Spin ½ and Matrices

The Dirac equation follows from the linearization of Einstein’s momentum-energy equation and leads to 4x4 matrices which contain the 2x2 Pauli matrices. The four vector free particle solution contains two spinor solutions, with the second containing p and E terms which convert the equation linear in E and p back into the quadratic momentum energy equation. In this note, we consider momentum p in the z direction and use the momentum energy relation without any quantum derivatives. In such a case, a two vector appears as on makes the equation linear with the second component containing E and p terms needed to make the equation quadratic again. A four vector is not needed, in other words there is no spin. It only becomes necessary if one considers p as having px,py,pz values. In such a case, one must obtain pp = pxpx + pypy + pzpz (in the momentum energy equation) and this requires matrices which act as an orthogonal basis i.e. the Pauli matrices if one wishes to use non-vectors throughout. (Here a matrix is being called a non-vector.) In this general situation, the two vector becomes a four vector with the Pauli matrices representing a kind of rotation of the spinor coordinates even though px, py and pz represent linear motion. We argue that one may have degenerate motion of the charge in the electron i.e. the turning motion (constrained motion) of the Pauli matrices linked to px, py and pz lead to charge motion which is degenerate with respect to the energy-momentum equation, but is discerned by a magnetic field