Theoretical Needs

MATH AND THE SEARCH FOR THE PERFECT ANSWER

GERMAN SCIENTIST SABINE HOSSENFELDER JOINED US LIVE

from Frankfurt, Germany, to discuss her concern that theoretical physicists have failed to make any major breakthroughs for more than four decades because they are obsessed with the goal that an accurate theory must be beautiful—at least to mathematicians. She argues that when this belief in beauty becomes too dogmatic, it conflicts with scientific objectivity and may be interfering with our ability to understand some of the big questions about the universe. From the April 20, 2021, online Humanities Member-Led Forum program “Sabine Hossenfelder: Lost in Math.” SABINE HOSSENFELDER, Research Fellow, Superfluid Dark Matter Group, Frankfurt Institute for Advanced Studies; Blogger, “Backreaction”; Author, Lost in Math: How Beauty Leads Physics Astray In Conversation with GEORGE HAMMOND, Author, Conversations With Socrates

GEORGE HAMMOND: I’m going to read something. We’re going to start at the latest news in theoretical physics, which is of course the research about the muon and whether the muon is upsetting things, because in the articles about this, whether this muon research at Fermilab is going to upset the Standard Model [of physics], several times I came across this quote, “Sabina Hossenfelder, a physicist at the Frankfurt Institute for Advanced Study, tweeted, ‘Of course, the possibility exists that it is new physics. But I wouldn’t bet on it.’ So why not? SABINE HOSSENFELDER: Well, maybe to give you some context, this muon has been there for 20 years. So it’s basically been with me throughout my whole career, and I’ve had some time to think about it. The thing with the muon g-2 anomaly is that it’s pitching an experiment against a terribly complicated calculation in the Standard Model of particle physics. There are many things that can go wrong with this calculation, and there have always been some parts in this calculation that have been kind of suspicious, that people have tried to improve upon. I suspect what’s going on is someone somewhere along the line is underestimating the uncertainty. So the uncertainty on the theoretical prediction is just too small, which is why the theoretical prediction is outside the measurement. HAMMOND: Well, it’s an interesting thing for outsiders to watch. In your book, Lost in Math: How Beauty Leads Physics Astray, I thought, first of all, this is a nice repeat of something the ancient Greek philosophers discussed, which is, “What do we pursue, beauty or truth?” Beauty versus truth. And you’re basically coming down all these years later in the physics area [saying] that we should not let beauty lead us astray. You have a lot of great examples of it. You talk about the Standard Model of fundamental particle physics and the fact that

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that hasn’t made much progress in the last few decades. You wonder whether it’s this obsession with beauty that is getting in the way.

One question I have about that is, both of the famous recent ones—besides the muons, the Higgs boson—they’re both particles that disintegrate fairly rapidly. So it’s kind of interesting why they’re in a group of fundamental particles, because fundamental particles aren’t supposed to disintegrate into smaller pieces, right? They’re supposed to be the fundamental pieces that everything is built from, or do I have that wrong? HOSSENFELDER: Well, they don’t disintegrate; they decay into other particles. In a nutshell, the reason is that they’re too massive. So they can decay into particles with smaller mass, and that’s what they do. But they’re still fundamental particles in the sense that they don’t have a substructure. They can decay, but they’re not made of smaller things. At least not for all we presently know; it could always be that some point, we’ll find some. In particle physics terms, the muon is a long-lived particle; you know, it’s no comparison to Higgs boson. So measuring something with the muon is vastly easier than measuring something with the Higgs. HAMMOND: You said something very interesting. You said that it decays into other particles, but it’s not made of something else. Isn’t the decay into something which is also fundamental particles—it doesn’t mean that it was made up of those particles in the first place? HOSSENFELDER: No, it doesn’t. What the decay means is that it interacts with those particles. So you can also in principle have the same process backwards. It’s like you have a path that forks—that’s basically how we describe it in terms of Feynman diagrams. So you have one particle coming in and two particles coming out, but you could have come in from any path to the junction and go out into two other ends. All these possible iterations actually do occur in nature. It’s just that some of them are more likely than others. The heavy particles are likely to decay into lighter ones; basically it’s because entropy increases. HAMMOND: For our listeners, [in terms of] something that they’re a little bit more used to, how does that sort of analysis compare to, say, water made up of hydrogen and oxygen? Both hydrogen and oxygen are atoms; they’re pretty long-lasting atoms. And even when they come together in water, that’s a fairly long-lasting molecule. But what you’re saying is that water is not really made up of or, if the analogy were correct—it’s not, but if the analogy were correct—is it like water is made up of or decays into hydrogen and oxygen, and hydrogen and oxygen can also go in the reverse order and become water, as opposed to that water is made up of it?

I’m using it as an analogy, because I understand that it’s not accurate about water, but in the case of these fundamental particles that are more massive, it would seem that you can look at it both ways. So there’s a reason why not, looking at it that way. Right? HOSSENFELDER: Yeah. You’re right. I understand that it can be confusing. So I needed to begin with water, luckily, usually doesn’t just decay, right? [Laughter.] The thing is with a water molecule, we can actually see the substructure. Like if you do a spectrograph or something like this, you can actually tell it’s not one homogeneous blob or something. That’s exactly what we don’t see for the elementary particles. For all we currently can tell, they don’t have any substructure. That’s why we call them elementary. HAMMOND: Got it.

I want to talk about what you find uncomfortable about the current ideas that makes you doubt them, or that makes you wonder where we’re going to go with them. Then I’d like to go back to what we really do know. Atomic theory I think is one of those things. So we’ll go back to the Greeks in a little while. But one of the things that you talk about is fine-tuned versus natural in analyzing theories. I thought it was a really great analysis for people who aren’t familiar with that. Could you explain why that approach, which is related to your beauty argument, makes you feel uncomfortable about what’s going on? HOSSENFELDER: Maybe I should start with briefly explaining what all this talk about naturalness and fine tuning us about. In theoretical physics, we deal with theories that are basically pieces of mathematics. And we use those to make predictions—for example, for the decay rate of some particles and stuff like that—and these theories all contain parameters. Some of those are dimensionless, so they don’t have units, they’re just numbers. All these dimensionless numbers in the theories are supposed to be “of order one,” which means close by one, but not exactly one, not very large, not very small. If that’s the case, then the theory is called natural. Theoretical physicists like theories that are natural. I call this a particular type of beauty.

A theory that’s fine-tuned is just a theory that isn’t natural. The idea is that you have to pick a particular number very, very precisely. Like, say you have a small number, 0.0003 or something; the idea is that you would have to fine tune it to the zero-zerozero-three. That’s where the idea comes from. Fine tuning arguments can be good arguments in certain circumstances. The usual example would be a pen balanced on its tip. You would find that very unnatural in the sense that you would have to very finely tune the position of the pen and all the motions of the molecules around them, that kind of thing. You would correctly conclude the probability that the pen just happens to be balanced on its tip is vanishingly small. Therefore it’s very likely it’s held in place by something.

So that’s how a good argument from naturalness or fine tuning works.

Now, the problem in the foundations of physics is some people use these finetuning arguments for cases where you cannot possibly know anything about the probability distribution. So when we’re talking about the constants of nature that appear in these theories—we might be talking about, say, the fine-structure constant or the mass of the Higgs boson, that kind of thing—then we just can’t make any statements about whether or not that particular value of the constant is likely.

That’s why I say these arguments are not scientific arguments, and if they’re not scientific arguments, well then what are they? Some people have told me that they’re metaphysical criteria. I think philosophically, that’s probably the right word to use, but I call them arguments from beauty, because that’s I think the way that physicists think about them and also where they originally came from. HAMMOND: In talking about those theories and different possibilities, you

make a really good point that there’s no proof for any theory. People in the popular press throw around the word proof all the time. I know scientists perfectly understand this, but I think it’s good for the rest of the people to understand proof is only in a few areas in mathematics. There is no physical theory that’s ever been proven. There’s been reliable evidence that we should keep looking in this direction.

Maybe explain the distinction there a little bit to people—this idea that scientists don’t believe that anything is proven about any of these theories. HOSSENFELDER: Yes, of course. It’s somewhat an issue about the colloquial use of language. You know, if we use words like proof or truth, they don’t mean exactly the same thing as we would mean with them in science. I would say that really the only thing you can strictly speaking prove are mathematical relations, and that cannot contain any statements about nature.

So if we have a theory of nature, like the Standard Model or the concordance model, which describes cosmology, that kind of thing, then we can collect evidence in favor of the theory. So we would say it’s been well confirmed, but we will never ultimately be able to prove that the theory is actually correct. There could always be a next measurement that contradicts the theory, and then we’ll have to go and look for a better theory. That’s how science works. HAMMOND: In your book, on the opposite end of the proofing, you said you cannot falsify a theory, but you can implausify it. I think that’s the other element of it. That’s another thing to explain about how science approaches this. Maybe you can expand out a little bit on that idea. HOSSENFELDER: That’s right. So there are some cases where they can actually falsify a theory, but they’re very, very rare, because if you get some kind of measurements that contradict your original theory, you can always twiddle with the theory. That’s something that scientists shouldn’t do, but they do it all the time and it’s not necessarily wrong, because you learn something from the measurement. So then you go back to the drawing board and you try to improve your theory.

The problem is that what happens is that it makes the theories more and more complicated, more and more arcane, you could say, and at some point what happens is the people just lose interest. If you have a theory that’s been fixed over and over and over again, maybe like the theory with the epicycles [an ancient Greek attempt to explain the movement of stars and planets to support their assumption that everything revolved around Earth—Ed.], then it becomes increasingly less plausible. And in particular, if there is a better theory that explains the data simpler, then practitioners will just go over and start using the better theory. HAMMOND: Your reference to the epicycle theory is at least something that a lot of people who pay attention to this, even from the outside, know about as an example where the mathematics was ever more elaborate and fairly accurate in predicting where the planets were going. Copernicus, when he first tried to rearrange it, his math was less accurate, until the elliptical idea [on orbits that were not perfectly circular]. Why don’t you explain how it’s a good example of where the math led in one direction and was more precise actually than the ideas coming back in another area. HOSSENFELDER: It’s a good example of how big a difference a simplification can make, even though people at the time thought that ellipses are ugly. One of the major reasons they kept doing these circles around circles around circles is that they had this idea that circles are the most perfect geometric shapes that you can think of. So certainly if you can do it with circles, then please do it with circles. But from what the calculation and effort is concerned, it just becomes very time consuming.

So from a purely pragmatic point of view, it made sense to switch to ellipsis. I think today, if you ask any astronomer “Are ellipses ugly?” They would just laught at you. So our conception of beauty has entirely changed there. HAMMOND: I think they would say it’s much more beautiful because it makes it clear, more beautiful. It’s what we were looking for—an answer. I think when we are looking for an answer, an answer that makes a lot of sense is beautiful to us all right away. It’s an interesting part of the argument between beauty and truth. It’s rather an extreme form of chutzpah to believe that our intuition and our ideas about beauty would be more accurate rather than trying to figure out what’s really out there.

You bring up the point really well in your book about how scientists move in a direction, and within a certain amount of time they’re so emotionally attached to their theories or whatever that they always want that theory to win.

It’s a normal human thing, but it doesn’t move us forward to trying to figure out what our reality is. We’re all trying to share an understanding of our shared reality. It’s not easy. In any case, we have millions of people working on it now, and we’re making more progress it seems more quickly, but some of the old ideas about how to go about it, emotional ideas, are getting in our way.

You mentioned I think Kurt Weill or somebody who said that if it came down to an argument between beauty and truth, he would probably lean toward beauty in his decisions. And there are plenty of famous physicists like that. Another thing that you bring up in your book is you mentioned that [Nobel Prize-winning German physicist] Max Planck said over a hundred years ago that there are 25 orders of magnitude between what we can see and what he thought that the substantive reality was. Why don’t you talk about that and how far we’ve progressed, because you’re talking about that we see the Higgs boson and the muon as homogenous and undividable. But we’re at what level of order of magnitude on that scale that he was talking about? HOSSENFELDER: That’s right. So Max Planck had the great insight you could say that there’s a particularly I’d dare to say natural set of units that you can assemble from the already known fundamental constants. Those are called the Planck units. There’s the Planck mass, and there’s a Planck time and a Planck length. The reason that Max Planck introduced them was because—and that was literally his argument—that aliens would also use them. [Laughter.] Because they are just composed of these fundamental constants that you can just extract from measurements.

Their relevance today is that you can estimate at which energies or distances the effects of quantum gravity should become important. That happens to be on the order of magnitude where the Planck energy is. So that’s about, 15 orders of magnitude beyond what even the Large Hadron Collider can probe. So it’s really a long, long way to go.