Unraveling the Spectrum of the Open String

TL;DR

This paper systematically constructs and analyzes the massive spectrum of open bosonic strings using light-cone quantization, revealing an ordered, recursive structure and providing tools for future amplitude computations.

Contribution

It introduces an explicit oscillator-based construction of the open string spectrum, including an algorithm to generate entire Regge trajectories from seed states.

Findings

Explicit oscillator realizations for string states

Development of an algorithm to build trajectories

Identification of recursive and regular spectrum structures

Abstract

We construct a large portion of the massive spectrum of the open bosonic string using light-cone quantization, providing explicit oscillator realizations for individual single-particle states as well as for full Regge trajectories. We show how combinations of transverse oscillators organize into irreducible SO(25) representations, and provide an algorithm for constructing them level by level. We then develop a general method to "climb" the spectrum-adding oscillators in a controlled way that generates entire Regge trajectories from a finite set of seed states. Remarkably, the coefficients determining each state's oscillator composition depend on the level in a simple way, allowing closed-form expressions for infinitely many states. Beyond individual trajectories, we explore internal regularities of the spectrum and establish relations among families of trajectories, extending the concept of a Regge trajectory to more general constructions. Our results expose a highly ordered and recursive structure underlying the open-string spectrum, suggesting that its massive excitations form an algorithmically constructible network. The framework presented here lays the groundwork for computing three-point amplitudes of arbitrary massive states, the essential building blocks of string interactions, which we tackle in upcoming work.