Questions in Weinberg The Quantum Theory of Fields Vol 1

In summary, Weinberg concludes that ζ_σ =ζ(-1)^(j-σ) in the next page, but doesn't explain why ζ_σ should be proportional to (-1)^j. I think k^2l Y_lm(k) is also a simple polynomial function, so why did he choose 'l' specially? In addition, I don't understand how we get k^(1/2)k'^(1/2) in k^(l+1/2)k'^(l+1/2) either.
  • #1
wphysics
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I have a few questions about the contents in Weinberg, the Quantum Theory of fields Vol1.

First one : At the end of page 77(Sec 2.6), we got -ζ_σ=ζ_σ±1. From this, we could refer that ζ_σ should be proportional to (-1)^σ. But, in the next page, Weinberg concluded that ζ_σ =ζ(-1)^(j-σ). As I said before, -σ in the exponential of (-1) can be easily understood. But, I don't know why ζ_σ should be proportional to (-1)^j.

Second one : It is about the equation below the equation (3.1.21) (page 112, sec 3.2)
The integral variable of this equation is dα, which is Ʃ_(n1,σ1,n2,σ2,...)∫d^3p1 d^3p2 ...
The paragraph below this equation explains how we get '0' in some cases by using the contour integration about E_α. But, in my opinion, dα does not include dE_α. How can we consider the contour integration about E_α ?

Third one : In the last paragraph of page 156(Sec 3.7), Weinberg said we note that k^l Y_lm(k) is a simple polynomial function of the three ..., so in order for M_... the coefficients M^j_l's'n',lsn must go as k^(l+1/2)k'^(l+1/2) when k and k' go to zero.
I think k^2l Y_lm(k) is also a simple polynomial function, so why did he choose 'l' specially? In addition, I don't understand how we get k^(1/2)k'^(1/2) in k^(l+1/2)k'^(l+1/2) either.
 
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  • #2
wphysics said:
I have a few questions about the contents in Weinberg, the Quantum Theory of fields Vol1.

First one : At the end of page 77(Sec 2.6), we got -ζ_σ=ζ_σ±1. From this, we could refer that ζ_σ should be proportional to (-1)^σ. But, in the next page, Weinberg concluded that ζ_σ =ζ(-1)^(j-σ). As I said before, -σ in the exponential of (-1) can be easily understood. But, I don't know why ζ_σ should be proportional to (-1)^j.

It's an arbitrary chose, since [itex]j[/itex] is constant (it labels the irreducible representation your particle transforms with) [itex](-1)^j[/itex] can be reabsorbed in [itex]\xi[/itex] (which can still depends on [itex]j[/itex]).

To answer the other points I'd need to read those contents again, I'll do this as soon as I'll have some more free time :smile:

Ilm
 
  • #3
First of all, thank you for your reply.

My further question about your answer.

Because we choose phase is proportional to (-1)^j, we can get Eq(2.6.26) in Page 80.
As you said, if this choice is arbitrary, we might choose (-1)^(2j) or something else (of course I am not sure...)
Then, Eq(2.6.26) is not true, but has a different proportional constant.
But, as far as I know, Eq(2.6.26) can be verified in experiments..

Why do we choose (-1)^j specifically?
 
  • #4
We choose [itex](-1)^j[/itex] so that [itex](-1)^{j-\sigma}[/itex] is real.

This chose is the most convenient, but is not necessary. Suppose we chose instead [itex]\xi(-1)^\sigma = \xi e^{i \pi \sigma}[/itex]. Then

[itex]
T^2 \Psi_{p, \sigma} =
T \xi e^{i \pi \sigma} \Psi_{\Pi p, -\sigma} =
\xi^* e^{-i \pi \sigma}T \Psi_{\Pi p, -\sigma} =
\xi^* e^{-i \pi \sigma} \xi e^{-i \pi \sigma} \Psi_{p, \sigma} =
e^{-2i \pi \sigma} \Psi_{p, \sigma} =
(-1)^ {2 \sigma} \Psi_{p, \sigma} = (-1)^ {2 j} \Psi_{p, \sigma}
[/itex]

where I used the fact that if [itex]2j[/itex] is odd (even) [itex]2\sigma=2(j - m), m \in \{0, ... , 2j\}[/itex] is odd (even) too.

Ilm

P.s. sorry I still didn't look for an answer to the other two questions :smile:
 

FAQ: Questions in Weinberg The Quantum Theory of Fields Vol 1

1. What is the main focus of "The Quantum Theory of Fields Vol 1" by Weinberg?

The main focus of this book is to provide a comprehensive and mathematical description of quantum field theory, which is the theoretical framework used to study the behavior of particles at a fundamental level.

2. Is "The Quantum Theory of Fields Vol 1" suitable for beginners?

No, this book is aimed at advanced students and researchers in the field of quantum field theory. It assumes a strong background in mathematics and physics.

3. What are some of the key topics covered in "The Quantum Theory of Fields Vol 1"?

Some of the key topics covered in this book include the quantization of classical fields, gauge theories, renormalization, and the path integral formulation of quantum field theory.

4. How is "The Quantum Theory of Fields Vol 1" different from other books on quantum field theory?

This book is known for its rigorous and mathematical approach to quantum field theory, which sets it apart from other more conceptual or introductory books on the subject.

5. Are there any prerequisites for reading "The Quantum Theory of Fields Vol 1"?

Yes, it is recommended to have a solid understanding of quantum mechanics, special relativity, and classical field theory before delving into this book.

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