Solution of the Dirac Equation

The positive energy solutions to the Dirac equation are
where p satisfies the mass shell condition and u is a Dirac spinor having the form, for r = 1, 2, and for a two-spinor ζ normalised so that
Proof: Observe that
So,
Hence,

Then, differentiation of the solution of the Dirac equation gives,
as required.

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Normalisation of Dirac Spinors

The Pauli spin matrices are Hermition, so (as above)
Then

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