I have a two component Weyl spinor transforming as [tex]\psi \rightarrow M \psi[/tex] where M is an SL(2) matrix which represents a Lorentz transformation. Suppose another spinor [tex]\chi[/tex] also transforms the same way [tex]\chi \rightarrow M \chi[/tex]. I can write a Lorentz invariant term [tex]\psi^T (-i\sigma^2) \chi[/tex] where [tex](-i\sigma^2) =\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) [/tex]. This is possible because [tex]M^T(-i\sigma^2)M=(-i\sigma^2) [/tex]. I understand everything up to here. My question is the following. For majorana neutrinos they write the Lagrangian as [tex]\psi^T (-i\sigma^2) \psi[/tex] where [tex]\psi[/tex] is the two component majorana field. The term is obviously Lorentz invariant, but when I expand it terms of the two components I get zero. [tex](\psi_1 \ \psi_2)\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)\left(\begin{array} (\psi_1 \\ \psi_2 \end{array} \right)=0[/tex].What mistake am I making here? Please help me out!(adsbygoogle = window.adsbygoogle || []).push({});

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# Weyl spinors

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