Schnabl's L_0 Operator in the Continuous Basis

TL;DR

This paper derives explicit expressions for Schnabl's ${ m L}_0$ operators in the continuous basis, simplifying their form and exploring their properties within string field theory, including the ghost sector and wedge states.

Contribution

It provides a new, simplified explicit form of ${ m L}_0$ operators in the continuous basis, extending to the ghost sector and connecting to wedge state representations.

Findings

Explicit expressions for ${ m L}_0$ and ${ m L}_0^ op$ in the continuous basis.

Simplification of the sum of these operators, revealing block diagonal structure.

Verification of commutation relations and relation to wedge states.

Abstract

Following Schnabl's analytic solution to string field theory, we calculate the operators ${\cal L}_0,{\cal L}_0^\dagger$ for a scalar field in the continuous $\kappa$ basis. We find an explicit and simple expression for them that further simplifies for their sum, which is block diagonal in this basis. We generalize this result for the bosonized ghost sector, verify their commutation relation and relate our expressions to wedge state representations.