Review: ‘t Hooft’s The Conceptual Basis of Quantum Field Theory

I should be preparing for my 10 minute lecture on classification of hadrons at the American Physical Society’s Northwest meeting on Saturday, but instead I am reading Gerard ‘t Hooft’s beautiful introduction to quantum field theory, The Conceptual Basis of Quantum Field Theory. I like the approach of this book because it concentrates on what I think of as the heart of quantum field theory, Feynman diagrams.

The author begins with classical field theory, but unlike most he defines the solutions to classical field theory with Feynman diagrams! The difference with the quantum theory is that there are no loops; he calls the loops the “quantum corrections” to the classical theory. So all the diagrams are tree diagrams looking something like this:

In the above, the points each correspond to a single point in spacetime. These points are labeled $x_j$. The $G_{49}$ in the above is a propagator, the Green’s function. Each of the legs carries one of these. The $g_{jkl}$ and $\lambda{jklm}$ are included for the 3-point and 4-point vertices. They are coupling constants. Uh, I’ve been sloppy in the above, please refer to the ‘t Hooft paper for details.

‘t Hooft’s approach to quantization is to discretize spacetime; that is, to replace the continuous points (which support differential equations) with a discrete set that is sufficiently dense to allow the differential equation to be modeled. My lecture on Saturday will be a little similar to this. Rather than working with an approximation to the classical field I will be looking at the solution to bound state problems on a finite number of points. I will begin with the hydrogen atom and then move to the equivalent problem on 2 and 3 discrete points. I discussed this on the blog a few weeks ago, quantum bound states, the hydrogen atom.

While ‘t Hooft’s paper is only 70 pages long, he manages to cover an amazingly large numbers of details. In addition to the stuff described above, he also discusses path integrals, the Dirac equation, Goldstone bosons, ghosts, gauge invariance and fixing, Yang-Mills, the Standard Model and Higgs, unitarity, dressed propagators, various regularization schemes, renormalization, BRST invariance, triangle diagrams and anomalies, asymptotic freedom, topological twists, vortices, magnetic monopoles, confinement, supersymmetry, naturalness, instantons, and Borel resummation. And finally, he ends with an opinion on superstring theory:

In its present form, superstring theory appears to have turned into a collection of wild ideas called M-theory, whose foundations are still extremely shaky. Some of the best minds of the world are competing to turn this theory into something that can be used to provide for reliable predictions and that can be taught in a text book, but this has not yet been achieved.

Filed under book review, physics

6 responses to “Review: ‘t Hooft’s The Conceptual Basis of Quantum Field Theory”

1. Doug

Hi Carl,

Note page 59, section 10 Topological Twists, item 10.1 Vortices.

This is very close to the mathematical dynamics terminology of V Kosloz, RAS.

2. Carl, it’s great that you will be giving such a lecture … as an amateur! Well, formally you are (unless things have changed) per the comment I quote below, and I applaud your being taken seriously by the physics community. However, maybe you can have some sympathy for the rest of the similar, little people down here who are trying to make some progress as amateurs. Sure, most of what they produce is rubbish (as is a lot of what “professionals” produce, look up for example “Why most published research findings are false” by John P. A. Ioannidis (2005.) However, the few who are capable of making advances as did you (aside from whether I turn out to be one myself) need better support – from someone, I’m not sure who. Well, how did you get taken seriously by the physics community?

# 8 | Carl Brannen | May 13, 2008 6:24 PM

I bet that the “jail mail” that public officials recieve is about as useful reading as the “green ink mail” that physicists receive from amateurs.

# 9 | Neil B. | May 13, 2008 8:33 PM

Some have suggested that instead of long prison terms in lieu of execution, we cripple violent criminals in ways that would make it harder for them to commit violent crimes again.

BTW Carl, I’d like to see a collection of the “green ink” mail that turned out to have useful or clever ideas about physics. I’m willing to bet there was some of it in there, and haven’t some amateurs or semi-amateurs done some important things lately? – heh – for example, you said of yourself: “If you’re interested in what I do, I’m an amateur working in elementary particles. My regular job involves stuff like driving forklifts and filling out environmental forms for Liquafaction Corporation’s ethanol plant.” I see that physicists have to reference your web page, that is great. But you still don’t have a bio in Wikipedia! Want me to write one up? LMK if you are interested, and someday maybe you can return the favor if you think I deserve it.

So, I wanted your thoughts on this issue. tx
(PS – keep an eye on that spelling …)

3. carlbrannen

Neil, LOL, please avoid putting up a page on Wikipedia about me; it would be in violation of the rules they have on “notability” or something like that.

Anybody can give 10 minute lectures at APS meetings. All you do is join up with APS, at a fee of about \$100 per year if I recall. I believe that their policy is to accept violent criminals as well as amateur physicists and anybody else who applies. Then you will begin getting emails announcing meetings, for which they typically charge a fee. You pays your money and you gives your lecture.

The less amateurish they think you are, the earlier in the day they will set up your lecture. I’m proud because this time I’m not giving the last lecture of the day.

The fee also allows you to attend the lectures and make pointless comments to the professionals in the “question and answer” periods.

It’s very nice that they allow amateurs to show up at meetings. Most amateur presentations are horrible. It’s nice to get a compliment after giving one.

4. Carl, yes I’ve heard that about APS. But you get sited by physicists, which you can’t buy! So I’m still interested in the story of how you came to be noticed and noted by them. I know roughly what you accomplished, but AFAIK (and what we all hear), isn’t easy (?) for an “amateur” to get even valid developments noticed .

5. carlbrannen

Neil,

I should write a blog post on this. Maybe next week, I have to get this lecture ready.

Basically, nobody, amateur or professional, has much time on their hands. So it’s just as difficult for the average professional to get attention as it is for the better amateurs. The “secret” is to write down something that either fits in well with stuff that is being worked on (and so is easy for the people doing that to understand), or make it so blindingly simple that it can be understood easily by anyone. I did a combination of these with my rewrite of Koide’s formula. I’ll be talking about this on Saturday.

A few years ago I got very lucky and found one of the clues to the unified field theory, the thing that unifies the forces and gravity and everything. Finding a clue like this is like getting a peek into what the mathematicians call “The Book” , but for physics.

Once you’ve gotten a peek into The Book, you automatically get more peeks. So I’ve learned a lot. The equations of modern physics are mostly accurate, but almost all of the philosophy of modern theory is complete crap. Because of this, showing people the simple and deep lessons from The Book is a waste of my time. Instead, I show them the simple stuff and leave the deep stuff unpublished.

6. Best wishes on your talk, wish I was there. APS is a valuable opportunity and a chance to contact other physicists.