Ghost story. I. Wedge states in the oscillator formalism

TL;DR

This paper proves that ghost wedge states in string field theory can be expressed as exponential operators in the oscillator formalism, aiding the understanding of Schnabl's solution.

Contribution

It demonstrates, using oscillator formalism, that wedge states are expressible as exponential operators, including the ghost sector, with a novel method for diagonalizing infinite matrices.

Findings

Wedge states can be written as exponential operators in oscillator formalism.

The relation is proven separately for matter and ghost sectors.

A method for diagonalizing infinite matrices is developed and validated.

Abstract

This paper is primarily devoted to the ghost wedge states in string field theory formulated with the oscillator formalism. Our aim is to prove, using such formalism, that the wedge states can be expressed as |n> = exp[{2-n}/2 ({\cal L}_0+{\cal L}_0^\daggert)]|0>, separately in the matter and ghost sector. This relation is crucial for instance in the proof of Schnabl's solution. We start from the exponentials in the rhs and wish to prove that they take precisely the form of wedge states. As a guideline we first re-demonstrate this relation for the matter part. Then we turn to the ghosts. On the way we face the problem of `diagonalizing' infinite rectangular matrices. We manage to give a meaning to such an operation and to prove that the eigenvalues we obtain satisfy the recursion relations of the wedge states.