For a real scalar field \phi(x), the standard interacting Lagrangian density is

    \[ \boxed{ \mathcal L = \frac12\,\partial_\mu\phi\,\partial^\mu\phi -\frac12m^2\phi^2 -\frac{g}{3!}\phi^3 -\frac{\lambda}{4!}\phi^4 } \]

Using natural units, \hbar=c=1, and the metric \eta_{\mu\nu}=\operatorname{diag}(+1,-1,-1,-1), let us expand each part.

1. Expand the spacetime derivatives

The repeated index \mu means a sum over time and all three spatial directions:

    \[ \partial_\mu\phi\,\partial^\mu\phi = \left(\frac{\partial\phi}{\partial t}\right)^2 - \left(\frac{\partial\phi}{\partial x}\right)^2 - \left(\frac{\partial\phi}{\partial y}\right)^2 - \left(\frac{\partial\phi}{\partial z}\right)^2. \]

Therefore, all the terms written explicitly are

    \[ \boxed{ \begin{aligned} \mathcal L={}& \underbrace{\frac12\left(\frac{\partial\phi}{\partial t}\right)^2}_{\text{time variation}} -\underbrace{\frac12\left(\frac{\partial\phi}{\partial x}\right)^2}_{\text{variation along }x} -\underbrace{\frac12\left(\frac{\partial\phi}{\partial y}\right)^2}_{\text{variation along }y} -\underbrace{\frac12\left(\frac{\partial\phi}{\partial z}\right)^2}_{\text{variation along }z} \\[4pt] &-\underbrace{\frac12m^2\phi^2}_{\text{mass term}} -\underbrace{\frac{g}{6}\phi^3}_{\text{cubic interaction}} -\underbrace{\frac{\lambda}{24}\phi^4}_{\text{quartic interaction}}. \end{aligned} } \]

Every field and derivative here is evaluated at the same spacetime point (t,x,y,z).

2. What each group does

TermsMeaning
\frac12\dot\phi^{\,2}-\frac12(\boldsymbol\nabla\phi)^2Kinetic terms: describe how the field varies through spacetime
-\frac12m^2\phi^2Mass term: together with the kinetic terms, determines free propagation
-g\phi^3/6Self-interaction coupling three field factors at a vertex
-\lambda\phi^4/24Self-interaction coupling four field factors at a vertex

The factorials 3! and 4! are conventions that simplify the interaction vertex factors.

3. Free field versus interacting field

Setting g=\lambda=0 gives the free Klein–Gordon Lagrangian:

    \[ \mathcal L_{\mathrm{free}} = \frac12\dot\phi^{\,2} -\frac12(\boldsymbol\nabla\phi)^2 -\frac12m^2\phi^2. \]

A common interacting example is \phi^4 theory, with g=0:

    \[ \boxed{ \mathcal L_{\phi^4} = \frac12\dot\phi^{\,2} -\frac12(\boldsymbol\nabla\phi)^2 -\frac12m^2\phi^2 -\frac{\lambda}{24}\phi^4 } \]

This theory has the symmetry \phi\to-\phi, which forbids the cubic term.

The corresponding classical field equation shows how these terms enter the dynamics:

    \[ \boxed{ \frac{\partial^2\phi}{\partial t^2} -\frac{\partial^2\phi}{\partial x^2} -\frac{\partial^2\phi}{\partial y^2} -\frac{\partial^2\phi}{\partial z^2} +m^2\phi +\frac{g}{2}\phi^2 +\frac{\lambda}{6}\phi^3 =0. } \]