Dirac spinor

In quantum field theory, the Dirac spinor is the bispinor in the plane-wave solution

of the free Dirac equation,

where (in the units ${\displaystyle \scriptstyle c\,=\,\hbar \,=\,1}$)

The Dirac spinor for the positive-frequency solution can be written as

where

The Dirac equation has the form

In order to derive the form of the four-spinor ${\displaystyle \scriptstyle \omega }$ we have to first note the value of the matrices α and β:

These two 4×4 matrices are related to the Dirac gamma matrices. Note that 0 and I are 2×2 matrices here.

The next step is to look for solutions of the form

while at the same time splitting ω into two two-spinors:

Using all of the above information to plug into the Dirac equation results in

This matrix equation is really two coupled equations:

Solve the 2nd equation for ${\displaystyle \scriptstyle \chi \,}$ and one obtains

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