Factor in equation (3.49) of Peskin and Schroeder

[jmlaniel]

My question concerns the 1/2 factor in the exponential of Eq. (3.49) of Peskin and Schroeder.

This equation concerns the Lorentz boost transformation of a spinor along the z-axis (or 3-direction).

According to Eq. (3.26):

$S^{03} = -\frac{i}{2}\begin{bmatrix}\sigma^3 & 0 \\0 & -\sigma^3\end{bmatrix}$

and Eq. (3.30):

$\Lambda_{1/2} = exp(-\frac{i}{2}\omega_{03} S^{03})$

Combining these two expressions and using the infinitesimal boost (according to Eq. (3.48) and Eq. (3.21)): $\omega_{03} = \eta$ (here $\eta$ is the rapidity):

$\Lambda_{1/2} = exp(-\frac{1}{4}\eta \begin{bmatrix}\sigma^3 & 0 \\0 & -\sigma^3\end{bmatrix})$

My problem is that Peskin and Schroeder have a 1/2 factor where my simple substitution gives a 1/4?

Am I misinterpresting the meaning of $\omega_{03}$? I am assuming that $\eta = \beta$ (v/c) from my understanding of Eq. (3.21)... this might be a problem?!?

Thanks for your help!

 

[strangerep]

[...] and Eq. (3.30):

$\Lambda_{1/2} = exp(-\frac{i}{2}\omega_{03} S^{03})$

In my version of P&S, eq(3.30) is
$$
\Lambda_{1/2} = exp(-\frac{i}{2}\omega_{\mu\nu} S^{\mu\nu})
$$
and you must sum over $\mu, \nu$. So you need an exponent involving
$$
(\omega_{03} S^{03} + \omega_{30} S^{30})
$$
Is that enough for you to figure out the rest...?

 

[jmlaniel]

Thanks Strangerep! I just completely forgot the summation over the indices... That was also enough for me to figure out the rest!