At the experimental level, those condensed matter Majorana degrees of freedom are the pioneering examples (assuming that the claims are true). The only other Majorana fields we know in the world around us are the neutrino fields but even though there are strong theoretical reasons to think that the neutrino species we know are Majorana and not Dirac, we can't really experimentally show it is the case.

Theoretical physics is full of Majorana fields, however. The world sheet fermions in string theory are Majorana fermions – well, in 2 dimensions, much like in 10 dimensions and any $8k+2$ dimensions, one may impose the Majorana and Weyl conditions simultaneously so we're working with Majorana-Weyl fermions.

Similarly, there are lots of hypothetical Majorana (but not Weyl at the same moment) fields in $d=4$ according to supersymmetry (and some other models of new physics). The superpartners to any bosonic field of the Standard Model – the Higgs and the gauge bosons – are Majorana fermions. Neutralinos may be the lightest example: they may be the lightest superpartners (LSPs) and in many models, they account for most of the dark matter. This dark matter would annihilate in pairs, something we expect for Majorana excitations that naturally carry a conserved ${\mathbb Z}_2$ quantum number.

I would slightly disagree that the absence of a $U(1)$ charge is equivalent to the Majorana condition. This identification of the two conditions holds in one direction and it is "economic" in the other but it doesn't have to be the case. The neutrinos could be Majorana but they could still refuse to carry any conserved $U(1)$ charges. Majorana degrees of freedom mean that the fields transform as Majorana spinors (spinor representations that obey a reality condition); more generally, these fermions are the canonical momenta to themselves so that one may write $\{\theta_a,\theta_b\}=\delta_{ab}$ anticommutators without any daggers while $\theta^\dagger=\theta$ for these components.

This post imported from StackExchange Physics at 2014-04-05 04:16 (UCT), posted by SE-user Luboš Motl