How to derive Eq. (3.110) in Peskin's book?

1. Feb 22, 2010

physlover

Hello,

I'm having trouble to derive Eq. (3.110) in Peskin's QFT book. In the book, it's said to use
$$u(\Lambda^{-1} \tilde{p})=\Lambda^{-1}_{1/2} u(\tilde{p})$$
But I ran into trouble to derive this. Here is my try:

The equation is equivalent to
$$u(p)=\Lambda^{-1}_{1/2} u(\Lambda p)$$
To make things simpler, I consider only a boost in axis-1 and a rotation in axis-3. In this way, the Lorentz transform on the momentum is given by Eq. 3.20 and 3.21 in the book, i.e.,
$$\tilde{p}^0 = p^0+\beta p^1$$
$$\tilde{p}^1 = p^1+\beta p^0 -\theta p^2$$
$$\tilde{p}^2 = p^2+\theta p^1$$
$$\tilde{p}^3 = p^3$$
Using Eq.3.37 in the book, and working on the left-hand portion of the spinnor
$$\left[ \Lambda^{-1}_{1/2} u(\tilde{p}) \right]_L =(1+i\theta\cdot \sigma/2+\beta\cdot \sigma/2) \sqrt{\tilde{p}\cdot \sigma} \xi$$
$$=\sqrt{(1+i\theta \sigma^3+\beta \sigma^1)\left[ (p^0+\beta p^1)-(p^1+\beta p^0 -\theta p^2)\sigma^1-(p^2+\theta p^1)\sigma^2 - p^3\sigma^3 \right]} \xi$$
Keeping the 1st order terms, I have
$$\left[ \Lambda^{-1}_{1/2} u(\tilde{p}) \right]_L = \sqrt{p\cdot \sigma +i\theta (p^0\sigma^3-p^3) -i\beta(p^2\sigma^3-p^3\sigma^2)} \xi$$
This is a little different from the left-hand portion of spinnor $$u(p)$$.

Could someone help me find out what's wrong with the above derivation?

Thanks.

2. Jun 27, 2012

AnZa85

Not sure I can help you, but the same law was used before while building solutions of Dirac equation in order to get u(P) from u(P-rest frame).