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.

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13 thoughts on “What is Quantum Field Theory? I

    • Yes, the trick is rather simple and boring. Write your formulas between dollar symbols and start inside with the keyword “latex” with no backslash or else. This is not a LaTeX keyword but a WordPress imprinting. If you omit “latex” you will get something like $\alpha$ rather than \alpha. The same if you put a backslash before the magical keyword.

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  1. I could not agree more with you. The old school RG is so much more confusing than the Wilsonian point of view… if people could stop copying the previous generation’s book and present things the way you just did, that would help making QFT understandable to students. Do you know any HEP QFT book that does that at its core ? (I mean, most QFT books introduce Wilson ideas at some point, but as an afterthought, and after spending hundreds of pages on regularization etc., e.g. P&S intro to QFT.)

    A short comment : removing Lambda at every order of perturbation theory (i.e., being able to send it to infinity if need be) is not equivalent to take the continuum limit non-perturbatively. The simplest example is a scalar theory in 4D, which is renormalizable, but does not have an UV fixed-point, and thus does not exist without a cut-off (unless you start at the gaussian fixed-point).

    Adam

    PS: have you tried https://wordpress.org/plugins/wp-latex/ ?

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  2. Adam,

    Books? Hmm. I remember being very confused by this aspect of the high-energy QFT, when I read Boguliubov and Shirkov’s book (from the 1950’s!), as a student. That book is otherwise excellent, doing everything from a very modern point of view (although I remember that the vertex correction has a number of serious canceling errors. I spent a lot of time fixing it). I read S.K. Ma’s “Modern Theory of Critical Phenomena”, which definitely helped. I am showing my age by mentioning these ancient tomes. I also found John Kogut’s first of two review articles in Rev. Mod. Phys. helpful. Polyakov’s book is a great place to learn all the basic ideas, but I think it helps to already know some field theory.

    Of the books written in the last 20 years, Tony Zee’s comes to mind.

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  3. This is thought-provoking. It would be interesting to see this at book-length, if you had time for such a thing at CUNY. I’m currently finding Anthony Duncan’s “The Conceptual Framework of QFT” thought-provoking, albeit differently.
    I take it your post is web-causally separated from today’s XKCD, http://xkcd.com/1489/, but perhaps there’s a zeitgeist. I take XKCD to deplore that the rules are so complicated, so that, yes, we can kludge a lot of rules together to do some very effective phenomenology, at the level of SM or at the level of “Bjoerken scaling, pseudoscalar mesons, …, etc.”, and with lots of experience indeed it makes quite a bit of sense, but the simplicity of Newtonian gravity or of GR apparently evades us. Perhaps that’s just how it is, at least as of now, but perhaps not.
    The phrase “This is a way to define some QFT’s regularization at all” requires, I think, “without”.
    I look forward to your “ideas [that] have their origins in “axiomatic field theory””. If we adopt something approximating an axiomatic framework, I find it troublesome to introduce \Lambda when we already have length scales given implicitly by the test functions that are introduced by Wightman to model experimental apparatus —but then we have \Lambda as well, which rather crudely conditions the effective dynamics globally depending on the experimental apparatus. Finally, I’m feeling good about Peter Woit having sent me to your blog!

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  4. Peter,

    Thanks for the correction. I did mean to include “without”. And for your appreciative comments.

    Wightman and Co. use test functions because Green’s functions have short-distance singularities AFTER the theory is defined. Such singularities (in bare two-point functions) also produce loop divergences in perturbation theory, but they are still present after renormalization. Even after renormalization, axiomatization or whatever to define the theory, they are still present. From the axiomatists’ viewpoint, even free fields need to be defined as distributions.

    The singular behavior signifies something physical, e.g., anomalous dimensions, in physical Green’s functions.

    Anyway, I hope I have the time and energy to do some of this justice in the future.

    I have not seen Tony Duncan’s book. Other people have also told me it’s good.

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  5. Mclaren,

    Did you mean the links in the Feb. 12 post? They are still working, but you need PROLA, the American Physical Society’s pay wall to access one (but you don’t need it for the other).

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  6. Maybe I should mention that there is also a drawback to keeping the cutoff finite. This means that one has to keep an infinite number of irrelevant couplings in the effective action, because the Wilsonian renormalization (semi-)group can generate such terms.

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  7. Gabor,

    By effective action, I assume you mean the result of integrating out degrees of freedom between \Lambda and a smaller cutoff \tilde\Lambda. I’m not sure if we disagree, so…

    1. For an asymptotically-free theory like QCD, there WILL be an infinite no. of irrelevant local couplings in the effective low-energy action, WHETHER OR NOT THE CUTOFF IS INFINITE. This is the effective strong-coupling theory, and it is genuinely non-local.

    2. For a theory like QED, we have no choice. We CAN’T take the original cutoff \Lambda to infinity. In that case the irrelevant couplings will not affect the comparison with experiments (at least not yet to the order we can calculate).

    Anyway, as I said, I am not sure we disagree.

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  8. Do u think Axiomatic quantum field theory approach is practical enough to deal with renormalization issue or learn qft? Does those axioms guarantee correctness?
    another thing is Axiomatic approach seems fail to construct any realistic 4 dimension qft ,do u know why?

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