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:

Equation 1

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:

Equation 2

Here are some definitions of terms:

matrix C
Charge conjugate of our spinor
Charge conjugate of the Pauli matrices, also their Adjoint

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:

Equation 3

Plugging this into equation 1 gives the following 4 equations for the variables A, B, C, and D:

Equations 4

This, like the Dirac equation, has 4 possible solutions:

Equations 5

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