Generalized states in SFT

Abstract

The search for analytic solutions in open string fields theory à la Witten often meets with singular expressions, which need an adequate mathematical formalism to be interpreted. In this paper we discuss this problem and propose a way to resolve the related ambiguities. Our claim is that a correct interpretation requires a formalism similar to distribution theory in functional analysis. To this end we concretely construct a locally convex space of test string states together with the dual space of functionals. We show that the above suspicious expressions can be identified with well defined elements of the dual.

Appendix

In this appendix we would like to briefly discuss the spectrum of operator $K_{1}^{L}$. The latter operator is defined by

$$\begin{aligned} K_1^L =&\frac{1}{2} K_1 - \frac{1}{\pi} \bigl({\mathcal{L}}_0+{\mathcal{L}}_0^\dagger \bigr) \end{aligned}$$

(73)

where K ₁=L ₁+L ₋₁ and

$$ \begin{aligned} & \mathcal{L}_0+{\mathcal{L}}_0^\dagger= 2 L_0 +\sum_{n=1}^\infty \ell_{2n} \bigl(L_{2n}+L_{2n}^\dagger\bigr), \quad \ell_{2n}= \frac{2(-1)^{n+1}}{4n^2-1} \end{aligned} $$

(74)

L _n represent the total (matter+ghost) Virasoro generators. For the sake of simplicity we restrict ourselves here to the matter part since it is enough to appreciate the complexity of the problem. In terms of oscillators $a_{n},a_{n}^{\dagger}$, n=1,2,… (we forget a ₀ because we are considering 0 momentum states), we can write in compact notation

$$\begin{aligned} K_1= a^\dagger\cdot F a, \qquad \mathcal{L}_0+\mathcal{L}_0^\dagger= a^\dagger A a^\dagger+ a^\dagger C a+a A a \end{aligned}$$

(75)

where F,A,C are ∞×∞ symmetric numerical matrices. Their explicit expressions can be found in [51, 52]. We recall here only the properties

$$\begin{aligned} AF+FA=0,\qquad [C,F]=0 \end{aligned}$$

(76)

which imply that

$$\begin{aligned} \bigl[K_1, {\mathcal{L}}_0+{\mathcal{L}}_0^\dagger\bigr]=0 \end{aligned}$$

(77)

The matrix C and the twisted matrix $\tilde{A}$ can be diagonalized on the basis v _n (κ) of F eigenvectors

$$\begin{aligned} \sum_{m=1}^\infty F_{nm} v_m(\kappa)= \kappa \, v_n(\kappa) \end{aligned}$$

(78)

The relevant eigenvalues ${\mathfrak{c}}(\kappa)$ and ${\mathfrak{a}}(\kappa)$ can be found again in [51, 52]. We have in particular

$$\begin{aligned} \bigl[K_1, a^\dagger\! \cdot v(\kappa)\bigr] = \kappa \, a^\dagger\! \cdot v(\kappa) \end{aligned}$$

(79)

This means that a ^†⋅v(κ)|0〉 is an eigenstate of K ₁ with eigenvalue κ. Due to (76), (77) any state of the form

$$\begin{aligned} f\bigl({\mathcal{L}}_0+{\mathcal{L}}_0^\dagger \bigr) a^\dagger \cdot v(\kappa)|0\rangle, \quad \mathrm{or} \quad g \bigl(a^\dagger A a^\dagger\bigr) a^\dagger \cdot v(\kappa)|0 \rangle \end{aligned}$$

(80)

where f,g are arbitrary analytic functions, are also eigenstates of K ₁ with eigenvalue κ. Differently from what happens for the matrices $F,\tilde{A}$ and C, which have common eigenvectors, the situation for the Fock space operators K ₁ and ${\mathcal{L}}_{0}+{\mathcal{L}}_{0}^{\dagger}$ is much more complicated: there is an infinite degeneracy corresponding to each eigenvalue κ of K ₁. What one should do next is extract from the infinite families (80) the eigenvectors of ${\mathcal{L}}_{0}+{\mathcal{L}}_{0}^{\dagger}$ and calculate the corresponding eigenvalues. Only in this way will one be able to compute the eigenvalues of $K_{1}^{L}$. Needless to say one should also consider the ghost part of $K_{1}^{L}$ (for which the results of [52–54] may be instrumental). Unfortunately these problems are still waiting for a solution.