Correct transformation of left-handed Weyl spinor In the book "Matthew D. Schwartz, Quantum Field Theory and the Standard Model", page 164, it says that a left-handed spinor transforms as
$$\psi_L \rightarrow e^{\frac{1}{2}(i\vec{\theta} - \vec{\beta})\vec{\sigma}} \psi_L.\tag{10.39}$$
In the book "Peskin, Schröder, An Introduction To Quantum Field Theory", page 44, it says that a lefthanded spinor transforms as
$$\psi_L \rightarrow e^{-\frac{1}{2}(i\vec{\theta} + \vec{\beta})\vec{\sigma}} \psi_L.\tag{3.37}$$
In the book "Anthony Duncan, The Conceptual Framework of Quantum Field Theory", page 76, it says that a spinor in the $(1/2,0)$ representation (I assume that is a lefthanded spinor) transforms as
$$\psi_L \rightarrow e^{-\frac{1}{2}(i\vec{\theta} - \vec{\beta})\vec{\sigma}} \psi_L.$$
All of them are different. Which transformation is correct?
If all of them are, why the different signs?
 A: As stated in some of the comments above, signs are a matter of convention, but they are not arbitrary ! Consistency is the key.
Since $e^{\pm i \frac{1}{2} \vec{\phi} \cdot \vec{\sigma} + \frac{1}{2} \vec{\beta} \cdot \vec{\sigma}}$ and $e^{\pm i \frac{1}{2} \vec{\phi} \cdot \vec{\sigma} - \frac{1}{2} \vec{\beta} \cdot \vec{\sigma}}$ (same sign for the rotation generators but a sign difference in the boost generators) are unitarily inequivalent, only one of them can be the transformation for a lefthanded spinor.
By consulting the sources I considered the most trustworthy (in the sense that they are quite explicit and comprehensive in the statement of their assumptions and conventions, and seem to use them consistently throughout) -
among them:


*

*Dreiner, H. K., Haber, H. E., & Martin, S. P. (2010). Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry. Physics Reports, 494(1-2), 1–196. doi:10.1016/j.physrep.2010.05.002 (http://arxiv.org/abs/0812.1594)

*Peskin Schröder, An Introduction to QFT

*Srednicki, Quantum Field Theory
I came to the conclusion that the following transformation is commonly agreed upon to be a active rotation (counterclockwise) with angle $\phi$ around the axis $\hat{\phi}$ and active boost with rapidity $\vec{\beta}$ for a lefthanded spinor:
$M_L = e^{-i \frac{1}{2} \vec{\phi} \cdot \vec{\sigma} - \frac{1}{2} \vec{\beta} \cdot \vec{\sigma}}$

In the case of "Anthony Duncan, The Conceptual Framework of Quantum Field Theory", he 


*

*first talks about passive rotations and boosts (page 110, formula 5.9, 5.10) 

*but then later seems to use a different conventions:
${\Lambda^\mu}_\nu = {\delta^\mu}_\nu + {\omega^\mu}_\nu$ for infinitesimal transformations (page 174, right after formula 7.19) and
$\vec{\phi} = (\omega_{23},\omega_{31},\omega_{12})$, $\vec{\beta} = (\omega_{10},\omega_{20},\omega_{30})$ (page 175, right after formula 7.35) 
which together implie that he is using passive rotations but active boosts.
The latter is then consistent with
$M_L = e^{-i \frac{1}{2} \vec{\phi} \cdot \vec{\sigma} + \frac{1}{2} \vec{\beta} \cdot \vec{\sigma}}$ (page 176, formula 7.36)
