In Chiral representation, a Majorana spinor looks like:
$$\psi=\begin{pmatrix}
\psi_L\\
-i\sigma^2\psi_L^*\end{pmatrix}$$
In this representation the Right handed field is the charge-conjugate of the left handed field. i.e., $(\psi_R)^c=\psi_L$, where $$\psi_R=\begin{pmatrix}
0\\
-i\sigma^2\psi_L^*\end{pmatrix}$$
and also $\psi^c=e^{i\phi}\psi$
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How does it look like in Majorana Representation, explicitly in the form of a column vector? What is the usefulness of Majorana representation?
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Can I use the condition $\psi^c=e^{i\phi}\psi$ to be the definition of a Majorana fermion?
Best Answer
Majorana spinors are used frequently supersymmetric theories. In the Wess-Zumino model - the simplest SUSY model - a supermultiplet is constructed from a complex scalar, auxiliary pseudo-scalar field, and Majorana spinor precisely because it has two degrees of freedom unlike a Dirac spinor. The action of the theory is simply,
$$S \sim - \int d^4x \left( \frac{1}{2}\partial^\mu \phi^{\ast}\partial_\mu \phi + i \psi^{\dagger}\bar{\sigma}^\mu \partial_\mu \psi + |F|^2 \right)$$
where $F$ is the auxiliary field, whose equations of motion set $F=0$ but is necessary on grounds of consistency due to the degrees of freedom off-shell and on-shell.
Yes, Majorana fermions are fermions whose charge conjugate are equal to the original field; my lecture notes suggest this is the defining property. Upon canonical quantization, one finds that Majorana fermions have real Fourier coefficients/operators in their expansion.