# Quantum Field Theory/Quantization of free fields

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## Spin 0 field[edit | edit source]

### Real and complex scalar fields. Klein-Gordon equation. Plane-wave (normal mode) solutions. Generation and anihilation operators. Hamiltonian. Commutation relations.[edit | edit source]

### Real and complex scalar fields.[edit | edit source]

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[edit | edit source]

Klein-Gordon equation is precisely the equation of motion for the spin-0 particle as derived above:

## Spin 1/2 field[edit | edit source]

### Dirac equation[edit | edit source]

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 .