What Should I Work On?

I haven’t done much blogging lately (or arguably ever). I have been pretty busy, went to a great workshop in Seattle and another in Madrid, and trying to get details of my travels next year straight (I am on sabbatical for a year). Let’s see if I can get back on track, blog-wise.

Recently a student who took my first-year physics class sent me an email. She is now at a high-powered university and is considering graduate school.

This student has strong interest in both physics and mathematics. She asked me a few difficult questions. Can she make contributions to physics by getting a math degree? Should she become a mathematician? Should she just get a degree in physics? Note: she has done some experimental work, but her inclination is towards theory. This is a talented student who probably has a bright future, so I took the question seriously.

I did not give this student complete answers to her questions, because I am not certain of what they are. I did offer some opinions instead. I offer these again below, after a little more contemplation.

I think that the question, “what should I work on?” is one that all people in our field, not just students, should be asking ourselves. Sometimes we continue an obsession with a mathematical or physical question, long after the reason for the question is obsolete.

Here are my attempts to answer the questions:
1. No, do not get a math Ph.D., if you really want to do theoretical physics. I repeat, do not do it. I know people who say math is better, mainly because you are far more likely to get an academic job. If not risking your future is the issue, I agree that math is better. I don’t see many effective advances in physics coming out of PURE mathematics, however. There is an important role for rigorous methods in theoretical physics, namely in traditional mathematical physics, where theorems are actually proven. Few math departments focus on this subject however. There are connections between analysis, algebra, differential geometry and algebraic geometry (whose usefulness is roughly in that order) and physics, but the typical math Ph.D. advisor will not direct you towards these connections. Or at least, I think he/she won’t. The only math people I have met who do this sort of thing are a minority. There is an emerging tradition of using physics to solve questions in mathematics. If you are interested primarily in physics, however, these may not be the questions YOU want to solve.

2. Having said all I did in 1., let me not dissuade you from doing mathematics. If you pursue a degree in math, do math and enjoy it.

3. Suppose you decide to go to graduate school in physics. Don’t focus just on high-energy theory and especially not exclusively on quantum gravity. The more specialized you are, the less likely you will do something of general interest. You will gain insight into your own research problems by being familiar with concepts in other subfields. Look at other areas of theoretical physics. These are probably more useful to you than pure mathematics, at least in the short term. Learn the math you need, as you need it.

As I wrote above, I think it is beneficial for all of us to ask what we should work on. By this I don’t only mean the choice of physics or mathematics, but what problems to attack. I can’t say much about the experience of others, but I have asked this of myself: and more than once. I did not completely change fields each time I tried to answer it, but I did change my approach on several occasions. During this evolution, I learned a lot of physics (and some mathematics too).

Why is the question “what should I work on?” important?

A. Let’s say you are a practicing theoretical physicist, studying some class of models. Why do you do it? Is it an attempt to answer one of the big questions? Is it because you hope it will lead to an answer of a big question? Or is it because you like it for its own sake? If you answered yes to the last question, I think you are in (figurative) trouble. I enjoy doing technical things, and love solving abstract problems. But it’s not enough. Solving a problem with no greater purpose gives satisfaction, but it is a hollow satisfaction.

B. Maybe you have a clear goal for your research. Macheteing your way through the technical rainforest may not lead you to the fabled City of Precious Metals and Wild Parties. If you have no map of the forest, you need to make one; you can only do this by pushing in many different directions and making trails.

C. Suppose you’ve published lots of papers on some problem and it’s going nowhere. You don’t have a clearer picture of the whole business than you did at the beginning. Know when to give up and try something new.

D. Don’t get too confident that you are on the right path. For example, if you and your friends are working on the same thing, it doesn’t make you or them right in doing so. Are you really doing something significant or are you experiencing the euphoria of conviviality? Don’t take publicity (including self-publicity) seriously.

E. At the back of your mind, remember that there are people out there who do experiments and make observations. What do you have to say to those people? You do not have to be a phenomenologist (god knows, I am not), but you should have some notion of what the scientific implications of your work are.

What is Quantum Field Theory? I

Although fraught with dangerous passes and poorly mapped in some places, quantum field theory (QFT) is a coherent subject. Some critics of QFT are modern-day Madame Blavatskys, channeling the spirits of dead physicists (Dirac, Pauli, Feynman, Heisenberg – you pick the ghost), who claimed to be confused by it all. The Nobel-laureate wraiths stand on the seance table, pointing grey diaphanous middle fingers to the heavens, incessantly demanding that we abandon everything and start from scratch.

The message here is that despite conceptual and technical problems, QFT is not nonsense. It is mathematically healthier than other problems in physics; some aspects of statistical mechanics come to mind. Nobody has been seized by the spirit of Gibbs or Langevin, telling us to start over.

This post is not highly technical. I hope to follow with succeeding technical posts about what QFT (the subject) and QFT’s (the examples) are, excrescences and all. These later posts will require more background from the reader.

There is nothing particularly original here. If you work on current topics in QFT, I expect you will be bored.

I believe the unease within the Zeitgeist stems from the way QFT is presented in many textbooks. Here is a quick summary of what a straw-man textbook author would have you do:

1. Write some Lagrangians.

2. Using some formalism (typically functional integration or canonical quantization with the interaction representation) set up the rules of perturbation theory.

3. Calculate some simple processes in QED, with tree Feynman diagrams. Rupture with pride in your understanding of decay rates and cross sections.

4. Now comes a glimpse of real QFT, including quantum processes, that is, loop Feynman diagrams.

5. Divergences! Duck and cover! See if the carpet under your desk has holes you can sweep infinities into.

6. With trepidation, regularize the loop diagrams, then do the integrals. Segregate divergences from finite parts. Introduce counterterms order-by-order.

7. Calculate physical quantities in QED. This part is fun, so savor it.

8. Learn about the renormalization group, via the methods of Topic 6. Calculate some beta functions and anomalous dimensions.

9. This is only if you have time, and the other students are still going to lectures. In a text oriented towards high-energy physics, you’ll find Standard-Model phenomenology, Bjoerken scaling, pseudoscalar mesons, anomalies, semiclassical methods, lattice gauge theory, gravity, etc. Or maybe you would turn to interesting problems of condensed-matter physics instead.

10. Ride into the sunset on your pony with your guitar. Go to a summer school on phenomenology, condensed-matter or string theory and forget that you were ever confused.

Topics 6. and 7. and 8. are close to the heart of the subject, but come across as heuristic and artificial, if not banal. Topics 9. are where the connection with high-energy physics is really made, but serious students will be nervous about applications if they don’t feel comfortable with the basics. No wonder those mental giants beyond this Veil of Tears are disturbed. No wonder our spirit guides are unhappy mediums. Topics 6. form a confusing enterprise, without any of the simplicity and consistency of physics before QED.

Now I’ll knock down the straw man. QFT was developed following 5., 6., and 7. in the thirties, forties and fifties, but this is an obsolete way to present the subject. It is often still the best way to formally prove statements in perturbation theory and to compute radiative corrections, but it completely obscures what QFT is.

I do think students should learn 6., but only after seeing the big picture. So what is that? I’ll try to describe this here with no mathematics. Later, I’ll try to put some flesh on the bones.

Here is how to think about it: ultraviolet regularize first, BEFORE calculating. In other words, put the cut-off into the action principle, instead of waiting until you calculate Feynman diagrams. There are many ways to do this. The lattice is the most effective way to keep global or gauge symmetry. A momentum cut-off is sometimes good enough to illustrate everything. Now you have a theory, depending on some parameters, namely masses and couplings, as well as an ultraviolet cut-off \Lambda, with dimensions of inverse centimeters. Next, calculate! One method is perturbation theory in couplings, but it is not the only method. Another powerful method, for some problems, is the 1/N expansion (I wrote about this here). A third method is the strong-coupling expansion (which has limited applicability, but this is a matter of practice, not principle). A fourth (very powerful) method is numerical simulation.

The meaning of the cut-off signifies a momentum scale at which the QFT breaks down. Perhaps it can be removed after calculating. Perhaps not.

Next calculate something dimensionful. It should be something defined at a momentum scale smaller than \Lambda, for example a low-energy cross section or a physical energy. This quantity will be some function of all the parameters, including \Lambda. Provided \Lambda is finite, so there are NO ultraviolet infinities. Suppose the symmetries the theory should possess are present with \Lambda.

Now see what happens for large \Lambda. Can you chose the other parameters so that some observable quantity at momentum scales smaller than \Lambda can be fixed? Then these parameters will change with \Lambda. This dependence illustrates how couplings “run” in the renormalization group.

If we can’t find any measurable distinction for different choices of \Lambda (remember, we’ve chosen functions of \Lambda for the other parameters), and no new parameters are needed, we say the theory is renormalizable. Renormalizability does not guarantee that we can take \Lambda to infinity. In fact, we can’t do this for some renormalizable theories, though it is generally believed we can for QCD. Renormalizability only means we can’t predict the value of the cut-off \Lambda from lower-momentum quantities.

Nothing in the procedure I outlined above is infinite. In practice, you need to see how quantities depend on \Lambda. Some will diverge as \Lambda \rightarrow \infty, but we don’t need to take that limit. It is useful to split quantities into “divergent” and “finite” parts, but nothing has to diverge.

I should mention that there is an alternative approach. This is a way to define some QFT’s without any regularization at all – and not necessarily with Lagrangians. The best-studied manifestations are conformal field theories and their deformations. There are also exact S matrices and form factors of non-conformal theories. Though they appear modern, these ideas have their origins in “axiomatic field theory”, which encompasses Gaarding and Wightman’s early ideas in the 1950’s, Lehman, Symanzik and Zimmerman’s reduction formulas, dispersion relations and more. This is another approach I hope to describe in a later post.

What is Strongly-Coupled Quantum Chromodynamics?

I don’t plan on filling my second post with a lot of mathematical symbols (although some of my posts may be more technical in the future). Instead, I want to scratch an itch.

The itch is an allergic reaction to statements I often hear about strongly-coupled quantum chromodynamics (usually called QCD). Basically these statements are that it’s all solved and anything remaining is a minor detail. I’ve tried to scratch this itch before (on Peter Woit’s blog, Not Even Wrong), but I never get satisfaction.

QCD is the theory of how quarks and gluons interact. In doing so, they produce all kinds of phenomena, in particular:

  1. the very existence of hadrons, the strongly-interacting particles (like the proton and neutron). Leaving out the details, these are bound states of quarks, held together by glue.
  2. how everything from electrons to nuclei behave in collisions, at high energies.

We understand the behavior of QCD at short distances (or high transverse momentum) very well. Experiments probing short distances are pretty convincing that the theory is right. This is because of the property called asymptotic freedom, which tells us that quarks and gluons interact very weakly at short distances. This is the weakly-coupled regimeOn the other hand very little is understood about why quarks are confined into hadrons or why the glue is massive (it is 99% of the hadrons’ mass!). This is the strongly-coupled regime.

Many physicists have tried to understand how confinement of quarks and the mass of glue (called the mass gap) follows from QCD. Even the Clay Mathematics Institute has gotten into the game, offering the weekly salary of someone who quits science to work on Wall Street.

My problem is with the claim that the strongly-coupled regime is understood, or nearly understood, a mere pimple on the beautiful wart of current theoretical ideas. Usually this claim is justified by arguing it is all just a string theory on a product of anti-DeSitter space and a five-sphere, with a few bells and whistles. But it’s a wrong claim.

The subject has a fascinating history, and I’m not going to summarize all of it here. Ken Wilson was the first person who saw how the strongly-coupled regime could be understood. It became clear that confinement and the mass gap could be true. Wilson also showed there was a strong-coupling expansion in which these phenomena were there. What is recovered is a kind of quark model, where hadrons form as color singlets. Unfortunately, extending this to genuine QCD is an open problem. The reason (as Wilson understood) is that this strong-coupling expansion has to be taken to many many orders to get the right strong-coupling description, where both asymptotic freedom and confinement are evident. And even that may not be good enough… Masses are multiples of an artificial scale, the lattice spacing. This scale has nothing to do with the QCD scale, emerging from dimensional transmutation.

The stringy models have the same trouble as Wilson’s. They are not guaranteed to describe real quarks or gluons. At best, they are phenomenological models. Just as with Wilson’s approach, the scale has nothing to do with the QCD scale.

Now there is a right strongly-coupled description of QCD, but we don’t know what it is. Wilson tells us how to find it. We start with QCD with a very large, nearly infinite ultraviolet cut-off. Then we integrate out all the short-wavelength degrees of freedom from the theory to get the strongly-coupled theory with a much smaller cut-off (say a few GeV). I wish I knew how to do this – it would solve the problem. This correct strongly-coupled description will be very complicated (with lots of features, called non-renormalizable operators). The probability of guessing it is zero.

Anyway, the message is this: We don’t yet understand strongly-coupled QCD.