Quantum Field Theory/Quantization of free fields

Spin 0 field edit

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

Real and complex scalar fields. edit

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

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

Spin 1/2 field edit

Dirac equation edit

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  .

Plane-wave (normal mode) solutions. Generation and annihilation operators. Hamiltonian. Anticommutation relations. edit

Spin 1 field edit

Massive spin 1 field. Additional (Lorentz) condition to eliminate spin-0. edit

Massless spin 1 field. Gauge invariance. Quantization within Coulomb (radiation) gauge. edit

Spin-statistics theorem. Discrete symmetries (C,P,T). CPT theorem. edit