Quantum Field Theory/Quantization of free fields

Spin 0 field

Real and complex scalar fields.

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

$\begin{array}{ccc}ℒ& =& \frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -\frac{1}{2}{M}^2{\varphi }^2\\ & =& -\frac{1}{2}\varphi ({\partial }_{\mu }{\partial }^{\mu }+M^2)\varphi \end{array}$

and the result is $(◻+M^2)\varphi (x)=0$.

The complex scalar field $\varphi$ can be considered as a sum of two scalar fields: ${\varphi }_1$ and ${\varphi }_2$, $\varphi =({\varphi }_1+i{\varphi }_2)/\sqrt{2}$

The Langrangian density of a complex scalar field is

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

Klein-Gordon equation

Klein-Gordon equation is precisely the equation of motion for the spin-0 particle as derived above: $(◻+M^2)\varphi (x)=0$

Spin 1/2 field

Dirac equation

The Dirac equation is given by:

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

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

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

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 $4\times 4$.