Personnal recollections about the birth of string theory

Invited contribution to the collective book “The Birth of String Theory”

💡 Research Summary

Eugene Cremmer’s recollection traces his personal and scientific journey from the late 1960s through the mid‑1970s, offering a vivid insider’s view of the formative years of string theory. After completing his Ph.D. on e⁺e⁻ annihilation under Michel Gourdin at Orsay, Cremmer found himself drawn to the emerging “dual model” program sparked by Veneziano’s amplitude. Early collaborations with Jean Nuyts produced a novel s‑t dual amplitude whose poles followed a logarithmic trajectory, but physical shortcomings led him back to the mainstream.

In the early 1970s, the community was busy elucidating multi‑loop dual amplitudes (Kaku‑Yu, Lovelace, Alessandrini). Cremmer turned to the study of multi‑loop operators and secured a two‑year fellowship at CERN (1971‑72), where he first met Joël Scherk. Scherk’s meticulous notes and clear calculations left a lasting impression. Together they examined one‑loop and higher‑loop diagrams, focusing on unitarity. In non‑planar orientable loops a new singularity appeared in the zero‑isospin (pomeron) channel, apparently violating unitarity. Lovelace showed that when the intercept equals one, the singularity factorizes; Cremmer and Scherk realized that in 26 space‑time dimensions the two sets of oscillators cancel via gauge conditions, removing the offending cut and leaving only poles. This implied a massless spin‑2 particle (intercept = 2) in the pomeron sector, which in modern language corresponds to the emergence of a closed‑string state alongside the open‑string (Veneziano) sector.

The next major step was the development of a string field theory. Building on earlier functional approaches (Godard et al., Gervais‑Sakita, Mandelstam), Cremmer and Scherk introduced an infinite‑component field theory for interacting relativistic strings. They defined a three‑string vertex as the overlap of three strings at a given time, allowing both “combining” (2 → 1) and “splitting” (1 → 2) processes. This vertex reproduced the ADDF three‑reggeon vertex and, when supplemented by a direct four‑string interaction, generated the correct four‑point amplitude. Their construction clarified the relationship between operator‑based dual models and the path‑integral formalism, and provided a systematic way to factorize both pomeron and reggeon poles in loop amplitudes, thereby preserving unitarity in 26 dimensions.

In 1974 the Orsay group moved to the École Normale Supérieure, forming the Laboratoire de Physique Théorique. Here Cremmer, Scherk, and John H. Schwarz began exploring compactification on a torus (Tⁿ). They showed that for open strings compactification merely quantizes momenta (p_i = n_i/R_i), while for closed strings an additional winding number m_i appears. The mass formula becomes M² = n_i²/R_i² + m_i²R_i²/α′², and loop integrals are replaced by discrete sums over these quantum numbers. Their analysis revealed a symmetry under n_i ↔ m_i and R_i ↔ α′/R_i, the first explicit statement of T‑duality. This duality later proved essential for constructing the heterotic string in 1985.

Scherk and Schwarz also proposed that dual models could be interpreted as a quantum theory of gravity unified with the other forces, provided some dimensions are compactified. Cremmer and Scherk demonstrated the consistency of this idea within the dual‑model framework, laying groundwork for later superstring developments. Their collaborative work continued until Scherk’s untimely death in 1980, after which Cremmer’s “Super‑gravity Era” began.

Overall, Cremner’s memoir highlights three pivotal achievements of the early string era: (1) the realization that 26‑dimensional consistency removes unphysical singularities and yields a massless spin‑2 graviton; (2) the formulation of a string field theory with explicit three‑ and four‑string vertices, bridging operator and functional approaches; and (3) the discovery of toroidal compactification and T‑duality, which opened the path to realistic model building and the heterotic string. These contributions, recounted with personal anecdotes and technical detail, illustrate how a small community of theorists transformed a phenomenological scattering model into the modern framework of string theory.