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Comments to the question (v4): Let us assume that the new and old Lagrangian density $${\cal L} ~\to~ {\cal L} + d_{\mu} {\cal X}^{\mu}\tag{A}$$ does not depend explicitly on the spacetime point $x$. Then Noether's 1st theorem states that the canonical stress-energy-momentum (SEM) tensor density $$ {\cal T}^{\mu}{}_{\nu} ~:=~ \frac{\partial{\cal ...


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I question your premise: in classical field theory point singularities are not necessary, let alone inevitable. The most simple and relevant example is the gravitational potential of the Earth: $$ \phi(r)\propto\begin{cases} r^2 & r<R_\oplus\\ \frac{1}{r} & r>R_\oplus \end{cases} $$ which is well-behaved and finite everywhere ($\phi\in\mathscr ...



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