How Do Local SU(2) Gauge Transformations Affect Field Components?

[Kali_89]

Hi all, (Also - if anybody could tell me how to get the latex to work on this page that'd be very handy!)

While not technically homework this is a problem I've found I'm stuck on during my revision. Any help would be greatly appreciated.

Homework Statement

"By demanding that the covariant derivative [itex] D^\mu \Psi [\latex] transforms in the same way as the fundamental doublet [itex] \Psi [\latex] under a local SU(2) gauge transformation, derive how the field components [itex] W_{\mu}^{i}, (i=1,2,3), [\latex] transforms under an infinitesimal such transformation. The Pauli matrix identity [ latex ] (\underline{\sigma} \cdot \underline{a})(\underline{\sigma} \cdot{b}) = \underline{a} \cdot \underline{b} + i \underline{\sigma} \cdot (\underline{a} \times \underline{b}) [\latex], may be assumed."

Homework Equations

I think that the following equations are going to have to be used:
[itex] \begin{align}
D_{\mu} &= \partial_{\mu} + igW_{\mu}^{a} t^{a}, \\
[t_{a},t_{b}] &= i C_{abc}t_{c}, \\
\end{align} [\latex]
where [itex] D_{\mu} [\latex] is the modified derivative, g is our coupling constant, W is the field in question, t are the generator matrices and C is the structure constant.

The Attempt at a Solution

Given that the covariant derivative transforms in the same way as the doublet I know we can write [itex] D_{\mu} \psi ----> D'_{\mu} \psi' = U(\underline{x}) (D_{\mu} \psi ) [\latex]. From this we can easily write [itex] D'_{\mu} [\latex]. I've found in my notes a general expression for U (incidentally, what is U? I see it has the same sort of form as the phase for transformations I've seen) as [itex] U(x) = \exp{(-ig\sum_{1}^{n^2 - 1} t_{k} \alpha_{k})} [\latex] where the n = 2 for the case of SU(2). Here I believe [itex] \alpha [\latex] is the phase but I'm not at all sure. The generators for SU(2) are half the Pauli matrices.

Now, using that we're taking an infinitesimal transformation we can Taylor expand U and so find that [itex] U(x) = I - igt_{k} \alpha_{k} [\latex]. At this point I've substituted most of the equations I've got together in order to try and find the following equation which I believe to be the final answer:
[itex] W'_{\mu}^{a} = W_{\mu}^{a} - \frac{1}{g} \partial_{\mu} \alpha_{a} - C_{abc} W_{\mu}^{c}. [\latex]

Substituting in I do seem to make some sort of headway but I've not really understood what I've been doing and also how to treat terms like [itex] \partial_{\mu} (\alpha_{a} t_{a} \psi) [\latex].

 

[Gregg]

Hi, use [tex]\text{$$}$$ or [itex]\text{$}$

 

[Kali_89]

Hi all, (Also - if anybody could tell me how to get the latex to work on this page that'd be very handy!)

While not technically homework this is a problem I've found I'm stuck on during my revision. Any help would be greatly appreciated.

1. Homework Statement
"By demanding that the covariant derivative [tex] D^\mu \Psi [/itex] transforms in the same way as the fundamental doublet $$\Psi$$ under a local SU(2) gauge transformation, derive how the field components $$W_{\mu}^{i}, (i=1,2,3),$$ transforms under an infinitesimal such transformation. The Pauli matrix identity $$(\underline{\sigma} \cdot \underline{a})(\underline{\sigma} \cdot{b}) = \underline{a} \cdot \underline{b} + i \underline{\sigma} \cdot (\underline{a} \times \underline{b})$$, may be assumed."

2. Homework Equations
I think that the following equations are going to have to be used:
$$\begin{align} D_{\mu} &= \partial_{\mu} + igW_{\mu}^{a} t^{a}, \\ [t_{a},t_{b}] &= i C_{abc}t_{c}, \\ \end{align}$$
where $$D_{\mu}$$ is the modified derivative, g is our coupling constant, W is the field in question, t are the generator matrices and C is the structure constant.3. The Attempt at a Solution
Given that the covariant derivative transforms in the same way as the doublet I know we can write $$D_{\mu} \psi ----> D'_{\mu} \psi' = U(\underline{x}) (D_{\mu} \psi )$$. From this we can easily write $$D'_{\mu}$$. I've found in my notes a general expression for U (incidentally, what is U? I see it has the same sort of form as the phase for transformations I've seen) as $$U(x) = \exp{(-ig\sum_{1}^{n^2 - 1} t_{k} \alpha_{k})}$$ where the n = 2 for the case of SU(2). Here I believe $$\alpha$$ is the phase but I'm not at all sure. The generators for SU(2) are half the Pauli matrices.

Now, using that we're taking an infinitesimal transformation we can Taylor expand U and so find that $$U(x) = I - igt_{k} \alpha_{k}$$. At this point I've substituted most of the equations I've got together in order to try and find the following equation which I believe to be the final answer:
$$W'_{\mu}^{a} = W_{\mu}^{a} - \frac{1}{g} \partial_{\mu} \alpha_{a} - C_{abc} W_{\mu}^{c}.$$

Substituting in I do seem to make some sort of headway but I've not really understood what I've been doing and also how to treat terms like $$\partial_{\mu} (\alpha_{a} t_{a} \psi)$$.

 

[fzero]

3. The Attempt at a Solution
Given that the covariant derivative transforms in the same way as the doublet I know we can write $$D_{\mu} \psi ----> D'_{\mu} \psi' = U(\underline{x}) (D_{\mu} \psi )$$. From this we can easily write $$D'_{\mu}$$. I've found in my notes a general expression for U (incidentally, what is U? I see it has the same sort of form as the phase for transformations I've seen) as $$U(x) = \exp{(-ig\sum_{1}^{n^2 - 1} t_{k} \alpha_{k})}$$ where the n = 2 for the case of SU(2). Here I believe $$\alpha$$ is the phase but I'm not at all sure. The generators for SU(2) are half the Pauli matrices.

You'll want to write down the expression for the infinitesimal variation $$\delta (D_\mu\psi)^a$$, expressing it in terms of $$\delta \psi^a$$ (which you know) and $$\delta W_\mu^a$$, which you're trying to determine.

Now, using that we're taking an infinitesimal transformation we can Taylor expand U and so find that $$U(x) = I - igt_{k} \alpha_{k}$$. At this point I've substituted most of the equations I've got together in order to try and find the following equation which I believe to be the final answer:
$$W'_{\mu}^{a} = W_{\mu}^{a} - \frac{1}{g} \partial_{\mu} \alpha_{a} - C_{abc} W_{\mu}^{c}.$$

That's close, but you can see that you have an extra free index on the $$CW$$ term that isn't on the LHS, so that term is wrong.

Substituting in I do seem to make some sort of headway but I've not really understood what I've been doing and also how to treat terms like $$\partial_{\mu} (\alpha_{a} t_{a} \psi)$$.

You can expand that term out using the product rule for derivatives. It also might help to put the matrix indices on the $$(t^a)_{bc}$$ and the index on $$\psi^a$$.

 

[paranormal]

As a scientist, it is important to carefully understand the concepts and equations involved in a problem before attempting to solve it. In this case, it seems that you are on the right track but may need some clarification on a few key points.

Firstly, U(x) is the gauge transformation matrix, which represents the change in the field under a gauge transformation. It is a function of the spacetime coordinates x and the transformation parameters α. The generators t_k are the matrices that generate the group transformations, and in the case of SU(2), they are the Pauli matrices.

Next, the covariant derivative D_μ is defined as the ordinary derivative plus a term involving the gauge field W_μ. In order for the covariant derivative to transform in the same way as the fundamental doublet, the gauge field must transform in a specific way. This is what you are trying to derive in this problem.

To solve this problem, you will need to use the Taylor expansion of the gauge transformation matrix U(x) and the commutation relations for the generators t_k. You will also need to use the Pauli matrix identity given in the problem statement. Remember that the generators t_k are matrices, so when taking derivatives with respect to the transformation parameters α, they will need to be treated as such.

Finally, when substituting and simplifying, be careful to keep track of the indices on the generators and gauge fields. This will help you to correctly identify the transformation rule for the gauge field.

I hope this helps to clarify some of the key points in this problem. It is important to carefully understand the concepts and equations involved in order to successfully solve it. Good luck with your revision!