Quantum Field Theory/Quantization of free fields

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

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

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

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

The Langrangian density of a complex scalar field is

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

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

The Dirac equation is given by:

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

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

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

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 ${\displaystyle 4\times 4}$.