# Transforming a Left-Handed Spinor into a Right-Handed Spinor

In pg. 125-126 of Symmetry and the Standard Model book, it was stated that for a spinor $$\psi_L$$ in the left-handed representation and a spinor $$\psi_R$$ in the right-handed representation, we can define $$\overline\psi_L=i\sigma_2{\psi_L}^*\tag{3.253}$$ $$\overline\psi_R=i\sigma_2{\psi_R}^*\tag{3.256}$$ so that $$\overline\psi_L$$ transforms like a right-handed spinor and $$\overline\psi_R$$ transforms like a left-handed spinor.

But then the author said in the last paragraph of the section below eq. (3.256) that

A spinor in either representation can be written as a spinor in the other representation by taking its complex conjugate and multiplying by $$\pm i \sigma_2$$ ($$-$$ when going from left to right-handed and $$+$$ when going from right to left handed).

But this sentence contradicts what the first two equations say. The first two equations say that to change representations, just multiply by $$+i \sigma_2$$ and take the complex conjugate of $$\psi_{R/L}$$. What am I getting wrong?

Also, the author performed a calculation in (3.256) where $$\psi_L=\overline\psi_R$$$$\psi_L=i\sigma_2{\psi_R}^*$$$$i\sigma_2(\psi_L)^*=-i\sigma_2(i\sigma_2{\psi_R}^*)^*$$

What is happening here? I expanded the steps and got $$\psi_L=\overline\psi_R \tag{3.256}$$ $$\psi_L=i\sigma_2{\psi_R}^*$$ $${\psi_L}^*=(i\sigma_2{\psi_R}^*)^*$$ $$i\sigma_2(\psi_L)^*=i\sigma_2(i\sigma_2{\psi_R}^*)^*$$

which is different from the author's result.

• I'd recommend not linking to a shady source. Also the author has strangely adopted notation on the second page which says $\psi_R$ is left-handed – Nihar Karve Nov 30 '20 at 9:27

OP has a point: What Ref. 1 meant to say is that one can go between the left & right Weyl spinor representations (i.e. complex conjugate representations) by using$$^1$$

$$\psi_R~=~f i\sigma_2\psi_L^{\ast}\qquad\Leftrightarrow\qquad\psi_L~=~f (-i\sigma_2)\psi_R^{\ast},$$

where $$f$$ is a phase factor that Ref. 1 chooses to be 1 by convention, cf. e.g. eqs. (3.253) & (3.285).

References:

1. M. Robinson, Symmetry and the Standard Model: Mathematics and Particle Physics, 2011; p. 125-126, eqs. (3.253) & (3.256).

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$$^1$$ This antilinear operation implements charge conjugation.