Klein-Gorden Field

A fundamental field, along with interactions which will describe the Higgs boson.

$\mathcal{L}=\frac{1}{2}{\mathring{\varphi }}^2-\frac{1}{2}(\nabla \varphi {)}^2-\frac{m^2{\varphi }^2}{2}$

$\Pi =\frac{\partial \mathcal{L}}{\partial \mathring{\varphi }}=\mathring{\varphi }$

$\mathcal{H}={\Pi }^2-\frac{1}{2}\Pi +\frac{1}{2}(\nabla \varphi {)}^2+\frac{m^2{\varphi }^2}{2}$

$H=\int d^{3}x\frac{1}{2}({\Pi }^2+(\nabla \varphi {)}^2+m^2{\varphi }^2)$

Now, quantising it using equal time commutation relations,

$[\varphi (x),\varphi (y){]}_{x^0=y^0}=0=[\Pi (x),\Pi (y){]}_{x^0=y^0}$

$[\varphi (x),\Pi (y){]}_{x^0=y^0}=i{\delta }^3(x-y)$.