The finetuned cube

The standard model of parenthood.

The kids are almost three years now and I spend a lot of time picking up wooden building blocks. That’s good for your health in many ways, for example by the following brain gymnastics.

When I scan the floor under the couch for that missing cube, I don’t expect to find it balancing on a corner - would you? And in the strange event that you found it delicately balanced on a corner, would you not expect to also find something, or somebody, that explains this?

When physicists scanned the LHC data for that particle, that particle you’re not supposed to call the god-particle, they knew it would be balancing on a corner. The Higgs is too light, much too light, that much we knew already. And so, before the LHC most physicists expected that once they’d be able to see the Higgs, they’d also catch a glimpse of whatever it was that explained this delicate balance. But they didn’t.

It goes under the name ‘naturalness,’ the belief that a finely tuned balance requires additional explanation. “Naturally” is the physicist’s way of saying “of course”. Supersymmetry, neatified to Susy, was supposed to be the explanation for finetuning, but Susy has not shown up, and neither has anything else. The cube stands balanced on the corner, seemingly all by itself.

Of course those who built their career on Susy cross-sections are not happy. They are now about to discard naturalness, for this would mean Susy could hide everywhere or nowhere, as long as it’s not within reach of the LHC. And beyond the LHC there’s 16 orders of magnitude space for more papers. Peter Woit tells this tale of changing minds on his blog. The denial of pre-LHC arguments is so bold it deserves a book (hint, hint), but that’s a people-story and not mine to tell. Let me thus leave aside the psychological morass and the mud-throwing, and just look at the issue at hand: Naturalness, or its absence respectively.

I don’t believe in naturalness, the idea that finetuned parameter values require additional explanation. I recognize that it can be a useful guiding principle, and that apparent finetuning deserves a search for its cause, but it’s a suggestion rather than a requirement.

I don’t believe in naturalness because the definition of finetuning itself is unnatural in its focus on numerical parameters. The reason physicists focus on numbers is that numbers are easy to quantify - they are already quantified. The cosmological constant is 120 orders of magnitude too large, which is bad with countably many zeros. But the theories that we use are finetuned to describe our universe in many other ways. It’s just that physicists tend to forget how weird mathematics can be.

We work with manifolds of integer dimension that allow for a metric and a causal structure, we work with smooth and differentiable functions, we work with bounded Hamiltonians and hermitian operators and our fibre bundles are principal bundles. There is absolutely no reason why this has to be, other than that evidence shows it describes nature. That’s the difference between math and physics: In physics you take that part of math that is useful to explain what you observe. Differentiable functions, to pick my favorite example because it can be quantified, have measure zero in the space of all functions. That’s infinite finetuning. It’s just that nobody ever talks about it. Be wary whenever you meet the phrase “of course” in a scientific publication – infinity might hide behind it.

This finetuning of mathematical requirements appears in form of axioms of the theory – it’s a finetuning in theory space, and a selection is made based on evidence: differentiable manifolds with Lorentzian metric and hermitian operators work. But selecting the value of numerical parameters based on observational evidence is no different from selecting any other axiom. The existence of ‘multiverses’ in various areas of physics is similarly a consequence of the need to select axioms. Mathematical consistency is simply insufficient as a requirement to describe nature. Whenever you push your theory too far and ties to observation loosen too much, you get a multiverse.

My disbelief in naturalness used to be a fringe opinion and it’s gotten me funny looks on more than one occasion. But the world refused to be as particle physicists expected, naturalness rapidly loses popularity, and now it’s my turn to practice funny looks. The cube, it’s balancing on a tip and nobody knows why. In desparation they throw up their hands and say “anthropic principle”. Then they continue to produce scatter plots. But it’s a logical fallacy called ‘false dichotomy’, the claim that if it’s not natural it must be anthropic.

That I don’t believe in naturalness as a requirement doesn’t mean I think it a useless principle. If you have finetuned parameters, it will generally be fruitful to figure out the mechanism of finetuning. This mechanism will inevitably constitute another incidence of finetuning in one way or the other, either in parameter space or in theory space. But along the line you can learn something, while falling back on the anthropic principle doesn’t teach us anything. (In fact, we already know it doesn’t work.) So if you encounter finetuning, it’s a good idea to look for a mechanism. But don’t expect that mechanism to work without finetuning itself - because it won’t.

If that was too many words, watch this video:


It’s a cube that balances on a tip. If your resolution scale is the size of the cube, all you will find is that it’s mysteriously finetuned. The explanation for that finetuned balance you can only find if you look into the details, on scales much below the size of the cube. If you do, you’ll find an elaborate mechanism that keeps the cube balanced. So now you have an explanation for the balance. But that mechanism is finetuned itself, and you’ll wonder then just why that mechanism was there in the first place. That’s the finetuning in theory space.

Now in the example with the above video we know where the mechanism originated. Metaphors all have their shortcomings, so please don’t mistake me for advocating intelligent design. Let me just say that the origin of the mechanism was a complex multi-scale phenomenon that you’d not be able to extract in an effective field theory approach. In a similar way, it seems plausible to me that the unexplained values of parameters in the standard model can’t be derived from any UV completion by way of an effective field theory, at least not without finetuning. The often used example is that hundreds of years ago it was believed that the orbits of planets have to be explained by some fundamental principles (regular polygons stacked inside each other, etc). Today nobody would assign these numbers fundamental relevance.

Of course I didn’t find a cube balancing on a tip under the couch. I didn’t find the cube until I stepped on it the next morning. I did however quite literally find a missing puzzle piece – and that’s as much as a theoretical physicist can ask for.