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
Alternatively, we can write
The equivalence between the two expression can be manifest once notice
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
This is the “real” Klein-Gordon equation — the equation of scalar field.