- #1

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*locality*is the condition that the Lagrangian density ##\mathscr{L}## is a functional of a field (or fields) and a finite number of its (their) spatial and temporal derivatives

*evaluated at a single spacetime point*##x^{\mu}=(t,\mathbf{x})##, i.e. it should be of the form $$\mathscr{L}=\mathscr{L}\left(\phi(t,\mathbf{x}),\frac{\partial\phi(t,\mathbf{x})}{\partial t},\frac{\partial\phi(t,\mathbf{x})}{\partial x^{i}}\right)$$ but why is this required?

Is it simply that we require physics to be local, i.e. the physical state of a system at particular point should only depend on what's "going on" in the immediate neighbourhood of that point. As one can reconstruct the behaviour of the fields in the neighbourhood of a spacetime point given the values of the field and a finite number of derivatives at that point (via a Taylor expansion), this leads us to require that the Lagrangian density should only depend on the value of the field and a finite number of its derivatives a that point?

Also, is it to impose that no two fields should be able to interact directly with one another if they are located at different spacetime points (however small their separation) since this would constitute action at a distance. As such, no interaction terms of the form ##\phi(x)\phi(y)##, where ##x^{\mu}\neq y^{\mu}## are allowed since otherwise this would imply that fields at different points in space and time could directly influence, with nothing mediating such an interaction.