To get to the point – what's the defining differences between them? Alas, my current understanding of a spinor is limited. All I know is that they are used to describe fermions (?), but I'm not sure why?
Although I should probably grasp the above first, what is the difference between Dirac, Weyl and Majorana spinors? I know that there are similarities (as in overlaps) and that the Dirac spinor is a solution to the Dirac equation etc. But what's their mathematical differences, their purpose and their importance?
(It might be good to note that I'm coming from a string theory perspective. Plus I've exhausted Wikipedia here.)
Best Answer
Recall a Dirac spinor which obeys the Dirac Lagrangian
$$\mathcal{L} = \bar{\psi}(i\gamma^{\mu}\partial_\mu -m)\psi.$$
The Dirac spinor is a four-component spinor, but may be decomposed into a pair of two-component spinors, i.e. we propose
$$\psi = \left( \begin{array}{c} u_+\\ u_-\end{array}\right),$$
and the Dirac Lagrangian becomes,
$$\mathcal{L} = iu_{-}^{\dagger}\sigma^{\mu}\partial_{\mu}u_{-} + iu_{+}^{\dagger}\bar{\sigma}^{\mu}\partial_{\mu}u_{+} -m(u^{\dagger}_{+}u_{-} + u_{-}^{\dagger}u_{+})$$
where $\sigma^{\mu} = (\mathbb{1},\sigma^{i})$ and $\bar{\sigma}^{\mu} = (\mathbb{1},-\sigma^{i})$ where $\sigma^{i}$ are the Pauli matrices and $i=1,..,3.$ The two-component spinors $u_{+}$ and $u_{-}$ are called Weyl or chiral spinors. In the limit $m\to 0$, a fermion can be described by a single Weyl spinor, satisfying e.g.
$$i\bar{\sigma}^{\mu}\partial_{\mu}u_{+}=0.$$
Majorana fermions are similar to Weyl fermions; they also have two-components. But they must satisfy a reality condition and they must be invariant under charge conjugation. When you expand a Majorana fermion, the Fourier coefficients (or operators upon canonical quantization) are real. In other words, a Majorana fermion $\psi_{M}$ may be written in terms of Weyl spinors as,
$$\psi_M = \left( \begin{array}{c} u_+\\ -i \sigma^2u^\ast_+\end{array}\right).$$
Majorana spinors are used frequently in 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.