Theory of Spinors

The groups SU(2) and SL(2, c) are revisited and their correspondence with the rotation group O(3) and the Lorentz group O(3, 1) is given. The group SL(2, c) is the covering group of the Lorentz group, therefore is more elaborate from a mathematical point of view. On the other hand, since they can be represented by two by two matrices with complex parameters, calculations are relatively easier compared to the four-by-four matrices of the Lorentz group. The transformation properties of spinors with regard to the SL(2, c) group is given and their correspondence between the components of a four-vector is made evident. Finding all possible irreducible representations of SU(2) is sketched out. It is also shown that E(2)-like groups, which leave the four-momentum of relativistic massless particles invariant, form a subgroup of SL(2, c). The spinor space of the Dirac equation is presented, and the symmetries of the Dirac equation both for the massive and the massless cases are discussed. It is shown that, thanks to those symmetry properties, a plane wave solution to the Dirac equation can be found easily.