Workshop on Weinberg's Theory of Quantum Fields

[Si]

Workshop on Weinberg's Quantum Theory of Fields

Hi, I've just joined this forum. I've been reading Weinberg's Quantum Theory of Fields Vol I for a few years on and off, and I'm now on Chapter 10. While I found it to be the most enlightening approach to Quantum Field Theory, I found parts of it heavy going. If there is anyone out there who, like me, has some answers to offer to some of the tricky bits but still many questions, do you fancy joining me on Weinberg Workshop here?

First question:

Chapter 2, page 68: Why does Weinberg say "It is very important that when Lambda_mu^nu is an arbitrary 3-dim rotation R, the Wigner rotation W(Lambda,p) is the same as R for all p"? Is this just for convenience, otherwise the equations get too complicated? I see this is true for the particular choice he uses, but I don't see why it has to be true.

 

[marcus]

Originally posted by Si
Hi, I've just joined this forum. I've been reading Weinberg's Quantum Theory of Fields Vol I for a few years on and off, and I'm now on Chapter 10. While I found it to be the most enlightening approach to Quantum Field Theory, I found parts of it heavy going. If there is anyone out there who, like me, has some answers to offer to some of the tricky bits but still many questions, do you fancy joining me on Weinberg Workshop here?

First question:

Chapter 2, page 68: Why does Weinberg say "It is very important that when Lambda_mu^nu is an arbitrary 3-dim rotation R, the Wigner rotation W(Lambda,p) is the same as R for all p"? Is this just for convenience, otherwise the equations get too complicated? I see this is true for the particular choice he uses, but I don't see why it has to be true.

The natural place at PF to take a question like that would seem to be Tom's QFT thread. You could help bring that thread back to life.

One problem that kind of workshop faces is finding a book that everybody wants to use-----they have tried using Warren Siegal "Fields" (arxiv.org/hep-th/9912205) which has the advantage of being online
and then one of the group, Rutwig, suggested Casalbuoni "Advanced Quantum Field Theory" which is also online
http://arturo.fi.infn.it/casalbuoni/lezioni99.pdf

And now you have a question stemming from Weinberg's book.
Well I would still post it in the QFT thread because someone there probably has Weinberg on the shelf and can look up the page reference you gave.

At the top of the list of topics in each forum are some hints for making symbols and also if you press the "quote" button at the bottom of a post you get a screen showing how that person made the various subscripts and greek letters etc. It may be helpful and you can always back off from the screen after learning the typography. Just for fun I will retype what you wrote about Lambda----see if it comes thru on your browser:

"It is very important that when Λ_μ^ν is an arbitrary 3-dim rotation R, ..."

 

[Si]

Purpose

Actually the question was not the most important part of my last posting (although I still need the answer, if anyone has it!), I just wanted to get the ball rolling.

Of course everybody has a favourite textbook. What I want is a thread for those people whose favourite is Weinberg's, like me, because I would like to help and get help. It is for the very reason that people cannot normally agree on one book to use that each thread should be dedicated to a single book, different for each thread, so people can choose the thread that suits them. QFT is just too vast and with too many approaches for it to have just one thread.

Secondly in Tom's thread most people seem interested in things beyond the foundations of QFT, and my thread is for people who want a more advanced knowledge of the basics (symmetries, the path integral, renormalization etc.).

Thanks for the advice - unfortunately it didn't come through quite right on my browser, but now I know what to do.

 

[marcus]

Originally posted by Si
...

Of course everybody has a favourite textbook. What I want is a thread for those people whose favourite is Weinberg's, like me, because I would like to help and get help. ...

...my thread is for people who want a more advanced knowledge of the basics (symmetries, the path integral, renormalization etc.).

this makes really good sense to me. I hope you succeed in getting a Weinberg-basics study-group going!

 

[Si]

I hope you and others will join. I would hope that those who don't have Weinberg Volume I but who want to join this thread can find it in a library or borrow from someone, although I understand this is a hassle.

My reason for choosing Weinberg: His approach feels much more "pure" than other books, particularly in that only simple physically inuitive axioms are introduced and only when they are needed. Thus your knowledge doesn't get entangled, the various theorems become more powerful since they can then be extended to other theories, and theorems can be obtained more completely and generally yet made simpler. His brief yet comprehensive style helps one avoid getting confused, although sometimes he is a bit too brief!

Examples: He doesn't say that the causality condition follows from something to do with measurements at spacelike separation not affecting each other, but because it is necessary to make the S-matrix Lorentz invariant. He doesn't quantize fields of classical field theories such as electromagnetism, as there is no real physical reason to do so. Rather he starts with "particles" (defined to be eigenstates of the generators of the Poincare group), then shows how fields arise from the need to satisfy the very obvious cluster decompostion principle. He gives complete proofs (e.g. he completes the theorem of Wigner, and shows that the physically-unintuitive Dirac equation is not an axiom, but turns out to be the only possibility for spin 1/2 fields) and derives results in a very general manner (e.g. derives LSZ reduction for fields of arbitrary spin and proves the spin-statistics theorem). You can obtain a lot of well known results in scattering theory (Ch. 3)
without using fields.

 

[jeff]

Originally posted by Si
...He doesn't say that the causality condition follows from something to do with measurements at spacelike separation not affecting each other...

He does mention this in the second last full paragraph on page 198.

Originally posted by Si
He doesn't quantize fields of classical field theories such as electromagnetism, as there is no real physical reason to do so. Rather he starts with "particles" (defined to be eigenstates of the generators of the Poincare group), then shows how fields arise from the need to satisfy the very obvious cluster decompostion principle.

Particle states arise since it's their masses and spins that label the irreducible representations of SO(3,1) under which they transform. The cluster decomposition principle is invoked to explain why and how the hamiltionian must be constructed from creation and annhilation operators acting on these states. But it's lorentz invariance that requires these operators be grouped together to form quantum fields that satisfy causality.

You're right that weinberg doesn't construct QED by quantizing maxwell, but he deduces it first from the gauge-invariance principle he shows any quantum theory of massless particles with spin must satisfy.

Here's a question for you. Can you verify the expression on page 548 in section 13.4?

 

[Si]

Originally posted by jeff
He does mention this in the second last full paragraph on page 198.

Although this is not really his reason for making it so. He gives a more formal argument - that it's needed for Lorentz invariance.

Particle states arise since it's their masses and spins that label the irreducible representations of SO(3,1) under which they transform. The cluster decomposition principle is invoked to explain why and how the hamiltionian must be constructed from creation and annhilation operators acting on these states. But it's lorentz invariance that requires these operators be grouped together to form quantum fields that satisfy causality.

Yes, I didn't mention LI + causality for brevity. CDP + LI + causality (+ anything else?) leads to fields.

You're right that weinberg doesn't construct QED by quantizing maxwell, but he deduces it first from the gauge-invariance principle he shows any quantum theory of massless particles with spin must satisfy.

Actually here I felt was one of Weinberg's weaker points. I arrived at the same question I do
from other QFT books when the author tries to derive the QED Lagrangian from the gauge invariance principle: Is the QED Lagrangian the only possibility (for Abelian fields)? Perhaps I missed something in Weinberg's argument.

Here's a question for you. Can you verify the expression on page 548 in section 13.4?

Probably not, as I am only on Chapter 12! However, I will try to look at it tonight.

 

[jeff]

Originally posted by Si
Probably not, as I am only on Chapter 12! However, I will try to look at it tonight.

Never mind.

 

[lethe]

Si: i might be willing to try weinberg with you. i have a copy.

 

[Si]

Great - how much have you read, if any? How did you find it? What textbooks have you found helpful?

By the way, chapter one is a historical account of QFT, not necessary for understanding the rest of the book.

 

[jeff]

Originally posted by lethe
Si: i might be willing to try weinberg with you. i have a copy.

Everyone's achilles heal around here is QFT. This really sucks because it lies at the intersection of virtually every subject in theoretical physics.

 

[Si]

Originally posted by jeff
Everyone's achilles heal around here is QFT. This really sucks because it lies at the intersection of virtually every subject in theoretical physics.

Definitely. Yet how easy it is to understand depends mostly on how well explained it is.

I've been trying to write a companion to Weinberg, to fill in the more technical steps and analyse in more depth the statements that Weinberg thinks is obvious. But it's a little messy.