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

TL;DR

This work proves that in a graph-based quantum setting with edges carrying internal degrees of freedom but no intrinsic geometry, the directional information must be encoded by quantum states in $\mathbb{C}^2$ with $SU(2)$ symmetry, and geometrically consistent representations project to $\mathbb{R}^3$ via the Bloch map. The authors establish a unique (up to rotations) Bloch projection, derive the Euclidean metric from the Killing form on $\mathfrak{su}(2)$, and show the ambient dimension saturates at $d=3$ regardless of local vertex valence. They also argue that higher gauge groups (e.g., $SU(N)$ with $N\ge3$) are incompatible with the directional encoding constraint. Numerical simulations confirm uniform Bloch-sphere coverage and dimensional saturation across graph topologies. The results illuminate how dimensional structure can emerge from information-theoretic and algebraic principles, with potential implications for quantum information, discrete geometry, and foundational questions about spacetime emergence.

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.

Figures (2)

• Figure 1: Bloch projection of 200 random states to $S^2 \subset \mathbb{R}^3$. Top: 3D visualization showing uniform sphere coverage. Bottom: Distribution of vector norms, peaked at unity for pure states.
• Figure 2: Dimensional saturation for vertices of different valences. Each panel shows $k$ Bloch vectors (blue arrows) from random qubit states, all lying in the same ambient $\mathbb{R}^3$ space. Increasing vertex valence adds vectors but does not increase spatial dimension.

Theorems & Definitions (20)

• Proposition 2: Directional Information Only
• proof
• Lemma 3: Qubit Minimality
• proof
• Theorem 4: Self-Consistent Dimensionality
• proof
• Definition 6
• Definition 7
• Lemma 8
• proof