In a previous post, I told you about how physicists use computers to study how the strong nuclear force combines certain elementary particles — specifically quarks and anti-quarks and gluons — into hadrons, such as protons and neutrons and pions. Computers can also be used to study certain other phenomena that, because they involve the strong nuclear force where it is truly “strong” [in the technical sense described here], can’t be studied using simpler methods of successive approximation. While computers aren’t a panacea, they do allow some important and difficult questions about the strong nuclear force to be answered with precision.
To do these calculations, physicists study an imaginary world, as I described;
all forces except the strong nuclear force are ignored, and
all particles are forgotten except the gluons and the up, down and strange quarks (and their anti-quarks).
On top of this, the up, down and strange quark masses are typically changed. They are taken larger, which makes the calculations easier, and then gradually reduced towards their small values in the real world.
The Notion of “Effective” Quantum Field Theories
There’s one more interesting method for understanding the strong nuclear force that I haven’t mentioned yet, and it too involves changing the quark masses — making them smaller, rather than larger! And weirdly, this doesn’t involve the equations of the quantum field theory for the quarks, antiquarks and gluons at all. It involves a different quantum field theory altogether — one which says nothing about the quarks and gluons, but instead describes the physics of the hadrons themselves. More precisely, its equations are useful for making predictions about the hadrons of lowest mass — called pions, kaons and etas — and it works for processes
with rather low energy — too low to affect the behavior of the quarks and anti-quarks and gluons inside the pions — and
at rather long distance — too long to detect that the pions have a lot of internal structure.
This year’s Nobel Prize, presumably to be given for the prediction of the particle known today as the “Higgs boson”, will be awarded next week. But in the meantime, the American Physical Society has made a large number of awards. A few of them are to people whose work I know about, so I thought I’d tell you just a little about them.
The J. J. Sakurai prize went to Professors Zvi Bern, Lance Dixon and David Kosower, for the work that I have already described on this website here and here. Dixon, a wide-ranging expert in particle physics, quantum field theory and string theory, was a young professor at the Stanford Linear Accelerator Center when I was a Stanford graduate student. He taught an excellent course on string theory, and provided a lot of scientific advice and insight outside the classroom. Bern and Kosower were young scientists using string theory to learn about how to do computations in quantum field theory, and their surprising results formed the starting point for my Ph. D. thesis (which has their names in its title.) The range of their work is hard to describe in a paragraph, but let’s just say that no one is surprised that they were awarded a prize of this magnitude.
The Dannie Heineman Prize for Mathematical Physics was awarded to my former colleague Greg Moore, a professor at Rutgers University. “For eminent contributions to mathematical physics with a wide influence in many fields, ranging from string theory to supersymmetric gauge theory, conformal field theory, condensed matter physics and four-manifold theory.” Allow me to translate:
condensed matter physics: the study of solids and liquids and their mechanical and electrical properties, and lots of other things too, in which quantum field theory is sometimes a useful tool
four-manifold theory: the mathematics of spaces which have four-spatial dimensions, or three-spatial dimensions and one-time dimension. These spaces are very interesting to mathematicians, and also, they’re interesting because we live in one.
This is not the complete range of Moore’s work by any means. Unfortunately this website doesn’t yet have pages that can put his work in proper context, but perhaps I’ll return to it later. But again, no surprise here to see Moore’s name on this award.
The Tom W. Bonner Prize in Nuclear Physics was awarded to experimental physicist William A. Zajc, currently chairman of the Columbia University physics department. Zajc has been heavily involved in one of the most surprising discoveries of the past fifteen years: that a hot dense fireball of quarks, anti-quarks and gluons (produced in the collision of two relatively large atomic nuclei) behaves in a very unexpected way, more like a very low viscosity liquid rather like than a gas. I’ve known him partly because of his interest in the attempts to apply string theory to certain quantum field theories that are perhaps relevant in the modeling of this novel physical system… something I’ll also probably be writing about in the relatively near future.
[This is the third post in a series; here’s #1 and #2.]
The quantum field theory that we use to describe the known particles and forces is called the “Standard Model”, whose structure is shown schematically in Figure 1. It involves an interweaving of three quantum field theories — one for the electromagnetic force, one for the weak nuclear force, and one for the strong nuclear force — into a single more complex quantum field theory.
We particle physicists are extremely fortunate that this particular quantum field theory is just one step more complicated than the very simplest quantum field theories. If this hadn’t been the case, we might still be trying to figure out how it works, and we wouldn’t be able to make detailed and precise predictions for how the known elementary particles will behave in our experiments, such as those at the Large Hadron Collider [LHC].
In order to make predictions for processes that we can measure at the LHC, using the equations of the Standard Model, we employ a method of successive approximation (with the jargon name “method of perturbations”, or “perturbation `theory’ ”). It’s a very common method in math and science, in which
we make an initial rough estimate,
and then correct the estimate,
and then correct the correction,
etc.,
until we have a prediction that is precise enough for our needs.
What are those needs? Well, the precision of any measurement, in any context, is always limited by having
a finite amount of data (so small statistical flukes are common)
imperfect equipment (so small mistakes are inevitable).
What we need, for each measurement, is a prediction a little more precise than the measurement will be, but not much more so. In the difficult environment of the LHC, where measurements are really hard, we often need only the first correction to the original estimate; sometimes we need the second (see Figure 2).
Fig. 2: Top quark/anti-quark pair production rate as a function of the energy of the LHC collisions, as measured by the LHC experiments ATLAS and CMS, and compared with the prediction within the Standard Model. The measurements are the colored points, with bars indicating their uncertainties. The prediction is given by the colored bands — purple for the initial estimate, red after the first correction, grey after the second correction — whose widths indicate how uncertain the prediction is at each stage. The grey band is precise enough to be useful, because its uncertainties are comparable to those of the data. And the data and Standard Model prediction agree!
Until recently the calculations were done by starting with Feynman’s famous diagrams, but the diagrams are not as efficient as one would like, and new techniques have made them mostly obsolete for really hard calculations.
The method of successive approximation works as long as all the forces involved are rather “weak”, in a technical sense. Now this notion of “weak” is complicated enough (and important enough) that I wrote a whole article on it, so those who really want to understand this should read that article. The brief summary suitable for today is this: suppose you took two particles that are attracted to each other by a force, and allowed them to glom together, like an electron and a proton, to form an atom-like object. Then if the relative velocity of the two particles is small compared to the speed of light, the force is weak. The stronger the force, the faster the particles will move around inside their “atom”. (For more details see this article. )
For a weak force, the method of successive approximation is very useful, because the correction to the initial estimate is small, and the correction to the correction is smaller, and the correction to the correction to the correction is even smaller. So for a weak force, the first or second correction is usually enough; one doesn’t have to calculate forever in order to get a sufficiently precise prediction. The “stronger” the force, in this technical sense, the harder you have to work to get a precise prediction, because the corrections to your estimate are larger.
If a force is truly strong, though, everything breaks down. In that case, the correction to the estimate is as big as the estimate, and the next correction is again just as big, so no method of successive approximation will get you close to the answer. In short, for truly strong forces,you need a completely different approach if you are to make predictions.
In the Standard Model, the electromagnetic force and the weak nuclear force are “weak” in all contexts. However, the strong nuclear force is (technically) “strong” for any processes that involve distances comparable to or larger than a proton‘s size (about 100,000 times smaller than an atom) or energies comparable to or smaller than a proton’s mass-energy (about 1 GeV). For such processes, successive approximation does not work at all; it can’t be used to calculate a proton’s size or mass or internal structure. In fact the first step in that method would estimate that quarks and anti-quarks and gluons are free to roam independently and the proton should not exist at all… which is so obviously completely wrong that no method of correcting it will ever give the right answer. I’ll get back to how we show protons are predicted by these equations, using big computers, in a later post.
But there’s a remarkable fact about the strong nuclear force. As I said, at distances the size of a proton or larger, the strong nuclear force is so strong that successive approximation doesn’t work. Yet, at distances shorter than this, the force actually becomes “weak”, in the technical sense, and successive approximation does work there.
Let me make sure this is absolutely clear, because the difference between what we think of colloquially as “weak” is different from “weak” in the sense I’m using it here. Suppose you put two quarks very close together, at a distance r closer together than the radius R of a proton. In Figure 3 I’ve plotted how big the strong nuclear force (purple) and the electromagnetic force (blue) would be between two quarks, as a function of the distance between them. Notice both forces are very strong (colloquially) at short distances (r << R), but (I assert) both forces are weak (technically) there. The electromagnetic force is much the weaker of the two, which is why its curve is lower in the plot.
Now if you move the two quarks apart a bit (increasing r, but still with r << R), both forces become smaller; in fact both decrease almost like 1/r², which would be your first, naive estimate, same as in your high school science class. If this naive estimate were correct, both forces would maintain the same strength (technically) at all distances r.
But this isn’t quite right. Since the 1950s, it was well-known that the correction to this estimate (using successive approximation methods) is to make the electromagnetic force decrease just a little faster than that; it becomes a tiny bit weaker (technically) at longer distances. In the 60s, that’s what most people thought any force described by quantum field theory would do. But they were wrong. In 1973, David Politzer, and David Gross and Frank Wilczek, showed that for the quantum field theory of quarks and gluons, the correction to the naive estimate goes the other direction; it makes the force decrease just a little more slowly than 1/r². [Gerard ‘t Hooft had also calculated this, but apparently without fully recognizing its importance…?] It is the small, accumulating excess above the naive estimate — the gradual deviation of the purple curve from its naive 1/r² form — that leads us to say that this force becomes technically “stronger” and “stronger” at larger distances. Once the distance r becomes comparable to a proton’s size R, the force becomes so “strong” that successive approximation methods fail. As shown in the figure, we have some evidence that the force becomes constant for r >> R, independent of distance. It is this effect that, as we’ll see next time, is responsible for the existence of protons and neutrons, and therefore of all ordinary matter.
Fig. 3: How the electromagnetic force (blue) and the strong nuclear force (purple) vary as a function of the distance r between two quarks. The horizontal axis shows r in units of the proton’s radius R; the vertical axis shows the force in units of the constant value that the strong nuclear force takes for r >> R. Both forces are “weak” at short distances, but the strong nuclear force becomes “strong” once r is comparable to, or larger than, R.
So: at very short distances and high energies, the strong nuclear force is a somewhat “weak” force, stronger still than the electromagnetic and weak nuclear forces, but similar to them. And therefore, successive approximation can tell you what happens when a quark inside one speeding proton hits a quark in a proton speeding the other direction, as long as the quarks collide with energy far more than 1 GeV. If this weren’t true, we could make scarcely any predictions at the LHC, and at similar proton-proton and proton-antiproton colliders! (We do also need to know about the proton’s structure, but we don’t calculate that: we simply measure it in other experiments.) In particular, we would never have been able to calculate how often we should be making top quarks, as in Figure 2. And we would not have been able to calculate what the Standard Model, or any other quantum field theory, predicts for the rate at which Higgs particles are produced, so we’d never have been sure that the LHC would either find or exclude the Standard Model Higgs particle. Fortunately, it is true, and that is why precise predictions can be made, for so many processes, at the LHC. And the success of those and other predictions underlies our confidence that the Standard Model correctly describes most of what we know about particle physics.
But still, the equations of the strong nuclear force have only quarks and anti-quarks and gluons in them — no protons, neutrons, or other hadrons. Our understanding of the real world would certainly be incomplete if we didn’t know why there are protons. Well, it turns out that if we want to know whether protons and neutrons and other hadrons are actually predicted by the strong nuclear force’s equations, we have to test this notion using big computers. And that’s tricky, even trickier than you might guess.
I spent the last two days at an extraordinary conference, “Dreams of Earth and Sky”, celebrating the life and career of an extraordinary man, one of the many fascinating scientists whom I have had the good fortune to meet. I am referring to Freeman Dyson, professor at the Institute for Advanced Study (IAS), whose career has spanned so many subfields of science and beyond that the two-day conference simply wasn’t able to represent them all.
The event, held on the campus of the IAS, marked Dyson’s 90th year on the planet and his 60th year as a professor. (In fact his first stay at the IAS was a few years even earlier than that.) The IAS was then still a young institution; Albert Einstein, John Von Neumann, Kurt Gödel and J. Robert Oppenheimer were among the faculty. Dyson’s most famous work in my own field was on the foundations of the quantum field theory of the electromagnetic force, “quantum electrodynamics”, or “QED”. His work helped explain its mathematical underpinnings and clarify how it worked, and so impressed Oppenheimer that he got Dyson a faculty position at the IAS. This work was done at a very young age. By the time I arrived to work at the IAS in 1996, Dyson had officially retired, but was often in his office and involved in lunchtime conversations, mostly with the astronomers and astrophysicists, which is where a lot of his late career work has been centered.
Retirement certainly hasn’t stopped Dyson’s activity. His mind seems to be ageless; he is spry, attentive, sharp, and still doing science and writing about it and other topics. When I went up to congratulate him, I was surprised that he not only remembered who I was, he remembered what I had been working on in 1992, when, as an unknown graduate student on the other coast, I had sent him a paper I had written.
By the way, it’s somewhat bizarre that Dyson never won a Nobel Prize. Arguably it is part of the nature of the awarding process, which typically rewards a specific, deep line of research, and not a polymath whose contributions are spread widely. Just goes to show that you have to look at the content of a person’s life and work, not the prizes that someone thought fit to award to him or her. Still, he has his share: Dannie Heineman Prize for Mathematical Physics 1965; German Physical Society, Max Planck Medal 1969; Harvey Prize 1977; Wolf Foundation Prize in Physics 1981; American Association of Physics Teachers, Oersted Medal 1991; Enrico Fermi Award 1995; Templeton Prize for Progress in Religion 2000; Henri Poincaré Prize 2012.
The thirteen talks and several brief comments given at the conference, all of which in one way or another related to Dyson’s work, were organized into sessions on mathematics, on physics and chemistry, on astronomy and astrobiology, and on public affairs. All of the speakers were eminent in their fields, and I encourage you to explore their websites and writings, some of which were controversial, all of which were interesting. For non-scientists, I especially recommend Stanford Professor Emeritus Sid Drell’s extremely interesting talk about nuclear disarmament (which he’s been working towards for decades), and a thought-provoking if disconcertingly slick presentation by Dr. Amory Lovins of the Rocky Mountain Institute on what he sees as a completely realistic effort, already underway, to wean the United States of its addiction to oil — with no net cost. Those with a small to moderate amount of scientific background may especially enjoy MIT Professor Sara Seager’s work on efforts to discover and study planets beyond our own solar system, Texas Professor Bill Press’s proposal for how to rethink the process of drug trials and approvals in the age of electronic patient records, Sir Martin Rees’s views on the state of our understanding of the universe, and Caltech’s Joseph Kirschvink’s contention that scientific evidence tends to favor the notion that life on this planet most likely started on Mars.
But really, if you haven’t heard about all the different things Freeman Dyson has done, or read any of his writings, you should not miss the opportunity. Start here and here, and enjoy!
Many happy returns, Professor Dyson; you have been an inspiration and a role model for several generations of young scientists, and may you have many more happy and healthy years to come!
Before we continue, a little terminology — trivial, yet crucial and slightly subtle.
Think about the distinction between the words “humanity” and “a human” and “humans”; or “higher education”, “university” and “universities”; or “royalty”, “king” and “kings”. In each case, the three words refer to the general case, the specific case, and a group of specific cases. Sometimes you even have to use context to figure out whether you’re dealing with the general case or a group, because “humans” or “kings” is sometimes used in place of “humanity” or “royalty”.
In a similar way:
The words “quantum field theory” (with no word in front) refers to the general case: the general type of equations that we use for particle physics and many other fields of study.
The words “a quantum field theory” refers to a particular case; an example of a set of equations drawn from the general case called “quantum field theory”.
The words “quantum field theories” refers to a group of particular cases: a set of examples, perhaps ones that share something in common. We might talk about “some quantum field theories”, or “all quantum field theories”, or “a few quantum field theories”.
Before I go on, I should probably point out that string theory/M theory is different, for a weird (and perhaps temporary) reason — it appears (currently) that there’s only one such theory that has fully consistent mathematics and contains a quantum version of Einstein’s theory of gravity. That was figured out in the 1990s.
However, in that theory you need another, similar distinction. Let me postpone a proper definition until later, but roughly, a “string vacuum” is a particular way that string/M theory could manifest itself in a universe; if you changed the string vacuum that we’re in (says string theory), we’d end up with different particles and forces and somewhat different laws governing those particles and forces. Many such vacua would have particles and forces described by a quantum field theory, along with Einstein’s gravity; many (perhaps most) other vacua might not have this property. And so, as before, we need to distinguish “string vacua in general”, “a string vacuum”, and “string vacua”, plural.
Now it’s time to talk about quantum field theory — the general case — in more detail.
Quantum Field Theory and Its Predictions
Does quantum field theory as a whole predict anything? In short, if you knew a universe had physics described by a quantum field theory, but you didn’t know which particular quantum field theory, what could you predict?
The answer is: almost nothing.
If you’ve been following this blog with care, or have read some books or articles about particle physics, you might well think: quantum field theories have fields; and fields have particles; so here’s a prediction: there should be some particles of some type.
Nope. You’ve learned your lesson well — everything I’ve told you on this website suggests that it is true — but the conclusion is false. I’ve been white-lying to you this whole time, and I have to apologize for that. Quantum fields do often have particles. But many quantum field theories are scale-invariant. A scale-invariant thing looks and behaves more or less the same in a microscope no matter how strong a lens you use. The wikipedia article on scale invariance actually has a nice animation of a scale-invariant process. Another example of something scale-invariant is a fractal. And in a scale-invariant quantum field theory (except for one in which there are no forces at all), there are no particles. I’ve told you that particles are well-behaved ripples in fields… well, it’s true, but in a scale-invariant quantum field theory with at least one force, any such ripples die away and turn into several ripples, which die away and turn into several ripples, which die away and turn into several ripples, which die away and turn into several ripples, which…
Click here to see an animation of a scale-invariant process, called the Wiener process. Animation from Wikimedia Commons (author Cyp). For more information, see http://en.wikipedia.org/wiki/File:Wiener_process_animated.gif
In short, no particles. This has been known for many decades (I’m actually not sure how far it goes back in the form I just described.) We observe this behavior in experiments — not in particle physics, where it isn’t relevant (at least currently), but in many solids, liquids, electrical conductors, magnets etc. that are undergoing “phase transitions”, such as melting, or spontaneous magnetization. In some cases we can also calculate this behavior, exactly or approximately. In some cases we can see this behavior emerge in computer simulations of real or imaginary materials. So we know there are many quantum field theories that don’t have particles.
[This might make you wonder if there are string/M theory vacua that don’t have strings. Good thinking… though actually there are other, unrelated ways to end up without strings… more on this later in the month.]
But If There Are Particles… A Prediction!
But here’s something important that we can predict. If we are studying a quantum field theory that does have particles (now we are talking not about all quantum field theories, but an interesting subset of them) then
the particles will come in types (in our world, electrons are examples of a type of particle, Higgs particles are an example of another, etc.); and
two particles of the same type will be identical in all their intrinsic properties. They will have the same electric charge, spin, mass [i.e. rest mass] etc.; if somebody swaps one for another, you won’t be able to tell the difference. (In our world, all electrons are identical.)
Furthermore, in a world with three (or more) spatial dimensions in which Einstein’s special relativity is true, then particles of each type will be either fermions or bosons. (The Pauli Exclusion Principle, which determines all of atomic physics and chemistry, is a consequence of the facts that all electrons are identical and that electrons are fermions.)
Again, these are general prediction, not of quantum field theory as a whole, but of the subset of quantum field theories that have particles in the first place. The Standard Model, the quantum field theory that seems to describe much of our world very well so far, is an example of one that has particles.
That these are the most important predictions of quantum field theory with particles was pointed out to me, when I was just out of graduate school 20 years ago, by none other than Freeman Dyson, who helped develop quantum field theory back in its infancy.
Moving Toward Particular Quantum Field Theories
Now, what about specific quantum field theories. What can we predict about them?
The answer, hardly surprising, is: it depends.
In some quantum field theories, we can predict an enormous amount, by doing some straightforward type of calculation.
In others, brute-force computer simulation of the quantum fields makes it possible to study the particles and forces described by that quantum field theory, and allows us to make some predictions.
In some quantum field theories, we can predict a smattering of things very well, using fancy math methods (for example, using fancy geometry, or supersymmetry, or string theory).
In still others… unfortunately, many of the most interesting… we have absolutely no idea what’s going on. Sometimes we have the equations we’re supposed to use, but none of our methods for calculation work for those equations. Occasionally we can make a good guess, but we can’t always check it.
In yet others, we don’t even know what the equations are that we should use to start studying them. In fact, the existence of some of these theories was, until relatively recently, unknown. We didn’t even suspect they existed. Maybe there are even more that we still don’t know about.
I’ll start describing these categories of quantum field theories in my ensuing posts. Fortunately for particle physicists, the Standard Model is mostly in the first category, with a little leakage into the second.
The Standard Model: A Quantum Field Theory Of Particular Significance
