Classical Theory: Scalar Field


This context will require knowledge on Lagrangian dynamics of fields. If you have not yet cover that in your previous learning, you will find Lagrangian Form of Fields helpful.

The Lagrangian of a scalar field is usually written as

\[\mathcal L = -\partial_\mu\varphi^*\partial^\mu\varphi - m^2\varphi^*\varphi\]

Alternatively, we can write

\[\mathcal L = \varphi^*(\partial^\mu\partial_\mu - m^2)\varphi\]

The equivalence between the two expression can be manifest once notice

\[\partial_\mu(\varphi^*\partial^\mu\varphi) = \varphi^*\partial^\mu\partial_\mu \varphi + \partial_\mu\varphi^*\partial^\mu\varphi = 0\]

The last equality is due to the fact that the left hand side is a surface term in the action integral.

Using the Lagrange equation, we get the equations of motion of scalar field

\[\begin{split}(\partial^\mu\partial_\mu - m^2)\varphi = 0\\ (\partial^\mu\partial_\mu - m^2)\varphi^* = 0\end{split}\]

This is the “real” Klein-Gordon equation — the equation of scalar field.