Quantum Field Theory/Quantization of free fields

Spin 0 field

Real and complex scalar fields.

The equations of motion for a real scalar field φ 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 φ can be considered as a sum of two scalar fields: φ₁ and φ₂, $\phi=\left(\phi_1+i\phi_2\right)/ \sqrt{2}$

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 ψ is a four-dimensional Dirac spinor. The γ 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 $4\times 4$.