[Physics] the “surface term”

Question

In Peskin's quantum field theory book, There is a sentence in page 17:

…

More generally, we can allow the action to change by a surface term, since the presence of such a term would not affect our derivation of the Euler-Lagrange equations of motion …

…

\begin{equation}
\mathcal{L}(x)\to\mathcal{L}(x)+\partial_\mu\mathcal{J}^\mu(x).\tag{2.10}
\end{equation}

What is the "surface term"? Is it just a partial derivative term as $\partial_\mu\mathcal{J}^\mu(x)$?

Answer 1

Yes, a surface term is a 4-divergence of a 4-vector. The reason is that the Action

$S= \int_\Omega d^4x \mathcal{L}$

defined in some Region of spacetime $\Omega$ corresponding to such a surface term can be converted by Gauss Theorem:

$S = \int_\Omega d^4x \partial^\mu J_\mu = \int_{\partial \Omega}d \sigma n^\mu J_\mu$.

Here, $n^\mu$ is the unit normal vector pointing out of the spacetime surface $\partial \Omega$.