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Dirac Equation

$$(i\gamma_\mu \partial^\mu - m ) \Psi =0 $$
  • $\partial _{\mu} $ denotes the partial derivative and $ \gamma_{\mu} \partial^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\gamma_{\mu} \partial ^{\mu} = \gamma_0 \partial^0 - \gamma_1 \partial^1 -\gamma_2 \partial^2 -\gamma_3 \partial^3$,
  • $m$ denotes the mass of the particle,
  • $\Psi$ is either the wave function of the spin $1/2$ particle if we use the Dirac equation in a particle theory, or describes the spin $1/2$ field if we work in a field theory,
  • $\gamma_\mu$ are the Dirac gamma matrices.

Why is it interesting?

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 289 in “The Emperors new Mind” by Penrose


Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.


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.

  • For a nice description, see section 7.4.3 “Diracology” in the book The Conceptual Framework of Quantum Field Theory by Duncan


The motto in this section is: the higher the level of abstraction, the better.
Common Question 1
Common Question 2




“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”

Contributing authors:

Jakob Schwichtenberg
equations/dirac_equation.txt · Last modified: 2018/03/13 10:25 by jakobadmin