Is equation 2.180 an example of equation 2.179 in Ryder's QFT?

[Jimmy Snyder]

I'm sorry, but the formulae pertinent to this question are too difficult for my spotty knowledge of tex. This means that the only people who can help me with this are those who have a copy of Ryder's QFT available.

Homework Statement

My problem comes from page 58 of Ryder's QFT. I think the text is asking me to show that equation 2.180 is an example of equation 2.179 with an appropriate choice of a^\alpha.

Homework Equations

Too complicated for me to reproduce. Please refer to the book if you have it.

The Attempt at a Solution

Actually, I have a solution for this by differentiating the Lorentz transformation equations with respect to v and treating x as a constant. For instance:
x' = \gamma(x + vt)
\frac{\partial}{\partial v}(x + vt)(1-v^2)^{-1/2}|(v = 0) = t(1-v^2)^{-1/2} + (x + vt)v(1-v^2)^{-3/2}|(v = 0) = t
I just feel funny about treating x as a constant when take the derivative with respect to v. If its wrong, what direction should I go? If it's right, why is it allowed to ignore the relation between x and \dot x?

 

[Dick]

I don't have Ryder handy, but see "Euler-Lagrange equations". This is calculus of variations. It gives you a differential equation and the correct relation between x and x' will fall out in the end since it's built into the derivation.

 

[Jimmy Snyder]

I don't have Ryder handy, but see "Euler-Lagrange equations". This is calculus of variations. It gives you a differential equation and the correct relation between x and x' will fall out in the end since it's built into the derivation.

Thanks Dick, I will look into this. The equation I am working with is a straightforward partial derivative. I hope your reply will not turn away people who do have a copy of the Ryder book.

 

[Dick]

Thanks Dick, I will look into this. The equation I am working with is a straightforward partial derivative. I hope your reply will not turn away people who do have a copy of the Ryder book.

Sorry. I'm looking at Ryder now, and it's not EL at all. But I'm not sure why you are worried either. v parametrizes the Lorentz transformation. x is just a variable in the transformation. They don't have much to do with each other. x isn't given to be a function of time, so you can't identify x' with v.

 

[Jimmy Snyder]

Sorry. I'm looking at Ryder now, and it's not EL at all. But I'm not sure why you are worried either. v parametrizes the Lorentz transformation. x is just a variable in the transformation. They don't have much to do with each other. x isn't given to be a function of time, so you can't identify x' with v.

Thanks Dick, I will ponder this for a while until I understand it. Are you saying that my derivation above is correct?

 

[Dick]

Thanks Dick, I will ponder this for a while until I understand it. Are you saying that my derivation above is correct?

It looks fine.

 

[Jimmy Snyder]

It looks fine.

Thanks again. I see what you mean about x being a variable in the transformation and not a function of time.