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.


Physics at CUNY I: Mighty Monkeys Climbing Ropes

I could have entitled this post “Research at CUNY”, but I prefer to speak for my own field.

I am thankful to have a long-term academic job, with tenure. I am ecstatic that I live in New York City (if only I could arrange for a geologic event to make some real mountains here without hurting anyone. I would need a promotion of some kind). I appreciate that I have a reasonably big office (which I manage to clutter, despite its size). I am glad that I like many of the students I know (or have known).

What I’m not glad about is the research environment. Before any released time, the CUNY contract mandates 21 hours of teaching per year. This is quite reasonable for an institution where faculty research is a sideline, but is too heavy a burden for those, like me, for whom it is a passion. That descends to 18 hours, for those who publish (still 9 contact hours per week, per semester). Committee work helps further. Advising each Ph.D. student results in a minuscule further reduction of 0.6 hours per semester. External grants result in no teaching reduction at all.

Don’t pity me. I am productive, despite the yoke. If you look at the last post, I even display a certain amount of pride in my work. I don’t write a lot of papers, but I think that what I do write is significant. The problem is that, like the proverbial monkey climbing the rope slung over the pulley, I have to drive myself very fast to ascend.

To those who say “your job is to teach, not do research,” I beg to differ. I actually love to teach (I really enjoyed my great students in relativity and quantum mechanics last semester). Furthermore, I find that my teaching has had a positive impact on my research. But as a scholar, I have the responsibility to give the community something new to teach in fifty years. If there is no research, scholarship recedes into a medieval state, in which we are monks and nuns, copying the same illuminated texts over and over and over and over and over…

CUNY can’t make up its mind as to whether it should be a real university or a teaching-only institution. It seems that it wants to be both, which is not feasible.

I won’t point fingers at particular administrators (or faculty) at CUNY Central or Baruch College (not in this post, anyway). Is that because I am a nice guy and don’t want to name names? Or is it because I am a chicken and frightened of these folks? Nope – it’s because everything is negotiated in secret first, then rubber-stamped. By keeping the decision-making in the dark, we never know who to give credit to or who to blame.

Other people at public universities will probably agree with what I say. Undoubtably the problems begin with government more than university administrations. The evolution of federal and state tax structure has stretched the budgets of public universities incredibly thin. They respond by making life harder for faculty, like us, or charging so much tuition that they are rapidly becoming private schools. The latter phenomenon is helping to deprive those students without upper-class backgrounds of a university education.

Since I came to Baruch, nearly a quarter of a century ago, we have grown quite a bit (though nowhere near enough). Two of our faculty came as RIKEN Fellows, which is a big deal (we get first-rate faculty essentially for free, for the first five years). Look at our publication record, if you disagree. I’ve seen quality growth in other departments around CUNY. CCNY, the boot-camp of science, has always had a great department, but the New York City College of Technology, Lehman College, The College of Staten Island have all become very good places (Hunter, Queens and Brooklyn will complain. Yes, those are good too).

Our physics is mighty. It sure would be nice to see some appreciation around here.


Now and then, a sense of accomplishment is welcome in my tiny existence. In the greater scheme of things it means little – as Kafka is reputed to have said to Brod, “there is hope, but not for us.” Still, it is nice, if only for a moment. If you are interested, you can find the reason here, or here.

Working on the confinement problem for Yang-Mills theories in lower dimensions, led me to the problem of the principal chiral model in one space and one time (1+1) dimensions. This is a model of an N by N unitary matrix field, of determinant one. Axel Cortes Cubero and I studied this problem at large N, when he was my Ph.D. student (he is now a postdoc at SISSA).

So what is large N? There is a special limit which some problems of high-energy or condensed-matter physics can be solved with. This is the large-N limit, where the number of species of particle is taken to infinity. Actually, there are a few of these large-N limits:

  • The large-N limit of an isovector field with N species. This sometimes goes by the name  “random-phase approximation”. Lots of interesting problems have been solved this way. The Feynman diagrams have a chain-like or linear structure.
  • ‘t Hooft realized that for models of matrices (like the principal chiral model above) the Feynman diagrams became planar. The 1/N-expansion is an expansion in the genus of Feynman diagrams. The original motivation was an approximation scheme for QCD, 1/N=1/3 being approximately zero.

It turns out that it is much harder to calculate anything in ‘t Hooft’s large-N limit than in isovector case. Perhaps that is because the problems ‘t Hooft was considering are potentially much more interesting.

Since the late 90’s, much of the activity in supersymmetric quantum field theory concerns ‘tHooft’s limit. I don’t want to just wax poetically about this, but there has been a lot of progress in superconformal field theories and their deformations.

The principal chiral model is in only two spacetime dimensions, whereas the supersymmetric theories above are in three or four spacetime dimensions. Nonetheless, this model is interesting, because it shares features with QCD that the supersymmetric theories do not. Most important, it is asymptotically free and has massive particles in its spectrum. Furthermore, the solutions Axel and I obtained are complete, going beyond both perturbation theory and strong coupling approximations. These solutions are valid at all length scales.

But there was a difficult snag in all of this. Axel’s and my methods heavily use integrability and the “form-factor bootstrap”. It’s pretty easy to see how correlation functions (or renormalized propagators) look at large distances. Unfortunately, it was not obvious how they would behave at short distances. Moreover, there is a prediction for the short-distance behavior from the perturbative renormalization group; the two-point function behaves like logarithm of the separation squared. I was not at all confident that the form-factor bootstrap in Mr. ‘t Hooft’s limit would obey this prediction (and behave like the logarithm squared). There was no straightforward way to check it, and there were other things for me to do. I felt, however, that I was failing to put the ideas to the test.

Last July I asked an elephant in the Copenhagen Zoo to step on my head, should I be unable to solve the problem. She did not agree, because we could not shake on it (her trunk was too far from the fence). The entire summer, I worked only on this problem. It turned out that the mathematics is related to ideas well-known to statistical mechanicians (Levy flights) and mathematicians (non-integer powers of Laplacians).

In the end, the answer fit the prediction. It’s kind of magical that it worked, since the prediction was made using very different mathematics.

For a few more days, I will feel good about myself. Then I’ll get tired of my hubris. O Goddess Nemesis, mine eyes search for thine sword and shield amongst the constellations.

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.

Setting Myself Up for Failure

This is my first attempt at blogging. I expect that most of my posts will concern theoretical physics, but not exclusively. On occasion I might write about a book, a person, a picture or my confusion.

There is enormous amount of blogging activity in physics, some of which is wonderful. I decided to add another blog to this already thriving ecosystem, because most of the activity focusses on research ideas far from what I am personally involved with. Perhaps I will fall on my face, but at least I will have tried.

If you’ve read to this point, this is me. The link is to my picture (with a little more hair than I currently have) and brief bio at the Niels Bohr Institute (the original Niels Bohr Institute at Blegdamsvej 17. Every physics institute in Copenhagen now is part of the “Niels Bohr Institute”, even if they already have perfectly good historical mascots, like H.C. Oersted). In fact, I am full time at Baruch College, which is a part of the City University of New York, located in Manhattan.Featured image