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

Abstract

Abstract For some classical solutions $$\Psi _\textrm{sol}$$ Ψ sol in Witten’s bosonic string field theory, it was proven that energy of the solution is proportional to the Ellwood invariant $$\textrm{Tr}(\mathcal {V}\Psi _\textrm{sol})$$ Tr ( V Ψ sol ) with $$\mathcal {V}=c\bar{c}\partial X^0\bar{\partial }X^0$$ V = c c ¯ ∂ X 0 ∂ ¯ X 0 . We examine the relation for solutions involving $$X^0$$ 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.