How to Compute the Little Group Matrix in Weinberg's QFT Trilogy?

[merius]

Hi. I am new to this discussion group, so forgive me if this is not the appropriate forum for this question. A friend of mine and I are going through the first volume of Weinberg's QFT trilogy. My question regards the first problem at the end of chapter two. I think I completely understand the procedure for computing a Lorentz transformed positive definite spin state, but I can't seem to compute the little group element corresponding to the Lorentz transformation given in this problem. The matrix multiplication is very lengthy and I really get caught up in the algebra. Is there an easier way to compute the little group matrix than his formula (W = L^(-1)(lambda*p)*Lambda*L(p))? I would like to hear from someone who has done this problem already and, if possible to see their solution. The little group for mass positive definite spin states is the 3-d rotation group and I know the rotation for this particular problem is about the x1 axis, but the algebra is ridiculous. I have spent a lot of time on this problem and would greatly appreciate some help. Thanks.
Merius

 

[jeff]

Originally posted by merius
I can't seem to compute the little group element corresponding to the Lorentz transformation given in this problem. The matrix multiplication is very lengthy and I really get caught up in the algebra. Is there an easier way to compute the little group matrix than his formula (W = L^(-1)(lambda*p)*Lambda*L(p))?

It's not so bad. Since Λ is a boost along the z-axis, it's only non-vanishing elements are

Λ¹₁ = Λ²₂
Λ³₃ = Λ⁰₀
Λ⁰₃ = Λ³₀.

Since p is in the y-direction, the only non-vanishing elements of L(p) are

L¹₁ = L³₃
L²₂
L⁰₂ = L²₀
L⁰₀.

One way to start off is by writing the little group element as

W(Λ,p) = L^-1(Λp)Q(p)

where Q(p) ≡ ΛL(p) and note that the above immediately yields

Q¹_ν = Λ¹₁L¹_ν
Q²_ν = Λ²₂L²_ν
Q⁰_ν = Λ⁰₀L⁰_ν + Λ⁰₃L³_ν
Q³_ν = Λ³₀L⁰_ν + Λ³₃L³_ν.

Noting that Λ¹₁ = Λ²₂ = L¹₁ = L³₃ = 1 simplifies things even further. Finally, L^-1(Λp) = L(p') in which p'ⁱ = -Λⁱ_νp^ν.

 

[R0man]

Hi Merius, welcome to the discussion group! The first volume of Weinberg's QFT trilogy can be quite challenging, so it's great that you have a friend to work through it with. As for your question about the little group element, I understand how the algebra can get overwhelming. I would suggest trying to break down the matrix multiplication into smaller steps and using properties of matrix multiplication to simplify the calculations. Another helpful resource would be to look for online forums or communities specifically for QFT students where you can ask for guidance from others who have also worked through Weinberg's problems. I'm sure there are many people who have encountered the same difficulties and would be willing to share their solutions with you. Best of luck with your studies!