I’ve been doing a lot of work recently just on a toy model, but I wanted to do a book review. I think I shall discuss Peskin and Schroeder’s An Introduction to Quantum Field Theory, since I have been studiously working out every problem in it just for fun (yeah, I’m weird, I do every problem in the books I read). (If you are a poor college student, you can find free lecture notes that cover essentially the entire book written very well by Dr Osborn…did I mention, it’s free?)
It’s neatly divided into three parts, dealing with a seeming historical introduction to the subject (starting with the Klein Gordon and Dirac equations, then getting to some QED); renormalization; and non-abelian gauge theory (respectively).
The writing style is very peculiar…kind of like Karl Marx’s Das Kapital: if you don’t studiously rewrite certain sections of this book word for word in terms you understand, you’ll miss a lot from it.
This is very bad, particularly because the section on Feynman diagrams is like this! So if you are reading this book to learn quantum field theory, I would urge you to first read Griffith’s Introduction to Elementary Particles to learn Feynman diagrams (its the best introduction for Feynman diagrams in my opinion) and Mandl and Shaw’s Quantum Field Theory.
Remarks on Section One
Unfortunately, the first two chapters appear to be this way as well…but, like Kapital, once you know Feynman diagrams and have rewritten the first two chapters in terms you understand (not that it’s particularly esoteric, but because there is so much information crammed into so few words), you can sail through the rest of the book easily.
It should be noted that the book is like any other modern book on the subject, there is a line somewhere remarking on how the canonical approach using commutators and the path integral approach using action are “complimentary”…so logically, we’ll only discuss path integrals and never ever mention the canonical approach ever again in this book. I was disappointed, particularly because I prefer the canonical method of quantization!
There are some topics which are not covered in near enough detail…the topic of bound states got roughly 2 pages. To be fair, 2 references were given for further investigation (one was an entry in a book, the other was G. T. Bodwin, D. R. Yennie, and M. A. Gregorio’s Recoil effects in the hyperfine structure of QED bound states in Review of Modern Physics, volume 57, Issue 3, pages 723 – 782 (1985)).
(If anyone knows a great book on the subject, I would be delighted if they would post it!)
The introduction to Feynman diagrams was frighteningly ambiguous. It was more of a deus ex machina than anything else. But I have found it most difficult to explain using Feynman diagrams and computing them to people who know nothing about them. This “introduction” included Wick’s theorem, which is frustratingly short.
The elementary processes of quantum electrodynamics are then introduced in the next chapter followed by radiative corrections in the next two chapters. I can’t really say one way or the other whether it was done well or poorly since I was already familiar with Feynman diagrams, but I would suspect there is much frustration if one didn’t know them well.
Remarks on Section Two
I think the section on Renormalization would scare the hell out of anyone who hasn’t read Griffith’s book I recommended above, or (if one is a mathematician) someone who doesn’t know combinatorics like the back of their hand.
Oddly enough, it begins with an introduction to the use of functional methods in Quantum Field Theory. I don’t see this as being bad, but I just didn’t think it tied in too well with the theme. I enjoyed the section though, I thought it was well written.
After that chapter, we get down to business and brass tacks: renormalization! It is mildly frustrating if one doesn’t know combinatorics (while trying to explain the “systematics” of renormalization, especially counting of divergences, to my friend he asked “Where are all these factorials coming from?”). Just remember n! = number of permutations of n objects, and “n choose k” is the number of different ways to choose k objects from n objects.
Goldstone’s theorem is covered mildly well, but the notion of spontaneous symmetry breaking was not. Remember, if there is a nonzero vacuum expectation value of a given field, then it’s spontaneously broken. This was dealt with better in Stefan Pokorski’s Gauge Field Theories in my opinion.
I admittedly skipped around and had a weird mathematical background prior to getting to the next chapter on Wilson’s approach and the Renormalization group. I don’t know how well one could learn about Wilson loops without knowledge of differential geometry (how can you know about Holonomy loops?) but meh I suppose it’s not the worst thing in the world.
The notion of running constants was introduced here as well, and they did an interesting job with it…using an analogy that Sidney Coleman came up with (comparing running constants to bacteria growing on a pipe). I was content with this section, but they could have done better with the renormalization group.
I kept jumping back and forth between analyzing what they were telling me; “It is a group, not it’s not, oh yes it is, no wait it isn’t, oh wait it is…hmmmm!” That was from my scratch notes on the section. (Consequently, I left more dazed and confused than when I came, so I referred to another book to see if it is a group or not and the answer is — as best as I can make sense of it — kind of but not really.)
So be warned about the section on renormalization, it’s frightningly weird! Or, if I may quote Dante:
“ABANDON ALL HOPE, YOU WHO ENTER HERE.
These words, dark in hue, I saw inscribed
over an archway. And then I said:
‘Master, for me their meaning is hard.’
And he, as one who understood:
‘Here you must banish all distrust,
here must all cowardice be slain.
…
‘We have come to where I said
you would see the miserable sinners
who have lost the good of the intellect.’”
Dante was talking about renormalization, err, I mean hell. It’s so hard to tell them apart…
Remarks on Section Three
But I must confess here that I have prior familiarity with Lie Algebras and Lie Groups, so chapter 15 Non-Abelian Gauge Invariance was relatively trivial. If you know nothing of Lie Algebras, I would highly advise reading Artin’s Algebra particularly the section on Lie Algebras (that’s where I learned it!). Then, given every matrix group (particularly the ones like O(n), SO(n), SL(n), U(n), Sp(n), SU(n)) find the lie algebra. Afterwards, read Daniel Bump’s Lie Groups. Then take two courses on Lie groups…And you’re set!
So in this way I cannot really comment too much on the coverage of the quantization of non-abelian gauge theories, since it seemed straightforward to me.
Naturally, isospin was given as the historical example…but a number of new tricks were introduced which may frighten some (e.g. Faddeev-Popov ghosts and BRST quantization).
The introduction to partons and QCD was frustrating to me, at least, since they more or less found it to be “god given” rather than cover the rich history behind strong force. Why, Regge theory isn’t even mentioned one bit!
The introduction to the standard model was kind of “god given” as well, just believe really really hard that there is no reason to derive it. But if one is versed in representation theory and Lie groups, by all means try your hand at doing Dr Baez’s homework problem sets on elementary particles here. It’s actually quite fun!
But it should be noted, armed with the tools previously covered like renormalization and radiative corrections, the standard model chapters just seem like an application to a bizarre gauge group SU(3)xSU(2)xU(1). There is a lot of Hocus Pocus with the Higgs mechanism, though…I would have liked to seen a better, more thorough section on this subject. What can one do?
Overall, the book is a great stepping stone to scarier texts like Ryder or Weinberg’s books…but as an introduction ab initio I would not recommend it.
Addendum: Spinors
I was disappointed with the introduction to spinors, but this is not uncommon for me. The only real introduction is Penrose and Rindler’s Spinors and Spacetime in my opinion. But they can be intimidating for those new to them, so I would urge them to read Griffith’s book first…he covers them in a way that is pragmatic for physicists. A mathematician would cry at how closed he is, as opposed to being as general as the Spin(n) group (which is a “better” definition since it’s far more vague and general). Just my two cents on how spinors are dealt with in this book…
![f(x) = f(x_{0}) + f'(x_{0})(x-x_{0}) + \mathcal{O}([x-x_{0}]^2)](http://l.wordpress.com/latex.php?latex=f%28x%29+%3D+f%28x_%7B0%7D%29+%2B+f%27%28x_%7B0%7D%29%28x-x_%7B0%7D%29+%2B+%5Cmathcal%7BO%7D%28%5Bx-x_%7B0%7D%5D%5E2%29&bg=ffffff&fg=444444&s=0 "f(x) = f(x_{0}) + f'(x_{0})(x-x_{0}) + \mathcal{O}([x-x_{0}]^2)")
. I then introduced the Fourier transform as a sort of generalization of the Fourier Series when we take the interval from [-
, 
giving us the interval 
which is unbounded. In general, when we have unbounded intervals, we use Fourier transforms; when we have bounded intervals, we have the Fourier series.
where 
are integers with 
being nonzero. With our fancy pants base 10 number system, we can use scientific notation to represent numbers like 
, or 
.
and 
.
where 
is the exponent part. We can translate it back to a number by the following relation
into base 2. We have 
. 
where 
means we’re using base b, and 3-2=1 so we need to add the equivalent of 1 in base 2 which is 1. Thus 10+1=11.
so the first bit is 0, we multiply by 2 again
so the second bit is 0
so the third bit is 1
the fourth bit is 0
the fifth bit is 0
the sixth bit is 1
the seventh bit is 0
the eighth bit is 0
the ninth bit is 0
the tenth bit is 0
the eleventh bit is 1
the twelfth bit is 1
the thirteenth bit is 1
.
if we just take the difference between the two numbers. And of course, after normalization we find 
in base 10.
? That’s what I am going to do for all practical purposes (if this bothers you, and it may possibly bother me later, then just treat it as two tuples).
and 
. These are three special “numbers”. Why? 
. If one of those numbers is negative, we get 
. There is a natural question that arises: if we use 
bits for the fractional part, what’s the smallest number greater than one that we can represent that? The answer is surprisingly simple: it’s 
.
.
numbers with this precisely, everything else will have roundoff error.
.
such that for any 
we have 
(Existence of Additive Identity).
is not associative, nor is there always an inverse element.
.
operator.
such that for any 
(i.e. we have the multiplicative identity).
has a multiplicative inverse. So it forms an Abelian group too. The lack of distributivity is what’s preventing this from being a nonassociative ring! Wo is me!
).
mod 4
mod 4
, when we square it we get 
which is 0 mod 4. If we have an odd number, 
, then we see 
mod 4.
digits) as a sort of “floating point” data type, that is we have
are 
is functionally differentiable at…”).
