[Paper Review] Novel String Field Theory and Bound State, Projective Line, and sharply 3-transitive group

H. B. Nielsen, Masao Ninomiya|ArXiv.org|Nov 4, 2021

Particle physics theoretical and experimental studies6 references4 citations

TL;DR

This paper proposes a novel framework in string field theory where bound states with infinitely many constituents—modeled as a projective line structure—exhibit high symmetry through sharply 3-transitive group actions. By identifying the cyclically ordered constituents with Möbius-invariant structures over a field (e.g., real or p-adic), the model suggests that scattering occurs via exchange of constituent bunches rather than individual constituents, leading to a solvable, free-theory-like behavior even in higher dimensions.

ABSTRACT

Using ideas from our long studied Novel String Field Theory we consider in this article a bound state of infinitely many constituents, something that at least very approximately could mean a hadron, since hadrons have typically very many constituents. Our main point, so far, is to speculate that there should be a very high degree of symmetry between the many consituents, since the constituents behave similarly at different places in the bound state. We assume speculatively that there is a group represented sharply 3-transitively as permutation of the constituents; one can namely only have {\em finite} number of elements sharply n-transitively permuted for n larger than 3. The scattering of such bound states will in the zero Bjorken $x$ limit (which is suggeted) only occur by exchange of parts of the system of constituents, quite like in our Novel String field Theory the "objects" are exchanged in bunches. The cyclically ordered chain of objects in this Novel String Field Theory are identified as a projective line structure. Also a p-adic field is a natural possibility.

Motivation & Objective

• To explore the possibility of describing hadron-like bound states with infinitely many constituents as highly symmetric systems.
• To generalize string field theory by treating constituents as non-interacting objects that exchange in bunches, inspired by the free-theory nature of the authors' prior work.
• To investigate whether a sharply 3-transitive group action on constituents can lead to a projective line structure over a field, enabling Möbius invariance and geometric consistency.
• To extend the framework to higher-dimensional quantum field theories by embedding initial and final state information into the Regge slope parameter.
• To incorporate p-adic structures as a natural alternative for the underlying field, motivated by p-adic Veneziano models.

Proposed method

• Model bound states as cyclically ordered chains of constituents, topologically equivalent to a circle, which is identified with a projective line $\mathbb{F} \cup \{\infty\}$ over a field $\mathbb{F}$.
• Assume a sharply 3-transitive group action on the constituents, which uniquely determines the structure as a projective line over a field, following Zassenhaus classification.
• Utilize Möbius transformations over the reals or p-adic numbers to preserve the anharmonic (cross) ratio of four points on the chain, ensuring invariance under symmetry operations.
• Define the correlation function $\langle p_1^\mu p_2^\nu \rangle = g^{\mu\nu} \cdot \text{coef.} \cdot (A,C;B,D)$, where the anharmonic ratio $(A,C;B,D)$ is invariant under Möbius transformations.
• Introduce the coefficient $\text{coef.} \propto 1/\alpha'$, linking the Regge slope $\alpha'$ to initial and final state data, breaking scale symmetry only through boundary conditions.
• Handle orientation dependence by restricting the Möbius group to orientation-preserving transformations, preserving consistency with the cyclic ordering condition.

Experimental results

Research questions

• RQ1 Can a bound state of infinitely many constituents with approximately equal Bjorken-$x$ values be modeled as a symmetric system with sharply 3-transitive group action?
• RQ2 Does the requirement of sharply 3-transitive symmetry on constituents uniquely lead to a projective line structure over a field $\mathbb{F}$?
• RQ3 How can Möbius invariance and the anharmonic ratio be used to define a consistent correlation function for momentum operators in the bound state?
• RQ4 Can the framework be extended to higher-dimensional quantum field theories by embedding the Regge slope $\alpha'$ in initial and final state data?
• RQ5 What role does the p-adic structure play in realizing a consistent, solvable string field theory with similar symmetry properties?

Key findings

• A bound state of infinitely many constituents can be modeled as a cyclically ordered chain that topologically forms a projective line $\mathbb{R} \cup \{\infty\}$, with Möbius symmetry preserved under orientation-preserving transformations.
• The momentum correlation function $\langle p_1^\mu p_2^\nu \rangle$ is proportional to the metric $g^{\mu\nu}$ times a coefficient and the anharmonic ratio $(A,C;B,D)$, which is invariant under Möbius transformations.
• The coefficient $\text{coef.} \propto 1/\alpha'$ is determined solely by initial and final state information, implying that the Regge slope $\alpha'$ emerges from boundary conditions rather than dynamics.
• The full Möbius group is broken to its orientation-preserving subgroup due to the cyclic ordering constraint, but this subgroup remains topologically distinct and consistent with the symmetry structure.
• The framework supports a solvable, free-theory-like behavior in string field theory, even in higher than 3+1 dimensions, by avoiding individual constituent scattering and instead allowing collective exchange of parts.
• The p-adic version of the Veneziano model is naturally incorporated, suggesting a possible path to consistent quantum field theories in higher dimensions via p-adic structures.