We know that we can describe a spin 1/2 massless particle using only a single Weyl field (lets say left-handed ψL). To introduce a mass term we have to use two spinor fields (one left-handed and one right-handed) and this gives the Dirac mass term.
The question is now that if we can describe a massive particle with a single Weyl field. Well yes, due to the fact that given a left-handed Weyl spinor, it is possible to construct a right-handed spinor ψR=iσ2ψ∗L. Thus, we can write the Dirac equation using iσ2ψ∗L
ˉσμi∂μψL=imσ2ψ∗L(1)
The known algebraic methods performed for the Dirac equation to prove that it implies a massive Klein-Gordon equation can be performed here without any problems. Thus, the above equation implies (◻+m2)ψL=0. Here we have constructed a mass term using only ψL and this is known as Majorana mass. The similarity with the Dirac mass can be seen by writting (1) in terms of the four component Majorana spinor ψm in the chiral representation
ψm=(ψLiσ2ψ∗L)
Now, equation (1) becomes
(iγμ∂μ−m)ψm=0
The Majorana mass has a very important physical difference when compared to the Dirac mass. We know that the Dirac action with a mass term is invariant under a global U(1) transformations of ψL and ψR (i.e. ψL→eiαψL,ψR→eiαψR).But for Majorana spinors, ψL and ψR are not independent, they are related by complex conjugation. So, if ψL transforms as ψL→eiαψLthen ψR transforms like ψR→e−iαψR. The Majorana equation (1) is not invariant under global U(1) symmetries. This fact implies that a spin 1/2 particle with a U(1) conserved charge cannot have a Majorana mass. All spin 1/2 particles with an electric charge cannot have a Majorana mass. Also leptons that have a Majorana mass violate the lepton number (because this is a U(1) symmetry).
One possible particle that could have a Majorana mass is the neutrino. But this is yet to be determined. (I didn't answer your questions point by point but I hope this clarifies some of them).
This post imported from StackExchange Physics at 2014-03-05 14:50 (UCT), posted by SE-user Leonida