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 .