Appropriate for Advanced Non-Experts

[This is the seventh post in a series that begins here.]

In the last post in this series, I pointed out that there’s a lot about quantum field theory [the general case] that we don’t understand. In particular there are many specific quantum field theories whose behavior we cannot calculate, and others whose existence we’re only partly sure of, since we can’t even write down equations for them. And I concluded with the remark that part of the reason we know about this last case is due to “supersymmetry”.

What’s the role of supersymmetry here? Most of the time you read about supersymmetry in the press, and on this website, it’s about the possible role of supersymmetry in addressing the naturalness problem of the Standard Model [which overlaps with and is almost identical to the hierarchy problem.] But actually (and I speak from personal experience here) one of the most powerful uses of supersymmetry has nothing to do with the naturalness problem at all.

The point is that quantum field theories that have supersymmetry are mathematically simpler than those that don’t. For certain physical questions — not all questions, by any means, but for some of the most interesting ones — it is sometimes possible to solve their equations exactly. And this makes it possible to learn far more about these quantum field theories than about their non-supersymmetric cousins.

Who cares? you might ask. Since supersymmetry isn’t part of the real world in our experiments, it seems of no use to study supersymmetric quantum field theories.

But that view would be deeply naive. It’s naive for three reasons.

1. Supersymmetric quantum field theories might be directly important in the real world. We don’t know that the world isn’t supersymmetric at shorter distances and higher energies than we can currently measure. We need to be able to calculate properties of supersymmetric quantum field theories so that we could recognize them for what they are, should experiments someday give us evidence for them.

2. Supersymmetric quantum field theories can give us analogies for aspects of the real world that we know about but don’t understand very well. The imaginary worlds that are described by certain supersymmetric quantum field theories sometimes bear a certain resemblance to the real world. Consequently it is sometimes possible to gain insights into the real world by studying these imaginary worlds… remembering that our understanding of the Standard Model is imperfect, because of the difficulty of working with its equations.

3. Supersymmetry enables us to discover entirely new features of quantum field theory that we did not know about, and that might be true of some non-supersymmetric theories too — and these too could be part of the real world. Remember the structure and features of the Standard Model, a rather complicated theory with many loose ends, still poses many puzzles for particle physicists. We don’t understand the number or strengths of the forces; we don’t understand why the particles come in the number they do and why their masses have the pattern that we observe; and there is the naturalness problem, among others. The imaginary worlds that are described by supersymmetric quantum field theories have a variety of interesting general features. Although supersymmetry was useful in making it possible for us to discover those features, some of those features are probably also true of non-supersymmetric quantum field theories for which we can’t calculate anything. So any one of these features might be an as-yet unknown aspect of the real world, even if the real world does not have supersymmetry at all.

So let me tell you a little about supersymmetry.

Supersymmetry: Some Basics

In a world with three spatial dimensions like ours, supersymmetric quantum field theory comes in three varieties

maximal supersymmetry

half-maximal supersymmetry

minimal supersymmetry

Technically these are called N=4, N=2 and N=1 supersymmetry. [There’s no N=3 because consistency requirements turn it back into N=4. Supergravity can go up to N=8, but that’s another story. With more or fewer spatial dimensions, the details are different.] These were discovered in the 1970s.

The more supersymmetry a quantum field theory has, the simpler the required mathematics becomes, and the more things about the quantum field theory that can be calculated. The calculations in question aren’t done using methods that I’ve described so far (successive approximation and computer simulation); they’re done using fancier mathematics that takes full advantage of the simpler equations. [For non-supersymmetric theories, no known fancy math can be used; we may well need even fancier mathematics that hasn’t yet been invented, but at this point we’re almost powerless.]

However, the more supersymmetry a quantum field theory has, the more remote it becomes from the real world. As I mentioned, the real world might have minimal supersymmetry, hidden from us but actually present, though so far that hasn’t been discovered to be the case. It is unlikely (but not impossible) that some form of half-maximal or maximal supersymmetry has a partial, complicated and subtle role to play in the real world, but that’s a much more complex issue than I want to cover here. Let’s just say that maximal supersymmetry has no simple and direct possible role in the real world, whereas minimal supersymmetry may perhaps have such a role.

Yet maximal and half-maximal supersymmetric quantum field theories have proven very useful in learning about the real world. And they’ve proven extremely useful in learning about minimal supersymmetric quantum field theories, which in turn (a) might be directly relevant in the real world, and (b) have been somewhat useful in learning about non-supersymmetric quantum field theories, which are definitely relevant in the real world.

So what kind of things can we gain from supersymmetry?

Learning Lessons From Supersymmetry

1) Supersymmetry has made it easier to make particle physics predictions in non-supersymmetric theories.

In supersymmetric theories, the method of successive approximation (or “perturbation theory”, in jargonese) is simpler. It turns out that one of the best ways to do successive approximation in the Standard Model — a somewhat complicated non-supersymmetric quantum field theory (QFT) — is

a) do a similar calculation in a maximally supersymmetric QFT

b) do a similar calculation in a minimally supersymmetric QFT

c) do a similar calculation in a simple non-supersymmetric QFT

d) add them up, with appropriate care

More generally, the structure of calculations in non-supersymmetric theories can be broken up into parts that already appear in supersymmetric theories (and these are relatively easy to calculate) and parts that are special to the non-supersymmetric case (which can be studied in the simplest possible contexts.) In short, supersymmetry gives us a useful way of organizing our real-world calculations, so that the simple parts are separated out and done first, and then the small number of hard parts can be focused on.

This is also the logic and strategy behind the now-(in)famous “amplitu-hedron” method that received a very enthusiastic press article recently. The article was certainly over the top — and the title, “A Jewel at the Heart of Physics”, was a bit misleading. It should have been entitled “A Jewel at the Heart of Maximally Supersymmetric Physics”, since — so far — the details have been most clear for maximal supersymmetric quantum field theories, which have very special features and are far from the real world. But, as I said above, understanding the maximally supersymmetric case can be step one, with minimally supersymmetric theories to come next, and non-supersymmetric ones to come last but not least. So it’s early days for the amplitu-hedron, and too early to declare joy and rapture as loudly as did the press article about it all, but there is good reason for cautious optimism that these breakthroughs in maximally supersymmetric contexts will affect calculations in the real world within the next decade.

2) Supersymmetry has taught us that the effective field theory that describes the low-energy and long-distance behavior of a quantum field theory can have features which we’ve not encountered — yet — in the real world.

In a previous post I emphasized that in the strong nuclear force, the effective field theory for the low-mass hadrons (pions and kaons) is a useful tool, but that its form had to be guessed — we can’t, even today, determine the equations for the effective quantum field theory from the quantum field theory of the strong nuclear force. The only way to figure out the right theory is to guess it starting from data. Well, in supersymmetric quantum field theories, using methods developed in the 90s (which in turn were founded on earlier work from the 80s), we often can determine the right effective quantum field theory to a greater or lesser degree, starting with the equations of the original theory! There’s still often some guesswork involved, but the math often provides the clues we need.

For instance, take the imaginary world that we use for calculation of the strong nuclear force, with three low-mass quarks and eight gluons, carrying three types of color) The supersymmetric version of this world (see Figure 2) has a superpartner particles for the quarks and for the gluons. And at long-distance and low-energy, it has pion-like and proton-like hadrons, for which an effective quantum field theory can be written down. The power of supersymmetry opens up the door to an understanding of many features of these theories, not just the ones I’ve mentioned here.

And in the process of learning how to do this, we also learned that quantum field theory can do many things that we didn’t previously know it could do. The initial wave of major discoveries about this were launched by the work in 1994 of Seiberg and Witten for half-maximal and maximal supersymmetric QFTs, and of Seiberg in minimal supersymmetric QFTs. I can tell you from having been there when it all was happening — these were heady days, where major insights into what quantum field theory was capable of were appearing every few weeks. [And then string theory got into the act, but that’s another story.]

For instance, we learned that the low-energy physics of a quantum field theory can involve gluon-like and quark-like particles that are neither some of the quarks and gluons of the original theory, nor made in a simple sense out of the quarks and gluons of the original theory. (This is in contrast to protons of the real world, which we understand as being made directly out of quarks and gluons.) They are, so to speak, “emergent”, arising in a complex and subtle way that even today we don’t entirely understand. See point (4) below.

By the way, curiously, computer simulations, such as are used for understanding the strong nuclear force, are not useful here! Supersymmetry is a complete disaster for computer simulations; the methods we have for simulating quantum field theories using computers do not work at all in the presence of supersymmetry. This is a very frustrating situation, but no one, despite years of work (and some very, very clever ideas) has yet crossed this barrier.

3) Study of supersymmetric quantum field theories has taught us that scale-invariant quantum field theories are far more common than we previously thought.

Thanks to the work of Seiberg, considerable evidence was obtained in the mid-90s that even for minimal supersymmetry, scale-invariant quantum field theories are very common. (A few such theories were known in the 70s, and more in the 80s, but they weren’t considered typical.) For instance, I emphasized that we have no idea how a quantum field theory like the real-world strong nuclear force, but with, say, 8 types of low-mass quarks, behaves when the forces become truly strong. Well, the supersymmetric version of this theory — with gluons and their superpartner particles (gluinos), along with 8 massless types of quarks and their superpartner particles (squarks) — turns out to be a scale-invariant theory at long distance. The supersymmetric version of the theory with a half-sextet and 7 half-quarks, is also scale-invariant, as we learned from Seiberg’s student Pouliot. Many properties of these theories can be calculated.

Today, thanks to these insights, it is widely believed that even non-supersymmetric scale-invariant quantum field theories are very common. This belief has motivated people to study the question using computers, and though the jury is still out, some evidence has already been obtained.

4) Often, two quantum field theories with entirely different fields are actually the same — sometimes exactly the same, and other times having the same long-distance low-energy behavior.

This phenomenon, known often as “duality” (but be careful, because the word “duality” is heavily over-used, and means different things in different contexts, even within physics), is one that deserves its own article. But suffice it to say that two (and sometimes more) quantum field theories, with entirely different sets of fields and entirely different equations, can sometimes have identical physical phenomena — depending on context, this may mean “exactly identical” or “identical only for low-energy and long-distance behavior”. Famous names associated with these developments, which go back to the 70s but became clear and widespread in the 90s, are Olive, Montonen, Sen, Seiberg, Witten; less famous names include yours truly and many others.

In fact, this is a grand generalization of the notion of effective field theories. An effective field theory is a quantum field theory whose equations may be used to describe the low-energy and long-distance phenomena of another quantum field theory. As occurs in the case of the strong nuclear force, the original theory and the effective field theory may have completely different fields and equations. For the strong nuclear force, the “original” theory is the theory of quark and gluon fields; the effective theory is a theory of pion and kaon fields, which describe the hadrons made from quarks and gluons. But the more general notion of duality is somewhat mind-blowing. Rather than the second set of fields being made out of the first set in the way a house is made from bricks, as for hadrons and quarks, it may happen that neither set of fields is made from the other in any simple-to-describe way.

5) Supersymmetry has allowed us to learn that there are quantum field theories for which no normal-looking equations can be written.

While using fancy mathematics to explore the various quantum field theories with half-maximal and minimal supersymmetry, some of my colleagues (initially Philip Argyres and Michael Douglas, but many more followed) came across theories whose properties simply aren’t consistent with any type of equations that we know how to write down.

Some clever trickery is required to find these things. In supersymmetric contexts, sometimes it is easier to figure out the answers to certain physical questions not by directly solving the equations that we use in quantum field theory but by an indirect route, using fancy math. The answers one obtains in this way suggest that certain quantum field theories that we didn’t know about not only exist but have properties that are clearly inconsistent with the types of quantum field theory equations that we normally write down. (For example, we learned of quantum field theories whose symmetries could never arise from ordinary quantum field theory equations.) In particular, such theories cannot be even partially specified by what fields they have. This tells us that we’ve somehow been too narrow-minded, and there must be ways of defining quantum field theories using methods with which we’re currently unfamiliar.

What we know about this subject may be just the tip of a large proverbial iceberg. We know there are many of these theories, but we may not have identified the majority of them, or understood anything about what makes them special. Although the ones we know about are supersymmetric, there is absolutely no reason to think there aren’t non-supersymmetric ones too.

A Moral

From this story, I hope you can get some insight into why particle physicists spend a lot of time thinking about imaginary worlds. First, certain imaginary worlds might actually turn out to be real; second, what happens in an imaginary world may be analogous to, but easier to study than, what happens in the real world; and third, imaginary worlds may reveal phenomena that we never previously thought of and might be relevant for solving real-world puzzles.

The developments of the past 20 years have given us great insight into how quantum field theory works, thanks in large part to what we’ve been able to learn about supersymmetric theories, which are simpler to understand. It may turn out that supersymmetry has nothing to do with particle physics at the LHC, and the solution to the naturalness problem of the Standard Model. But for understanding how quantum field theory works overall, this is beside the point: supersymmetry as a tool for studying quantum field theory is here to stay. As a teacher of quantum field theory, I will say this: anyone wanting to understand quantum field theory fully must study supersymmetry. It’s not optional.

Continued here…