Sunday 17 June 2018

dirac matrices - Are Lifshitz and Berestetskii right in this case?


In the Quantum electrodynamics book (look at the problem) its authors Lifshitz and Berestetskei claim that operator of charge conjugation $\hat {C} = -\alpha_{2}$ in Majorana basis transforms as $\hat {C}^{M} = \alpha_{2}$. Here (they started from the standart (Dirac) representation of the gamma-matrices) $$ \alpha_{2} = \begin{pmatrix} 0 & \sigma_{y}\\ \sigma_{y} & 0\end{pmatrix}, \quad \beta = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}, $$ $$ \hat {C}^{M} = \hat {U}^{+} \hat {C} \hat {U}, \quad \hat {U} = \frac{1}{\sqrt{2}}(\alpha_{2} + \beta ) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & \sigma_{y}\\ \sigma_{y} & -1\end{pmatrix} = \hat {U}^{+}. $$ I tried to get their result, but I only got $$ \hat {C}^{M} = \frac{1}{2}\begin{pmatrix} 1 & \sigma_{y}\\ \sigma_{y} & -1\end{pmatrix}\begin{pmatrix} 0 & \sigma_{y}\\ \sigma_{y} & 0\end{pmatrix}\begin{pmatrix} 1 & \sigma_{y}\\ \sigma_{y} & -1\end{pmatrix} = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}, $$ because $\sigma_{y}^{2} = 1$.


Where is the mistake?


An edit.


I found the mistake. I used the wrong definition of transformation of charge conjugation operator under unitary spinor transformation. The correct one is $\hat {C}^{'} = \hat {U}\hat {C}\hat {U}^{T}$.




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...