We compute semiclassical corrections to the energy density of kinks in $ϕ^4$ theory and of solitons in the sine-Gordon model in $1+1$ dimensions, using local and covariant renormalization techniques from quantum field theory in curved spacetimes. For the semiclassical correction to the energy, we recover the known results. Our analysis highlights a subtlety in the definition of a conserved stress tensor for scalar field theories in $1+1$ dimension.

20 pages; Mathematica notebooks containing the relevant calculations as ancillary files; v3: final version