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 , 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 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 , for example a low-energy cross section or a physical energy. This quantity will be some function of all the parameters, including . Provided is finite, so there are NO ultraviolet infinities. Suppose the symmetries the theory should possess are present with .

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

If we can’t find any measurable distinction for different choices of (remember, we’ve chosen functions of for the other parameters), and no new parameters are needed, we say the theory is renormalizable. Renormalizability does not guarantee that we can take 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 from lower-momentum quantities.

Nothing in the procedure I outlined above is infinite. In practice, you need to see how quantities depend on . Some will diverge as , 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.