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 is precisely the equation of motion for the spin-0 particle as derived above:
Spin 1/2 field
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 anihilation operators. Hamiltonian. Anticommutation relations.
Spin 1 field
Massive spin 1 field. Additional (Lorentz) condition to eliminate spin-0.
Massless spin 1 field. Gauge invariance. Quantization within Coulomb (radiation) gauge.
Spin-statistics theorem. Discrete symmetries (C,P,T). CPT theorem.