Quantum Field Theory, “Easier Than Easy”

by David Kordahl

The book under review.

I began reading Anthony Zee’s most famous book, Quantum Field Theory in a Nutshell, at Muncher’s Bakery in Lawrence, Kansas, where, as a would-be quantum field theorist in 2010, Zee’s book taught me to evaluate Gaussian integrals. Zee made it all seem almost trivial, but his fast style belied the true expectation that his book would be read slowly, pen in hand, the reader studiously working their way from one line to the next. You couldn’t escape the sense that Zee was a very clever man, if not a very sympathetic teacher. This was a book whose readers would select it. If they couldn’t proceed, well, who was really to blame?

I never did become a quantum field theorist, though that’s hardly Zee’s fault. (At that point, I barely had the patience to sit and eat a donut.) Thankfully, Zee has now published an even swifter book, Quantum Field Theory, As Simply as Possible, which readers of this column will be happy to know I actually finished.

On the first page, Zee comments wryly that popular physics books jumped straight from quantum mechanics to string theory—so this book fills the quantum field theory gap. Now, if you are not a physicist, you may not know what quantum field theory is. This review is for you. Unfortunately, Zee’s new book probably isn’t. For whom then, is QFT, as Simply as Possible (henceforth: QFT, ASAP) written? My own answer is that it’s perfect for a past version of myself, just way too late for that bakery.

Paparazzi photo of A. Zee (from Wikipedia).

Zee is now at the stage in his academic career where he has written about most of his interests not only in research articles, but in books for students. His QFT in a Nutshell was followed by Einstein Gravity in a Nutshell and Group Theory in a Nutshell for Physicists—each of which is referenced liberally in the endnotes to each chapter. So when I now refer to an old 3QD piece of mine on the way to explaining “what the heck is a quantum field theory,” know that this is homage, and not mere hackery.

In a piece for 3QD on quantum entanglement and what it isn’t, I wrote that “the nineteenth century saw the rise of field theory, which posited that the effects formerly interpreted as inexplicable long-distance communiques could better be understood by introducing local entities hanging out in space, that is, by introducing fields. […] These fields were eventually understood as dynamical entities, both reacting to and causing changes in the motions of charged particles.”

Okay, so if a field is an object that has some definite properties at each point in space, which is coupled to other objects and reacts accordingly, what’s a quantum field, and how does it differ from regular old quantum mechanics?

Wikipedia visualization of quantum jello.

Regular old quantum mechanics made great strides forward for science by treating things that were once thought of as particles (e.g., electrons) more like classical fields. The quantum-mechanical model of a hydrogen atom has electron globs wiggling around a proton nucleus, in what the physicist John P. Ralston, in How to Understand Quantum Mechanics, has called a “continuum ‘jello’ theory.”

The problem with the “quantum jello” model for particles first arose due to the tensions between quantum mechanics and relativity. Or, to quote Zee, “Quantum field theory emerges from the union of special relativity and quantum mechanics, and so I am obliged to start by telling you a bit about both.”

At the heart of Einstein’s special theory of relativity is the insight that in a world where signals take time to propagate from one place to another, there’s no naturally obvious way to label all events as taking place either “before” or “after” all other events. Yet there are nice ways of relabeling all such points in the arena where these events take place—call it “space-time”—such that viewers in such relabelings will still agree essentially on what happened at the end of the day.

Quantum field theory manages to attach to these space-time points the quantum objects—the quantum fields—in such a way as to keep this relabeling consistent. In the process, everything is put on the same level. Electrons? They’re excitations of a quantum electron field. Photons (light particles)? Ditto: excitations of the quantum electromagnetic field. Quarks, gluons, muons, and neutrinos? You guessed it—all excitations of quantum fields. But what exactly does that mean?

This question of how exactly how one should think about the “thing-ness” of quantum fields is something I’m personally confused about, but my sense is that Zee does not put much stock in such questions. In fact, he seems to reserve a certain amount of playful hostility for those who do. In Chapter IV.4, “The road to gauge theory,” after presenting a particularly strange, though important, piece of physics arcana, he comments, “Isn’t it funny that Nature12 works this way?”

Here, then, is Note 12: “Of course, we could debate whether Nature actually works in this way or whether physicist’s [sic] description of Nature works this way. Chaired professorship of philosophy at an elite university, here I come!”

Feynman diagrams—physically suggestive cartoons that encode the calculation of a probability amplitude—are ubiquitous in QFT.

One might quibble, but it seems obvious to me that the broad questions of “how Nature actually works” are exactly what a popular science book should strive to address. A useful point of comparison, here, is Sean Carroll’s recent book, The Biggest Ideas in the Universe: Space, Time, and Motion, reviewed last year for 3QD by Ashutosh Jogalekar. Carroll’s book is patient and unafraid to state the obvious. Carroll intuits exactly what readers need, and gives it to them straight.

Zee, on the other hand, has written an everything-and-the-kitchen-sink popular approach to QFT, which is as exhausting as it is exhilarating. Part V, “Quantum field theory and the four fundamental interactions,” is the best popular introduction to the “Standard Model” of particle physics that I’ve ever read—though, come to think of it, Sean Carroll’s treatment of those same topics, in The Particle at the End of the Universe, was significantly easier to read.

Every responsible reviewer should give readers a block quote to sample, so here is one from Chapter V.3, “The weak and electroweak interactions”:

In one of the most surprising developments in particle physics, the muon was found to have its own “private” neutrino, denoted by νμ. […] For our purposes here, the one fact you need to know is that the muon decays into its lighter cousin the electron by μ → e + νe + νμ. May I test your potential ability as a particle phenomenologist, the subset of particle theorists who relate experimental observation to theory? See if you can draw the Feynman diagram for this decay process. Hint: it is easier than easy.

A figure illustrating Zee’s jokey sensibility.

Sorry, but I’m not sure “It is easier than easier” really qualifies as a hint. But Zee actually has included (just) enough scaffolding to do this problem, by that point in the book, and the type of reader Zee is looking to court is likely to gobble it up.

The thing is, there is an eager audience for such information, as Zee himself appreciates. The closing pages of QFT, ASAP constitute another advertisement for himself. In a section titled “Some unsolicited advice,” Zee prods the reader, “don’t you think that it is high time to tackle a real textbook on quantum field theory, now that you have made it this far?” He suggests his own QFT in a Nutshell, but appends a comically long list of prerequisites the reader would need to credibly tackle that book. After stretching out his hand, he yanks it right back.

Luckily, students today have more resources than I did over a decade ago. And to be frank? Zee’s books aren’t the easiest to learn from. I recently took a look at a book titled Student-Friendly Quantum Field Theory, written by an obscure college educator, and I’m planning to read it this summer. I’ll admit, though, that it was Zee who made me interested take this up again, and I had a lot of fun reading his new book.