[Paper Review] String Theory: The Early Years

John H. Schwarz|ArXiv.org|Jul 14, 2000

Computational Physics and Python Applications64 references3 citations

TL;DR

This paper traces the foundational development of string theory from 1968 to 1972, emphasizing Lenny Susskind's pivotal contributions to S-matrix theory, duality, and the bootstrap program. It details how duality symmetries, finite energy sum rules, and the emergence of string-like world-sheets from dual resonance models led to the formulation of a quantum theory of extended objects, culminating in the identification of critical spacetime dimension 26 and the resolution of ghost states via the Virasoro algebra, establishing the early framework of string theory.

ABSTRACT

Lenny Susskind has made many important contributions to theoretical physics during the past 35 years. In this talk I will discuss the early history of string theory (1968-72) emphasizing Susskind's contributions.

Motivation & Objective

• To document the early development of string theory from 1968 to 1972, focusing on the intellectual foundations of the S-matrix program and duality.
• To highlight Lenny Susskind's central contributions to the duality program, including finite energy sum rules, duality diagrams, and the interpretation of Regge trajectories.
• To explain how the dual resonance model evolved into a consistent quantum theory of extended objects (strings), including the resolution of ghost states via the Virasoro algebra.
• To clarify the emergence of critical spacetime dimension 26 and the role of subsidiary conditions in ensuring unitarity and causality in loop amplitudes.
• To establish the historical continuity between early S-matrix ideas and the modern formulation of string theory, particularly through the work of Susskind and others.

Proposed method

• The paper reconstructs the historical progression of ideas from phenomenological S-matrix theory to the dual resonance model, using archival insights and personal recollections from the author.
• It analyzes the Veneziano amplitude and its N-particle generalizations as the core mathematical framework for duality and Regge trajectory behavior.
• The method involves tracing the physical interpretation of duality diagrams as string world-sheets with quark lines as boundaries, linking them to electrostatic analogs on a disk or sphere.
• It examines the role of the Virasoro algebra in eliminating negative-norm ghost states by imposing subsidiary conditions on the Hilbert space.
• The paper uses loop amplitude calculations—particularly nonplanar and torus diagrams—to demonstrate the necessity of 26 spacetime dimensions for unitarity.
• It compares and contrasts approaches by Susskind, Neveu, Scherk, and others in interpreting fishnet diagrams and their connection to string world-sheets before the advent of large-N field theory.

Experimental results

Research questions

• RQ1 How did the duality between s-channel and t-channel processes lead to the formulation of the dual resonance model?
• RQ2 What role did Lenny Susskind play in developing finite energy sum rules and duality diagrams in the context of hadronic scattering?
• RQ3 How did the requirement of an exponentially growing density of states lead to the Hagedorn temperature and its physical implications?
• RQ4 In what way did the Virasoro algebra resolve the problem of negative-norm ghost states in the dual resonance model?
• RQ5 Why is the critical spacetime dimension 26 necessary for unitarity in one-loop string amplitudes?

Key findings

• The Veneziano amplitude provided the first explicit realization of duality and Regge trajectory behavior in hadronic scattering, later understood as the tree-level amplitude for open strings.
• N-particle generalizations of the Veneziano amplitude were consistently factorized into spectra of harmonic oscillators, suggesting a full quantum field theory underlying the dual model.
• The Virasoro algebra, with its central charge (later identified as 26), provided subsidiary conditions that eliminated negative-norm ghost states, ensuring unitarity and causality.
• The requirement of unitarity in nonplanar loop amplitudes led to the conclusion that the spacetime dimension must be 26 and the intercept must be 1, corresponding to the critical dimension for bosonic string theory.
• The divergence in one-loop planar amplitudes was traced to the tachyon propagating in the closed-string channel, with the divergent piece identified as a counterterm, later interpreted as a Pomeron-state contribution.
• The first closed-string loop amplitude was computed by Shapiro on a torus, confirming the presence of poles corresponding to closed-string states in the nonplanar channel.