Quantum Field Theory/Quantization of free fields
Spin 0 field 
Real and complex scalar fields. Klein-Gordon equation. Plane-wave (normal mode) solutions. Generation and anihilation operators. Hamiltonian. Commutation relations. 
Real and complex scalar fields. 
The equations of motion for a real scalar field can be obtained from the following lagrangian densities
and the result is .
The complex scalar field can be considered as a sum of two scalar fields: and ,
The Langrangian density of a complex scalar field is
Klein-Gordon equation 
Klein-Gordon equation is precisely the equation of motion for the spin-0 particle as derived above:
Spin 1/2 field 
Dirac equation 
The Dirac equation is given by:
where is a four-dimensional Dirac spinor. The 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 four-dimensional Minkowski space, however, it turns out that the matrices have to be at least .