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Quantum Mechanics with Basic Field Theory by Bipin R. Desai

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35 Dirac equation in a relativistically invariant form

The Dirac equation is re-expressed in a covariant form so that the equation is manifestly invariant under Lorentz transformations. The matrices α and β are replacedby the imagesμ matrices and the spinor wavefunctions are appropriately defined. The Lorentz transformation of spinors and their bilinear products are obtained. We also derive the Gordon decomposition rule.

35.1 Covariant Dirac equation

 

Let us express the Dirac equation in a fully relativistic form. We start with the equation in the momentum space,

and multiply it on the left by β. Since β2 = 1, we have

We introduce the matrices

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