Quantum Field Theory as Simply as Possible

5.0 out of 5 stars Tired of popularizations? Buy this book.

The following is my review published in Amazon — the other 25 reviews are interesting and somewhat divergent– here’s a link to them all — https://www.amazon.com/Quantum-Field-Theory-Simply-Possible/dp/0691174296#customerReviews.

 

 

I’ve put in two further thoughts I left out to keep the Amazon review to a reasonable length (Addendums 1 and 2).

 

Disclaimer: I have probably spent more time with Quantum Field Theory As Simply As Possible (QFTASAP) than most as I was a lay reader providing commentary to Dr. Zee as he was writing it. I came to know Dr. Zee after he responded to some questions I had while going through another of his books, Group Theory in a Nutshell for Physicists. We’ve corresponded for years about math, physics, medicine, biology and life both turning out to be grumpy old Princeton Alums (1960 and 1966). If this makes me a friend of his despite never having met and living on opposite coasts — and if this precludes this review appearing on Amazon, so be it.

Should you buy the book? It depends on two things: (1) your ability and (2) your background. Dr. Zee spends a lot of time in the preface describing the rather diverse collection of people he is writing for. “I am particularly solicitous of the young, the future physicists of the world. .. delighted beyond words if some college students, or even a few high school students are inspired by this book .. “

Addendum 1  In retirement, I met one such high school student whil auditing an abstract algebra course at a local college.  He was simultaneously doing his German homework while listening to the lecture with one ear.  He did not go on in physics but did get a double summa in math and physics at an Ivy League university (not Princeton despite my attempts to get him to go there).

 

I fall into another class of reader for QFTASAP which the author mentions, “scientists, engineers, medical doctors, lawyers and other professionals . … Quite a few are brave enough to tackle my textbooks. I applaud these older readers, and address them as I write.”

So you need to know the background I bring to the book to put what I say in perspective. I am a retired neurologist. I did two years of graduate work in chemistry ’60 – ’62 before going to medical school. All grad students in chemistry back then took quantum mechanics, solving the Schrodinger equation to see where atomic (and molecular orbitals) come from.

At my 50th reunion, I met a classmate I didn’t know as an undergraduate, Jim Hartle, a world class relativist still writing papers with Hawking, so I decided to try and learn relativity so I’d have something intelligent to say to him if we met at another reunion. I studied his book on Gravitation (and Dr. Zee’s). Unfortunately COVID19 has stopped my attendance at reunions. To even begin to understand gravitation (which is what general relativity is about), you must first study special relativity. So my background was perfect for QFTASAP, as quantum field theory (QFT) tries to merge special (not general) relativity and quantum mechanics.

Do not despair if you have neither background as Dr. Zee starts the book explaining both in the first 99 pages or so. His style is very informal, with jokes, historical asides, and blinding clarity. As a retired MD I can’t speak to how accurate any of it is, but the publisher notes that his textbooks have been used at MIT and Cal Tech, which is good enough for me.

So quantum mechanics and special relativity gets you to base camp for the intellectual ascent to QFT — which takes the rest of the book (to page 342). If this sounds daunting, remember thAt physics majors and physics grad student’s QFT courses last a year (according to Dr. Zee). So read a little bit at a time.

The big leap (for me) was essentially abandoning the idea of force and thinking about action (which was totally unfamiliar). Much of the math, as a result is rather unfamiliar, and even if you took calculus, the integrals you will meet look like nothing you’ve ever seen.
e.g. d^4x e psibar (x) gamma^mu psi(x) A_mu(x) all under the integral sign
Fortunately, on p. 154 Dr. Zee says “I have to pause to teach you how to read this hieroglyphic.” This is very typical of his informal and friendly teaching style.

QFTASAP contains all sorts of gems which deepened my understanding of stuff I’d studied before. For example, Dr. Zee shows how special relativity demolishes the notion of simultaneity, then he goes even farther and explains how this implies the existence of antiparticles. Once you get integrals like the above under your belt, he gives a coherent explanation of where and how the idea of the expanding universe comes from and how it looks mathematically.

There is much, much more: gauge theory, Yang Mills, the standard model of particle physics etc. etc.

To a Princetonian, some of the asides are fascinating. One in particular tells you why you or your kid should want to go there (spoiler alert — not to meet the scion of a wealthy family, or an heiress, not to form connections which will help you in your career). He mentions that there was an evening seminar for physics majors given by a young faculty member (33) about recent discoveries in physics. In 1964 the same young prof (James Cronin) said that he had discovered something exiting — in 1980 he got the Nobel for it. For my part, it was John Wheeler (he of the black hole, wormholes etc) teaching premeds and engineers (not future physicists) freshman physics and bringing in Neils Bohr to talk to us. So go to Princeton for the incredible education you will get, and the way Princeton exposes their undergraduates to their very best faculty.

Addendum #2 — As a Princeton chemistry major, my undergraduate adviser was Paul  Schleyer , Princeton ’52, Harvard PhD ’56. We spent a lot of time together in his lab, and would sometimes go out for pizza after finishing up in the lab of an evening. For what working with him was like please see — https://luysii.wordpress.com/2014/12/15/paul-schleyer-1930-2014-a-remembrance/ and https://luysii.wordpress.com/2014/12/14/paul-schleyer-1930-2014-r-i-p/

Contrast this with Harvard where I did chemistry graduate work from ’60 – ’62.   None of the 7 people who were in the department back then who later won the Nobel prize later (Woodward, Corey, Hoffmann, +4 more) did any undergraduate teaching.  I did most of the personal teaching the Harvard undergrads got — 6 hours a week as a teaching assistant in the organic chemistry lab.  I may have been good, but I was nowhere as good as I would be if I stayed in the field for 8 more years.   I thought the Harvard students were basically cheated. 

I guess every review should have a quibble, and I do; but it’s with the publisher, not Dr. Zee. The whole book is one mass of related concepts and is filled with forward and backward references to text, figures and diagrams. Having a page to go to instead of Chapter III, 1 or figure IV.3.2 would make reading much easier. Only the publisher could do this once the entire text has been laid out.

One further point. QFTASAP clarified for me the differences between the (substantial) difficulties of medicine and the (substantial) difficulties of theoretical physics. When learning medicine you are exposed to thousands of unrelated (because we don’t understand what lies behind them) facts. That’s OK because you don’t need to remember all of them. Ask the smartest internist you know to name the 12 cranial nerves or the 8 bones of the wrist. The facts of theoretical physics are far fewer, but you must remember, internalize and use them — that’s why QFTASAP contains all these forward and backward references.

There is a ton more to say about the book and I plan to write more as I go through the book again. If interested, just Google Chemiotics II now and then. QFTASAP is definitely worth reading more than once.

 

Helpful

General relativity at last

I’ve finally arrived at the relativistic gravitational field equation which includes mass, doing ALL the math and understanding the huge amount of mathematical work it took to get there:  Chistoffel symbols (first and second kind), tensors, Fermi coordinates, the Minkowski metric, the Riemann curvature tensor (https://luysii.wordpress.com/2020/02/03/the-reimann-curvature-tensor/) geodesics, matrices, transformation laws, divergence of tensors, the list goes on.  It’s all covered in a tidy 379 pages of a wonderful book I used — “The Geometry of Spacetime” by James J. Callahan, professor emeritus of mathematics at Smith college.  Even better I got to ask him questions by eMail when I got stuck, and a few times we drank beer and listened to Irish music at a dive bar north of Amherst.

Why relativity? The following was written 8 years ago.  Relativity is something I’ve always wanted to understand at a deeper level than the popularizations of it (reading the sacred texts in the original so to speak).  I may have enough background in math, to understand how to study it.  Topology is something I started looking at years ago as a chief neurology resident, to get my mind off the ghastly cases I was seeing.

I’d forgotten about it, but a fellow ancient alum, mentioned our college president’s speech to us on opening day some 55 years ago.  All the high school guys were nervously looking at our neighbors and wondering if we really belonged there.  The prez told us that if they accepted us that they were sure we could do the work, and that although there were a few geniuses in the entering class, there were many more people in the class who thought they were.

Which brings me to our class relativist (Jim Hartle).  I knew a lot of the physics majors as an undergrad, but not this guy.  The index of the new book on Hawking by Ferguson has multiple entries about his work with Hawking (which is ongoing).  Another physicist (now a semi-famous historian) felt validated when the guy asked him for help with a problem.  He never tooted his own horn, and seemed quite modest at the 50th reunion.  As far as I know, one physics self-proclaimed genius (and class valedictorian) has done little work of any significance.  Maybe at the end of the year I’ll be able to read the relativist’s textbook on the subject.  Who knows?  It’s certainly a personal reason for studying relativity.  Maybe at the end of the year I’ll be able to ask him a sensible question.

Well that took 6 years or so.

Well as the years passed, Hartle was close enough to Hawking that he was chosen to speak at Hawking’s funeral.

We really don’t know why we like things and I’ve always like math.  As I went on in medicine, I liked math more and more because it could be completely understood (unlike medicine) –Why is the appendix on the right and the spleen on the left — dunno but you’d best remember it.

Coming to medicine from organic chemistry, the contrast was striking.  Experiments just refined our understanding, and one can look at organic synthesis as proving a theorem with the target compound as statement and the synthesis as proof.

Even now, wrestling with the final few pages of Callahan today took my mind off the Wuhan flu and my kids in Hong Kong just as topology took my mind off various neurologic disasters 50 years ago.

What’s next?  Well I’m just beginning to study the implications of the relativistic field equation, so it’s time to read other books about black holes, and gravity.  I’ve browsed in a few — Zee, Wheeler in particular are written in an extremely nonstuffy manner, unlike medical and molecular biological writing today (except the blogs). Hopefully the flu will blow over, and Jim and I will be at our 60th Princeton reunion at the end of May.  I better get started on his book “Gravity”

One point not clear presently.  If mass bends space which tells mass how to move, when mass moves it bends space — so it’s chicken and the egg.  Are the equations even soluble.

Relativity becomes less comprehensible

“To get Hawking radiation we have to give up on the idea that spacetime always had 3 space dimensions and one time dimension to get a quantum theory of the big bang.”  I’ve been studying relativity for some years now in the hopes of saying something intelligent to the author (Jim Hartle), if we’re both lucky enough to make it to our 60th college reunion in 2 years.  Hartle majored in physics under John Wheeler who essentially revived relativity from obscurity during the years when quantum mechanics was all the rage. Jim worked with Hawking for years, spoke at his funeral and wrote this in an appreciation of Hawking’s work [ Proc.Natl. Acad. Sci. vol. 115 pp. 5309 – 5310 ’18 ].

I find the above incomprehensible.  Could anyone out there enlighten me?  Just write a comment.  I’m not going to bother Hartle

Addendum 25 May

From a retired math professor friend —

”

I’ve never studied this stuff, but here is one way to get more actual dimensions without increasing the number of apparent dimensions:

Start with a 1-dimensional line, R^1 and now consider a 2-dimensional cylinder S^1 x R^1.  (S^1 is the circle, of course.)  If the radius of the circle is small, then the cylinder looks like a narrow tube.  Make the radius even smaller–lsay, ess than the radius of an atomic nucleus.  Then the actual 2-dimensional cylinder appears to be a 1-dimensional line.

The next step is to rethink S^1 as a line interval with ends identified (but not actually glued together.  Then S^1 x R^1 looks like a long ribbon with its two edges identified.  If the width of the ribbon–the length of the line interval–is less, say, than the radius of an atom, the actual 2-dimensional “ribbon with edges identified” appears to be just a 1-dimensional line.

Okay, now we can carry all these notions to R^2.  Take S^1 X R^2, and treat S^1 as a line interval with ends identified.  Then S^1 x R^2 looks like a (3-dimensional) stack of planes with the top plane identified, point by point, with the bottom plane.  (This is the analog of the ribbon.)  If the length of the line interval is less, say, than the radius of an atom, then the actual 3-dimensional s! x R^2 appears to be a 2-dimensional plane.

That’s it.  In general, the actual n+1-dimensional S^1 x R^n appears to be just n-space R^n when the radius of S^1 is sufficiently small.

All this can be done with a sphere S^2, S^3, … of any dimension, so that the actual k+n-dimensional manifold S^k x R^n appears to be just the n-space R^n when the radius of S^k is sufficiently small.  Moreover, if M^k is any compact manifold whose physical size is sufficiently small, then the actual k+n-dimensional manifold M^k x R^n appears to be just the n-plane R^n.

That’s one way to get “hidden” dimensions, I think. “

High level mathematicians look like normal people

Have you ever had the pleasure of taking a course from someone who wrote the book? I did. I audited a course at Amherst from Prof. David Cox who was one of three authors of “Ideals, Varieties and Algorithms” It was uncanny to listen to him lecture (with any notes) as if he were reading from the book. It was also rather humbling to have a full professor correcting your homework. We had Dr. Cox for several hours each weak (all 11 or 12 of us). This is why Amherst is such an elite school. Ditto for Princeton back in the day, when Physics 103 was taught by John Wheeler 3 hours a week. Physics 103 wasn’t for the high powered among us who were going to be professional physicists (Heinz Pagels, Jim Hartle), it was for preMeds and engineers.

Dr. Cox had one very useful pedagogical device — everyone had to ask a question at the beginning of class, Cox being of the opinion that there is no such thing as a dumb question in math.

Well Dr. Cox and his co-authors (Little and O’Shea) got an award from the American Mathematical sociecty for their book. There’s an excerpt below. You should follow the link to the review to see what the three look like along with two other awardees. http://www.ams.org/publications/journals/notices/201604/rnoti-p417.pdf. Go to any midsize American city at lunchtime, and you’d be hard pressed to pick four of the five out of the crowd of middle aged men walking around. Well almost — one guy would be hard to pick out of the noonday crowd in Williamsburg Brooklyn or Tel Aviv. Four are extremely normal looking guys, not flamboyant or bizarre in any way. This is certainly true of the way Dr. Cox comports himself. The exception proving the rule however, is Raymond Smullyan who was my instructor in a complex variables course back in the day– quite an unusual and otherworldly individual — there’s now a book about him.

Here’s part of the citation. The link also contains bios of all.

“Even more impressive than its clarity of exposition is the impact it has had on mathematics. CLO, as it is fondly known, has not only introduced many to algebraic geometry, it has actually broadened how the subject could be taught and who could use it. One supporter of the nomination writes, “This book, more than any text in this field, has moved computational algebra and algebraic geometry into the mathematical mainstream. I, and others, have used it successfully as a text book for courses, an introductory text for summer programs, and a reference book.”
Another writer, who first met the book in an REU two years before it was published, says, “Without this grounding, I would have never survived my first graduate course in algebraic geometry.” This theme is echoed in many other accounts: “I first read CLO at the start of my second semester of graduate school…. Almost twenty years later I can still remember the relief after the first hour of reading. This was a math book you could actually read! It wasn’t just easy to read but the material also grabbed me.”
For those with a taste for statistics, we note that CLO has sold more than 20,000 copies, it has been cited more than 850 times in MathSciNet, and it has 5,000 citations recorded by Google Scholar. However, these numbers do not really tell the story. Ideals, Varieties, and Algorithms was chosen for the Leroy P. Steele Prize for Mathematical Exposition because it is a rare book that does it all. It is accessible to undergraduates. It has been a source of inspiration for thousands of students of all levels and backgrounds. Moreover, its presentation of the theory of Groebner bases has done more than any other book to popularize this topic, to show the powerful interaction of theory and computation in algebraic geometry, and to illustrate the utility of this theory as a tool in other sciences.”

A book recommendation, not a review

My first encounter with a topology textbook was not a happy one. I was in grad school knowing I’d leave in a few months to start med school and with plenty of time on my hands and enough money to do what I wanted. I’d always liked math and had taken calculus, including advanced and differential equations in college. Grad school and quantum mechanics meant more differential equations, series solutions of same, matrices, eigenvectors and eigenvalues, etc. etc. I liked the stuff. So I’d heard topology was cool — Mobius strips, Klein bottles, wormholes (from John Wheeler) etc. etc.

So I opened a topology book to find on page 1

A topology is a set with certain selected subsets called open sets satisfying two conditions
l. The union of any number of open sets is an open set
2. The intersection of a finite number of open sets is an open set

Say what?

In an effort to help, on page two the book provided another definition

A topology is a set with certain selected subsets called closed sets satisfying two conditions
l. The union of a finite number number of closed sets is a closed set
2. The intersection of any number of closed sets is a closed set

Ghastly. No motivation. No idea where the definitions came from or how they could be applied.

Which brings me to ‘An Introduction to Algebraic Topology” by Andrew H. Wallace. I recommend it highly, even though algebraic topology is just a branch of topology and fairly specialized at that.

Why?

Because in a wonderful, leisurely and discursive fashion, he starts out with the intuitive concept of nearness, applying it to to classic analytic geometry of the plane. He then moves on to continuous functions from one plane to another explaining why they must preserve nearness. Then he abstracts what nearness must mean in terms of the classic pythagorean distance function. Topological spaces are first defined in terms of nearness and neighborhoods, and only after 18 pages does he define open sets in terms of neighborhoods. It’s a wonderful exposition, explaining why open sets must have the properties they have. He doesn’t even get to algebraic topology until p. 62, explaining point set topological notions such as connectedness, compactness, homeomorphisms etc. etc. along the way.

This is a recommendation not a review because, I’ve not read the whole thing. But it’s a great explanation for why the definitions in topology must be the way they are.

It won’t set you back much — I paid. $12.95 for the Dover edition (not sure when).