Quantization of electron-positron field

The Dirac equation (7) can be considered as an equation for the components of the classical electron-positron field I a(a ), 4 c(a ) (a is the spinor index). The Lagrangian for this classical field can be constructed as  

The Dirac equation (7) then follows from the variational principle for the action functional  

From the invariance of the action S with respect to translations in 4-coordinate space follow the expressions for the energy and momentum densities of the electron-positron field  

The field (x), l (a ) is classical in the sense that no particle creation and annihilation is yet introduced. 

For the quantization of the electron-positron field the expansion of the arbitrary solution of the Dirac equation (x) with the fidl set of stationary 

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