Energy from Ellwood invariant for solutions involving $X^0$ variables

Abstract

For some classical solutions $\Psi_\mathrm{sol}$ in Witten's bosonic string field theory, it was proven that energy of the solution is proportional to the Ellwood invariant $\mathrm{Tr}(\mathcal{V}\Psi_\mathrm{sol})$ with $\mathcal{V}=c\bar{c}\partial X^0\bar{\partial}X^0$. We examine the relation for solutions involving $X^0$ variables. As a result, we obtain that the relation may not hold for such solutions. Namely, there is a possibility that the energy is not proportional to the Ellwood invariant.