Weinberg's QFT -- Two moving observers see a W-boson differently....

• Fedor Indutny
In summary: Your Name]In summary, the Wigner rotation matrix can be computed using a simple Lorentz transformation matrix, which is a rotation in the spin space with an angle determined by the relative velocity between the two observers. This can be expressed using the Pauli matrices.
Fedor Indutny

Homework Statement

Suppose that observer $\cal O$ sees a W-boson (spin one and mass $m \neq 0$) with momentum $\textbf{p}$ in the y-direction and spin z-component $\sigma$. A second observer $\cal O'$ moves relative to the first with velocity $\textbf{v}$ in the z-direction. How does $\cal O'$ describe the W state?

Homework Equations

How does one compute the Wigner rotation matrix for this? It seems that the calculations quickly become cumbersome and unwieldy, without producing any meaningful result.

The Attempt at a Solution

I'm quite sure that the result should be rotation around x-axis in the spin space. The question is the angle of the rotation, and its dependence on the $\textbf{v}$.

I started it by looking up formulas from the p. 68:

$$U(\Lambda) \Psi_{p,\sigma} = \sqrt{\frac{(\Lambda p)^0} {p^0}} \sum_{\sigma'}D_{\sigma' \sigma}^{(j)}(W(\Lambda,p))\Psi_{\Lambda p,\sigma'} \\ W(\Lambda,p) = L^{-1}(\Lambda p)\Lambda L(p)$$

Where $L(p)$ for mass positive-definite particle is chosen to be:

$$L^i_k(p) = \delta_{ik} + (\gamma - 1)\hat{p}_i \hat{p}_k \\ L^i_0(p) = L^0_i(p) = \hat{p}_i \sqrt{\gamma^2 - 1} \\ L^0_0(p) = \gamma \\ \hat{p}_i \equiv p_i /|\textbf{p}| \\ \gamma \equiv \sqrt{\textbf{p}^2 + M^2}/M$$

In the mentioned problem I took $\vec{p}$ to be $(E, 0, p_y, 0)$, $L(p)$ is then:

$$L(p) = \begin{bmatrix} \gamma_p & 0 & \beta_p & 0 \\ 0 & 1 & 0 & 0 \\ \beta_p & 0 & \gamma_p & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$

Where $\beta_p = \sqrt{\gamma_p^2 - 1}$.

$\Lambda$ is then:

$$\Lambda = \begin{bmatrix} \gamma & 0 & 0 & \beta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \beta & 0 & 0 & \gamma \end{bmatrix}$$

Again $\beta$ is defined similarly.

$L(\Lambda p)$ is:

$$L(\Lambda p) = \begin{bmatrix} \gamma_{\Lambda p} & 0 & \beta_{\Lambda p} \frac{p_y}{\tau} & \beta_{\Lambda p} \frac{\beta E}{\tau} \\ 0 & 1 & 0 & 0 \\ \beta_{\Lambda p} \frac{p_y}{\tau} & 0 & (\gamma_{\Lambda p} - 1) (\frac{p_y} {\tau})^2 + 1 &(\gamma_{\Lambda p} - 1) \frac{p_y \beta E}{\tau^2} \\ \beta_{\Lambda p} \frac{\beta E}{\tau} & 0 & (\gamma_{\Lambda p} - 1) \frac{p_y \beta E}{\tau^2} & (\gamma_{\Lambda p} - 1) (\frac{\beta E}{\tau})^2 + 1 \end{bmatrix}$$

Where:

$$\gamma_{\Lambda p} \equiv \sqrt{p_y^2 + \beta^2 E^2 + M^2}/M$$

Inverting this matrix should be the same thing as changing the sign of $\Lambda p$, so it will flip the signs of off-diagonal matrix elements.

And here is when I gave up, the result of multiplication is so huge that I can't really figure out anything out of it. It is clear however that $W_{00} = 1$ and $W_{0i}=W_{i0}=0$, since it should leave the $\vec{k}$ invariant, everything else doesn't seem to help that much.

Any suggestions appreciated!

(Note that it is very likely that I made mistake somewhere in the computations above, but it doesn't seem to be making things more complex than they might be)

Thank you!

Thank you for your post! It seems like you have put a lot of effort into your attempt at solving this problem. However, I would like to suggest a simpler approach to computing the Wigner rotation matrix.

First, let's define the Wigner rotation matrix as R(\textbf{v}), where \textbf{v} is the relative velocity between the two observers \cal O and \cal O'. This matrix is given by:

R(\textbf{v}) = \begin{bmatrix}
\gamma & 0 & 0 & -\beta\gamma \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
-\beta\gamma & 0 & 0 & \gamma
\end{bmatrix}

where \gamma = 1/\sqrt{1-\beta^2} and \beta = |\textbf{v}|. This matrix is simply a Lorentz transformation matrix that takes the four-momentum of the W-boson in the \cal O frame and transforms it to the \cal O' frame.

Next, let's consider the spin state of the W-boson in the \cal O frame. This can be represented by a spinor \Psi_{\textbf{p},\sigma}, where \textbf{p} is the momentum of the W-boson and \sigma is its spin z-component. The spinor in the \cal O' frame can then be written as:

\Psi'_{\textbf{p},\sigma} = R(\textbf{v})\Psi_{\textbf{p},\sigma}

This is simply a rotation of the spinor in the spin space, with an angle determined by the velocity \textbf{v}. The explicit expression for this rotation can be obtained using the Pauli matrices, but I will not go into the details here.

I hope this helps! Let me know if you have any further questions or need more clarification.

1. What is Weinberg's QFT?

Weinberg's QFT is a mathematical framework used to describe the behavior of particles and their interactions in the field of quantum mechanics. It is based on the principles of quantum field theory and is used to study the fundamental forces of nature.

2. What is a W-boson?

A W-boson is a type of subatomic particle that is responsible for the weak nuclear force, one of the four fundamental forces in nature. It is an elementary particle with a spin of 1 and carries a positive or negative electric charge.

3. What does it mean for two moving observers to see a W-boson differently?

In Weinberg's QFT, the behavior of particles is described in terms of fields, which can be observed differently by different observers. This means that if two observers are moving at different velocities, they may perceive the same particle, such as a W-boson, to have different properties.

4. How does Weinberg's QFT explain the behavior of W-bosons?

Weinberg's QFT uses mathematical equations and principles to describe the interactions of subatomic particles, including W-bosons. This framework allows scientists to predict and understand the behavior of particles in different situations and interactions.

5. What are the practical applications of Weinberg's QFT?

Weinberg's QFT has many practical applications in fields such as particle physics, cosmology, and condensed matter physics. It has been used to make predictions about the behavior of particles and to develop new technologies, such as particle accelerators and medical imaging devices.

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