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The equations of motion for a real scalar field φ can be obtained from the following lagrangian densities
and the result is .
The complex scalar field φ can be considered as a sum of two scalar fields: φ_{1} and φ_{2},
The Langrangian density of a complex scalar field is
KleinGordon equation is precisely the equation of motion for the spin0 particle as derived above:
The Dirac equation is given by:
where ψ is a fourdimensional Dirac spinor. The γ matrices obey the following anticommutation relation (known as the Dirac algebra):
Notice that the Dirac algebra does not define a priori how many dimensions the matrices should be. For a fourdimensional Minkowski space, however, it turns out that the matrices have to be at least .
