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The Dirac equation is the correct equation of motion that describes free spin $1/2$ particles.
In fact, Dirac's equation for the electron must be rated, alongside the Maxwell and Einstein equations, as one of the Great Field Equations of physics.page 289 in “The Emperors new Mind” by Penrose
Gamma Gymnastics:
There are many important rules for the $\gamma$ matrices that appear in the Dirac equation. These rules are important for many practical calculations.
“A great deal more was hidden in the Dirac equation than the author had expected when he wrote it down in 1928. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. It should be added, however, that it was Dirac who found most of the additional insights.” Weisskopf on Dirac
Niels Bohr: “What are you working on Mr. Dirac?” Paul Dirac: “I’m trying to take the square root of something”
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