Why do we use two ways to write the kinetic term in a Lagrangian? I have just started reading Schwartz's book on QFT and I see from the first few chapters that he writes the kinetic part of the Lagrangian in a way I find strange.
As an example, for the massless scalar field he writes it as:
\begin{equation}
\mathcal{L}_{K} = \frac{1}{2}\phi\Box\phi
\end{equation}
I get that I can rewrite this in the "normal" way plus a four-divergence that vanishes when we vary the action, such that the equations of motion remain invariant.
My question is why do we write the Lagrangian in this form?
I did skim a bit ahead in the book and saw that in the path integral chapter this seems like a convenient way to have the Lagrangian such that the we can compute the generating functional by just using the formula for the Gaussian integrals. Is this the only reason?
 A: The correct starting point is always$^1$
$$ S~=~\frac{1}{2}\int\! d^4x \left( \mp\partial_{\mu}\phi\partial^{\mu}\phi  + \ldots \right)
~=~\frac{1}{2}\int\! d^4x \left( \dot{\phi}^2 + \ldots \right),\tag{1}$$
since the kinetic term should be positive definite. When authors write
$$ S~=~\frac{1}{2}\int\! d^4x \left( \pm \phi\Box\phi + \ldots\right)
~=~\frac{1}{2}\int\! d^4x \left( - \phi\ddot{\phi} + \ldots\right) ,\tag{2}$$
it is because they have integrated (1) by parts.
(This fact also apply to other fields than a scalar field $\phi$.)
If we want to study extended field configurations, like solitons, etc, we cannot drop boundary terms. However, for many other applications$^1$ in field theory, the boundary terms vanish, so that (1) reduces to (2).
TL;DR: Why do authors write (2) instead of (1)? Answer: Convenience.
--
$^1$ It should be stressed that for certain tasks, such as, e.g.,

*

*finding the functional derivative of the action, or


*in order for $\Box$ to be a self-adjoint operator,
it is important to impose appropriate boundary conditions (BC).
