We know that for Real scalar field \(\phi(x)=\phi(t,\vec{x})\) ;

For free scalar field \(\phi(x)\) lagrangian density \(\mathcal{L}\); the EOM turns out to be Klein-Gordan equation.

\[ \begin{align*} \mathcal{L} = \frac{1}{2}\delta _{\mu} \phi \delta ^{\mu} \phi - \frac{1}{2} m^2\phi ^2 \\ \implies (\delta ^2 + m^2)\phi =0 \text{K.G. equation} \\ \end{align*} \]

1. Poincare (inhomogeneous lorentz symmetry) \[ x^{\mu } \to \lambda^{\mu }_{ \nu} + b^{\mu } \\ \Lambda \eta \Lambda ^{T}=\eta \\ \]
2. Discrete symmetry \[ \phi \to -\phi \]
3. Axionic shift symmetry (only for m=0) \[ \phi (x) \to \phi (x) + \alpha \]

\[ \mathcal{L}=\delta _{\mu }\phi ^{*}\delta ^{\mu }\phi -m^2\phi ^{*}\phi \\ \]

If we write \(\phi=\frac{1}{\sqrt{2}} (\phi _{1}+i \phi _{2})\)
It reduceds to sum of two real scalar field lagrangians.