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In field theory a typical Lagrangian (density) for a "free (scalar) field" ##\phi(x)## is of the form $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi -\frac{1}{2}m^{2}\phi^{2}$$ where ##m## is a parameter that we identify with the mass of the field ##\phi(x)##.

My question is, what is the motivation for including this mass term? Is it simply that in doing so one can reproduce the Klein-Gordon equation or are there some other physical arguments? Furthermore, is it in some sense a "self-interaction" term? Classically the Lagrangian of a free particle would just have a kinetic term so how does this Lagrangian describe a free field?

My question is, what is the motivation for including this mass term? Is it simply that in doing so one can reproduce the Klein-Gordon equation or are there some other physical arguments? Furthermore, is it in some sense a "self-interaction" term? Classically the Lagrangian of a free particle would just have a kinetic term so how does this Lagrangian describe a free field?

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