Dirac Equation for Massive 2-Spinors
It is possible to write an equation for 2-spinors that properly yields the Klein-Gordon equation when squared AND contains elements for a rest energy. Here is the solution:
The sigma are the Pauli matrices with the zero element just being the identity. We can use some basic math to derive the following from equation 1:
Here are some definitions of terms:
In this way, the rest energy acts as a sort of coupling between the spinor and it’s own charge conjugate. We can solve this equation in free space by doing the following:
Plugging this into equation 1 gives the following 4 equations for the variables A, B, C, and D:
This, like the Dirac equation, has 4 possible solutions: