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The action is the integral of the Lagrangian over time, or the integral of the Lagrangian Density over both time and space.
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votes
Discretization of action in path integral
For
\begin{align}
x\rightarrow\frac{x_{k+1}+x_{k}}{2},
\end{align}
it is often called mid-point prescription. It reveals itself in the operator ordering problem, and is important to lead to gauge inv …