Dimensional Constraints from SU(2) Representation Theory in Graph-Based Quantum Systems

TL;DR

This paper demonstrates that internal degrees of freedom on graph edges without geometric properties inherently encode directional information, leading to a minimal quantum dimension of 2 and an emergent 3D geometry via SU(2) representation theory.

Contribution

It establishes a fundamental link between SU(2) representation theory and emergent 3D geometry from purely informational constraints on graph-based quantum systems.

Findings

Edges encode only directional information as qubits.

Emergent geometry corresponds to the Bloch sphere in R^3.

Dimensional constraints are robust and unique for SU(2).

Abstract

We investigate dimensional constraints arising from representation theory when abstract graph edges possess internal degrees of freedom but lack geometric properties. We prove that such internal degrees of freedom can only encode directional information, necessitating quantum states in $\mathbb{C}^2$ (qubits) as the minimal representation. Any geometrically consistent projection of these states maps necessarily to $\mathbb{R}^3$ via the Bloch sphere. This dimensional constraint $d=3$ emerges through self-consistency: edges without intrinsic geometry force directional encoding ($\mathbb{C}^2$), whose natural symmetry group $SU(2)$ has three-dimensional Lie algebra, yielding emergent geometry that validates the hypothesis via Bloch sphere correspondence ($S^2 \subset \mathbb{R}^3$). We establish uniqueness (SU($N>2$) yields $d>3$) and robustness (dimensional saturation under graph topology changes). The Euclidean metric emerges canonically from the Killing form on $\mathfrak{su}(2)$. A global gauge consistency axiom is justified via principal bundle trivialization for finite graphs. Numerical simulations verify theoretical predictions. This result demonstrates how dimensional structure can be derived from information-theoretic constraints, with potential relevance to quantum information theory, discrete geometry, and quantum foundations.