Peskin & Schroeder - page 5 - Computation of ##\mathcal{M}##

[spaghetti3451]

The following is taken from page 5 of Peskin and Schroeder. It talks about the computation of $\mathcal{M}$ for the annihilation reaction $e^{+}e^{-}\rightarrow \mu^{+}\mu^{-}$.

Even for this simplest of QED processes, the exact expression for $\mathcal{M}$ is not known. Actually this fact should come as no surprise, since even in nonrelativistic quantum mechanics, scattering problems can rarely be solved exactly. The best we can do is obtain a formal expression for $\mathcal{M}$ as a perturbation series in the strength of the electromagnetic interaction, and evaluate the first few terms in this series.

Feynman has invented a beautiful way to organize and visualize the perturbation series: the method of Feynman diagrams. Roughly speaking, Feynman diagrams display the flow of electrons and photons during the scattering process. For our particular calculation, the lowest-order term in the perturbation series can be represented by a single diagram, shown below. The diagram is made up of external lines (representing the four incoming and outgoing particles), internal lines (representing virtual particles, in this case one virtual photon), and vertices. Straight lines are used for fermions and wavy lines are used for photons. The arrows on the straight lines denote the direction of negative charge flow, not the direction of momentum. A $4$-momentum vector is assigned to each external line. The momentum $q$ of the one internal line is determined by momentum conservation at either of the vertices: $q=p+p'=k+k'$. A spin state (either up or down) must also be associated with each external fermion.

I have the following questions regarding the above two paragraphs:

1. Why is the perturbation series called a formal expression?
2. What is the original Hamiltonian and the interaction Hamiltonian for this problem?
3. What is the expression for the strength of the electromagnetic interaction?
4. Why is the perturbation series obtained in the strength of the electromagnetic interaction? Can't we obtain the perturbation series in the order of some other quantity?

 

[DrDu]

to 2: The Hamiltonian for the interacting system is not known for QED in 3+1 dimensions. An interaction hamiltonian does not exist (this is the content of Haag's theorem)
to 1: The perturbation series is known to be divergent (I think there is a simple argument by Dyson as to why).

 

[Vanadium 50]

Failexam, you have posted three questions on the first five pages (really three, since it starts on page 3). This is one question per page, which means we have another 800 questions to look forward to. Is it possible that this book is too advanced for you?

 

[JorisL]

You are getting ahead of yourself.
Question 3 and 4 will be answered along the way through the book.
You'll probably learn a good deal about question 2 up to the point you can either find out what the answer is or find other resources on that issue.

The only "valid" question might be the one about formal expressions.
I don't know if they define exactly what they mean with that in the book, the picture should be clear once you've banged your head for hours getting through the topic.

Remember that section is an "invitation" meant to get you motivated to learn from the book.

 

[spaghetti3451]

Failexam, you have posted three questions on the first five pages (really three, since it starts on page 3). This is one question per page, which means we have another 800 questions to look forward to. Is it possible that this book is too advanced for you?

I feel lost reading Peskin and Schroeder, and also random questions always pop up in my head. I just can't keep control of the random questions. I feel like I have to get them answered, or at the very least, have the questions written down.

I was hoping that these threads would serve as a useful guide to advanced undergraduates wishing to learn QFT from Peskin and Schroeder w/o prior experience to the subject.

These threads could be added to https://www.physicsforums.com/threads/pf-is-a-resource-for-questions-about-peskin-schroeder.400073/ which has been very helpful to me.

I'll be posting more questions on Peskin and Schroeder, but if that somehow crosses the limit on what's to be asked on the forum, then I'll stop my queries once that matter is communicated to me.

As for the number of questions, the above are handpicked from my notes. The number of questions in my handwritten notes is enormous and covers all possible queries that could come to a reader.

 

[spaghetti3451]

You are getting ahead of yourself.
Question 3 and 4 will be answered along the way through the book.
You'll probably learn a good deal about question 2 up to the point you can either find out what the answer is or find other resources on that issue.

The only "valid" question might be the one about formal expressions.
I don't know if they define exactly what they mean with that in the book, the picture should be clear once you've banged your head for hours getting through the topic.

Remember that section is an "invitation" meant to get you motivated to learn from the book.

Thanks! I guess a lot of the material that I read I have to take on faith the first time. Things are going to make sense only once my understanding of the subject has matured after a couple of years.

I never faced difficulties learning quantum mechanics, for example, from Griffiths or Gasiorowicz. QFT is hard.

 

[Vanadium 50]

I feel lost reading Peskin and Schroeder, and also random questions always pop up in my head. I just can't keep control of the random questions.

That is a good sign that the book is too advanced for you. Learning is not a race; it's better to learn slowly than to not learn quickly.

 

[spaghetti3451]

That is a good sign that the book is too advanced for you. Learning is not a race; it's better to learn slowly than to not learn quickly.

I've chased way too many textbooks on QFT. In fact, my laptop has at least 30 textbooks on QFT, ranging from Zee to Peskin to Itykson to Weinberg to whatnot.

My goal is to read one textbook and one textbook only , and the textbook I've chosen is P&S. Because it's hard, I'll only read 4 pages a day. Although questions keep popping up now, I hope that when I actually do courses on QFT in grad school, it's going to be a lot easier on me. In those courses, things are just naturally going to fall in place as I've already seen and read P&S by then. That's the purpose behind trying to study P&S now.

 

[spaghetti3451]

That is a good sign that the book is too advanced for you. Learning is not a race; it's better to learn slowly than to not learn quickly.

The other motivation behind wanting to read P&S is my belief that this is always going to be a hard textbook regardless of how advanced I am.

The only courses I'll be doing before QFT now are courses in:

Advanced Classical Mechanics; Advanced Quantum Mechanics; Advanced Classical Electromagnetism; Advanced Statistical Mechanics.

Textbooks on these subjects I can read and understand easily.

 

[DrDu]

I'd rather say that who doesn't come up with 800 questions while reading about QFT is either a genius or a complete idiot.

 

[haushofer]

My main lesson of studying QFT (with P&S) was that learning is not a linear process. So you should develop the ability to read on while having questions, and to find a middleway between rigour and conceptual understanding. I completely agree with DrDu.

 

[MathematicalPhysicist]

My goal is to read one textbook and one textbook only , and the textbook I've chosen is P&S.

Wrong approach, I understand that you don't have enough time now reading several more books and you want to concentrate on one book; but in the long run you should understand that by reading several books will help you master this field and its esoteric extensions.

I myself used Srednicki and Radovanovic problem book, and by reading these books I know that there are still some subjects which weren't dealt in these books and are dealt in other books such as Weinberg and P&S and others.

Also in String theory you won't find everything you want in just one book.

Be patient, wisdom comes slowly with age and accumulating knowledge.

 

[spaghetti3451]

My main lesson of studying QFT (with P&S) was that learning is not a linear process. So you should develop the ability to read on while having questions, and to find a middleway between rigour and conceptual understanding. I completely agree with DrDu.

Is QFT the hardest thing in theoretical physics?

Is string theory even harder?

 

[spaghetti3451]

Wrong approach, I understand that you don't have enough time now reading several more books and you want to concentrate on one book; but in the long run you should understand that by reading several books will help you master this field and its esoteric extensions.

I myself used Srednicki and Radovanovic problem book, and by reading these books I know that there are still some subjects which weren't dealt in these books and are dealt in other books such as Weinberg and P&S and others.

Also in String theory you won't find everything you want in just one book.

Be patient, wisdom comes slowly with age and accumulating knowledge.

I will be reading more textbooks on QFT in the future (hopefully), once I get the basic gist of the major concepts of QFT. I'm hoping it's only going to get easier reading QFT textbooks once I finish the first one.

 

[MathematicalPhysicist]

Is QFT the hardest thing in theoretical physics?

Is string theory even harder?

There's also turbulence in fluid dynamics which I heard is very hard.

As been said here, be patient you won't understand everything in your first reading; even if you ask on every page questions it's best if you gather all your questions in one thread as I did in the summer of 2014 from Srednicki.
Quite a thrilling endeavour is reading QFT.

 

[spaghetti3451]

There's also turbulence in fluid dynamics which I heard is very hard.

As been said here, be patient you won't understand everything in your first reading; even if you ask on every page questions it's best if you gather all your questions in one thread as I did in the summer of 2014 from Srednicki.
Quite a thrilling endeavour is reading QFT.

Well, I'm preparing a LaTeX document with skeleton notes and my all associated questions.

The list of questions is piling up to be enormous.

 

[DrSuage]

QFT IS HARD. That having been said, its almost certainly not beyond you if you are willing to put in serious effort. If you've gotten far enough to find P&S and to be taking the other courses you mentioned, you will be able to learn QFT.

However, there are some things you must bear in mind:

Firstly and most importantly: Peskin and Schroeder is NOT a book that is to be read.

It is an instruction manual that will tell you how to learn QFT. In other words, it does not give you the answers, it shows you how to go about coming up with the answers yourself. Your task is to go through each chapter and "fill in the blanks" i.e do all the in between steps. When Peskin says "and it is easy to show that..." he does not mean that it is actually easy to show, he means that it can be shown without resorting to mathematical techniques which the reader is unlikely to have come across. Indeed it may take you 3-4 pages of nasty algebra to get the desired result. Every single time a formula appears you must work out how it was derived on a scrap of paper (unless its a postulated thing).

Secondly: Do not expect yourself to get through the material quickly. Peskin himself said that he once tried to teach all of the material to a graduate level class in one year and it was a disaster - there is simply too much to get through. P&S will serve you as a reference text long into a research career
At Imperial College the chapters 2, 3 and 4 are covered in a 26 lecture final year undergraduate course. Admittedly, it moves at a fairly slow pace (Cambridge part iii will do 2-7 in a single 24 lecture course). This works out at roughly 8 hours of lectures per chapter (giving 2 hours of the introductory motivational stuff). And that's not even factoring in the time an undergrad would spend doing the homework problems etc.

If you are studying by yourself - I'd try and do one chapter a month (assuming your doing your undergrad courses concurrently and so you'll have to spend time on those as well.)
Break it down into small subsections.

The introductory chapter is motivational - i.e. don't worry if you don't totally understand what they're doing - they're just trying to get you interested and give you a flavour of what doing QFT is about and give you some perspective in case you get bogged down in the formalism of the first 3 chapters and forget why your doing what your doing.

As for string theory - I'm currently taking my first course in that. I can't really say whether its hard or not - it depends what you find hard. TBH my course has hardly contained any "classic" string theory (i.e 70's stuff like the bosonic string) because our lecturer is focusing mainly on the representation theory so far. I'm going to stick with it for another week or two and see what happens but I don't think I'm going to try and do a phd in strings - its too competitive for the likes of me - I'm sticking to phenomenology.

Good luck learning QFT - if your trying to do it by yourself that shows real initiative and I think you have every chance of success. Don't give up, and don't let people put you off.

 

[vanhees71]

QFT is hard, but the trick is to do the calculations carefully yourself. Start with simple $\phi^4$ theory and try to follow the derivation of the Feynman rules (in both the canonical operator quantization and the path-integral method with generating functionals for Green's functions and proper vertex functions) and then start calculating scattering amplitudes (S-matrix elements) and cross sections at tree level. The next step is to evaluate loop diagrams and learn about renormalization. The latter is the hardest part to understand.

After that you can attack the "fun part" of the subject, which are symmetries, most importantly local gauge symmetries. For that, you should be familar with the path-integral quantization. Again go slowly and start with the Abelian case, i.e., QED, including a good understanding of Ward-Takahashi identities and the renormalizability. Then you can finally go to the non-Abelian case. I'd start with QCD, which is an un-Higgsed SU(3) gauge theory. At this step you should also learn about the renormalization group and asymptotic freedom. The most simple approach is the background-field gauge, where you have simple Ward-Takahashi identities instead of the more involved Slavnov-Taylor identities based on BRST symmetry of the gauge-fixed action. Finally you can learn about the Anderson-Higgs-et-al mechanism and electroweak theory (Quantum Flavor Dynamics aka. Glashow-Salam-Weinberg model).

The most important thing is, not to get frustrated to soon. It takes quite some time to learn all the calculational techniques. My favorite book (at the moment) as an introduction is

M. D. Schwartz, Introduction to Quantum Field Theory and the Standard Model, Cambridge University Press (2014)

It's much more carefully written than Peskin/Schroeder which is sometimes a bit too sloppy, even committing mortal sins like having dimensionful arguments in logarithms (and, even worse, that even in the chapter on the renormalization group!).

 

[DrDu]

QFT is hard, but the trick is to do the calculations carefully yourself. Start with simple $\phi^4$ theory and try to follow the derivation of the Feynman rules (in both the canonical operator quantization and the path-integral method with generating functionals for Green's functions and proper vertex functions) and then start calculating scattering amplitudes (S-matrix elements) and cross sections at tree level. The next step is to evaluate loop diagrams and learn about renormalization. The latter is the hardest part to understand.

I am not sure what you can learn from this.
After all, you want to describe interacting particles and a $\phi^4$ theory is (most probably) trivial.

 

[vanhees71]

You learn the techniques of calculating perturbative S-matrix elements with the most simple theory available.

 

[samalkhaiat]

I've chased way too many textbooks on QFT. In fact, my laptop has at least 30 textbooks on QFT, ranging from Zee to Peskin to Itykson to Weinberg to whatnot.

My advise to you: never learn theoretical physics from books written phenomenologists even if you want to be one.

In my opinion, Mandl & Shaw “Quantum Field Theory”, and Greiner & Reinhardt “Field Quantization” are the best textbooks for a beginner.

 

[DrSuage]

I am not sure what you can learn from this.
After all, you want to describe interacting particles and a $\phi^4$ theory is (most probably) trivial.

Phi-4 teaches you how the whole apparatus works without distracting you with all the other weird and wonderful bits and bobs. It is ESSENTIAL for seeing the big picture of what the hell it is your even trying to do for the first 7 chapters! I would also make sure you get Yukawa theory down since it will allow you to see the connection between particle exchange and potentials in a nice way. Have a look at David Tong's notes on the Cambridge website - he discusses an even easier version of Yukawa theory with the fermions replaced by a complex scalar field. If you get all of those down in a couple of months, its not hard to scale up to QED.

 

[vanhees71]

My advise to you: never learn theoretical physics from books written phenomenologists even if you want to be one.

In my opinion, Mandl & Shaw “Quantum Field Theory”, and Greiner & Reinhardt “Field Quantization” are the best textbooks for a beginner.

It depends. I don't know about Mandl and Shaw, but W. Greiner and Joachim Reinhardt (both at my university ;-)) are for sure phenomenologists. Nevertheless their book (part of an entire theoretical-physics course-book series).

Also Peskin and Schroeder is not bad. Unfortunately it's sometimes too sloppy and it has a lot of typos. The only book mentioned, I explicitly would not recommend at all is Zee's "QFT in a Nutshell". It has too much material for the little nushell ;-)).

I'd rather advice to to the contrary to first look at QFT books written by phenomenologists, because it's likely to get the essence of the theory with regard to physics from them. If interested, you can later read more specialized books on the math side of the subject although it's pretty frustrating, because after all these years of struggle with all kinds of "axiomatic QFT" so far nobody has come up with a mathematically rigorous foundation of the theory applicable to realistic situations (i.e., the Standard Model).

 

[MathematicalPhysicist]

I'd rather advice to to the contrary to first look at QFT books written by phenomenologists, because it's likely to get the essence of the theory with regard to physics from them. If interested, you can later read more specialized books on the math side of the subject although it's pretty frustrating, because after all these years of struggle with all kinds of "axiomatic QFT" so far nobody has come up with a mathematically rigorous foundation of the theory applicable to realistic situations (i.e., the Standard Model).

Well, it shouldn't stop us from trying; I mean string theory has been around without any empirical evidence so far (something like more than 30 years).

 

[vanhees71]

Of course, one should not stop trying!

 

[MathematicalPhysicist]

Of course, one should not stop trying!

Unless it's theoretically impossible like theorems in algebra that tells you cannot find a solution in radicals to quintic polynomials and above.

 

[vanhees71]

But that's then also a solution from a mathematical point of view! For physics it's of little relevance in some sense, because many (if not most) physicists believe anyway that QFT is an effective theory anyway, i.e., valid only up to an energy, and thus you can introduce some cutoff (e.g., by putting it on a lattice), and in this sense everything is well-defined.

It's a little bit like with Haag's theorem which says that the interaction picture used to derive the most successful models of physics (the Standard Model) in fact doesn't exist. Nevertheless one can work with it in the sense of a heuristic tool to derive perturbative QFTs, leading to these successful models.