Quantum Field Theory/Quantization of free fields

Spin 0 field

Real and complex scalar fields.

The equations of motion for a real scalar field $\phi$ can be obtained from the following lagrangian densities

${\begin{matrix}{\mathcal {L}}&=&{\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {1}{2}}M^{2}\phi ^{2}\\&=&-{\frac {1}{2}}\phi \left(\partial _{\mu }\partial ^{\mu }+M^{2}\right)\phi \end{matrix}}$

and the result is $\left(\Box +M^{2}\right)\phi (x)=0$.

The complex scalar field $\phi$ can be considered as a sum of two scalar fields: $\phi _{1}$ and $\phi _{2}$, $\phi =\left(\phi _{1}+i\phi _{2}\right)/{\sqrt {2}}$

The Langrangian density of a complex scalar field is

${\mathcal {L}}=(\partial _{\mu }\phi )^{+}\partial ^{\mu }\phi -M^{2}\phi ^{+}\phi$

Klein-Gordon equation

Klein-Gordon equation is precisely the equation of motion for the spin-0 particle as derived above: $\left(\Box +M^{2}\right)\phi (x)=0$

Spin 1/2 field

Dirac equation

The Dirac equation is given by:

$\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi \left(x\right)=0$

where $\psi$ is a four-dimensional Dirac spinor. The $\gamma$ matrices obey the following anticommutation relation (known as the Dirac algebra):

$\left\{\gamma ^{\mu },\gamma ^{\nu }\right\}\equiv \gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2g^{\mu \nu }\times 1_{n\times n}$

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 $4\times 4$.