable quartic interaction term to the Dirac free Lagrangian.

Heisenberg’s Weltformel at the 1958 Planck Centenary in West Berlin. Source: DPA

With only a single fundamental self-interacting spin 1/2 particle, the theory seems far removed from the physics we know. Sure, it could be that all the physics we know, with leptons, hadrons, and electromagnetic forces, could be obtained effectively, from non-trivial bound states made from this fundamental particle. It could be, but most likely it is not the case and so Heisenberg’s crazy conjectures should be easy to disprove. But here comes the catch: the theory is non-renormalizable, and there existed no reliable way to extract predictions from it at the time. It is impossible to falsify something that is not even consistent in the weakest sense available. Heisenberg could argue: maybe the theory is just non-renormalizable at the perturbative level, maybe the singular behavior of the propagator is just a feature of the free theory… Heisenberg could exploit the fact that there were strong doubts about the consistency of QFT anyway, with the Landau pole, and Dyson’s argument about the necessary divergence of perturbative expansions.

Interestingly, it is partly to conclusively disprove Heisenberg’s proposal that rigorous approaches to quantum field theory were developed. Working at the same institute as Heisenberg but deeply skeptical of his theory, Harry Lehman, Kurt Symanzik, and Wolfhart Zimmermann laid the basis of axiomatic field theory. The Källen-Lehmann  (K-L) spectral representation theorem, showing as a corollary that an interacting propagator cannot be more regular than a free propagator, provided a no-go theorem disproving Heisenberg’s speculations.

But Heisenberg could fight back. It was understood at the time that Quantum Electrodynamics contained (at least in some formulations) quantum states with negative norm, the so called “ghosts”. Maybe such ghosts could be exploited to bypass the K-L theorem, yielding cancellations of divergences in the expression of the interacting propagator. This speculation lead to an intense fight with Wolfgang Pauli in 1957, the “battle of Ascona”. Pauli argued that ghosts, if exploited in this fashion, would never “stay in the bottle”, and would necessarily make the theory inconsistent. After 6 weeks of intense work, Heisenberg came up with a toy model combining a unitary S-matrix (hence consistent in the sense required) and containing ghosts.

So Heisenberg’s theory was not easy to unequivocally kill, which of course does not make it correct. Heisenberg tried extracting predictions from his theory using new (unreliable) approximation methods, giving essentially random results. Hence he had no option but to fall back on beauty, the only justification for his theory being its radically simple starting point. Nothing ever came from his line of research which no one ever pursued after him. Blum ended his talk with a timely warning: One still needs to beware of falling into Heisenberg’s trap.

In a previous post, I made the simple point that theoretical physicists put too much trust in notions of beauty and mathematical simplicity because of survival bias: we remember the few instances in which it worked, but forget the endless list of neat constructions by excellent physicists that eventually proved empirically inadequate. I did not know of Heisenberg’s theory, but I gladly add it to the list.

Blum’s talk was a teaser for an article he told me is about to be finished. More generally, his study of Heisenberg’s Weltformel is the first step in an inquiry into theorist’s attempts at coming up with a theory of everything from post-empirical arguments (see a well explained description of his group program). This is a timely research program.

One does not need to think too hard to see the obvious parallel between Heisenberg’s story and current attempts at coming up with a theory of everything (or of quantum gravity). One easily finds popular theories that are not manifestly fitting known physics but also not obviously not fitting known physics. They could be correct, but we cannot know for lack of proper non-perturbative tools. Should we trust them only because they are so hard to conclusively disprove and obey some (quite subjectively) appealing principles?

Update 30/05/2019: There is now a book about this story.

Anil Ananthaswamy

I have really enjoyed Anil Ananthaswamy’s latest book: Through two doors at once: The Elegant Experiment That Captures the Enigma of Our Quantum Reality. It is very well written and one reads through it like a novel. But, most importantly, it gets the physics right, and the subtleties are not washed away with metaphors. Accurate and captivating, the book strikes a balance rarely reached in popular science books.

The foundations of quantum mechanics is a difficult branch of physics, and almost every narrative shortcut that was invented to convey its subtlety is, strictly speaking, a bit wrong. Further, foundations is an unfinished branch of physics: different group of experts disagree about what the main message of quantum mechanics is and what should be done to make progress in understanding. This makes it hard to popularize the subject without writing incorrect platitudes or pushing one orthodoxy.

Anil’s strategy is to use the simplest experiment illustrating quantum phenomena: the double slit experiment. He discusses the results and shows why they are so counter-intuitive. However, the simple double slit experiment is not enough to go to the bottom of the mystery. Anil thus very progressively refines the experimental setup to gradually add the subtleties that prevent naive stories from explaining away the weirdness of quantum theory. As in a police investigation, Anil interviews the experts of the main interpretations of quantum mechanics, and guides the reader through the explanations they give for each setup. The reader can then decide for herself which story she finds most appealing.

Crucially, I think the different interpretations are presented fairly. Anil does not take a side. I personally much prefer “non-romantic and realist” interpretations of quantum theory: I find accounts of the world where stuff simply moves, be it with non-local laws of motion, far more convincing than alternatives (where there are infinitely many worlds, or where “reality” has a subjective nature). The “realist” view is well represented in the book (which is rare, because it is not “hype”), but I was not annoyed by the thorough discussion of the other possibilities. More radical proponents of one or the other interpretation may however be annoyed by this attempted neutrality.

Anil’s writing style is very enjoyable. He does not make the all too common mistake of using cheap metaphors which are dangerous in the context of quantum mechanics where they provide a deceiving impression of depth and understanding. In this book, you actually learn something. Sure you do not become an expert in foundations, but you get an accurate sense of what motivates researchers in the field. This is both nice in itself, and if you want to keep on digging with a more specialized book. Even though I already knew the technical content of the book, I found the inquiry captivating. I definitely recommend Through two doors at once, especially to my friends and family who want to quickly yet genuinely understand the sorts of questions that drive me.

Disclaimer: I have provided minor help for the rereading of an almost finished draft of the book.