Cubic Matrix, Generalized Spin Algebra and Uncertainty Relation

TL;DR

The paper investigates a generalization of spin algebra using three-index objects (cubic matrices) and shows that a triple commutation relation among these objects can imply a novel, triple-based uncertainty principle. It develops a cubic-matrix framework, defines triple products and triple commutators, and introduces a cubic-spin algebra with generators J^a and K^a, including a fundamental identity and hermitian 3-index matrices S^a. A key result is a generalized uncertainty relation of the form $\delta A\,\delta B\,\delta C \geq \tfrac{1}{6}|\langle [A,B,C]\rangle_c|$, built on a proposed cubic-matrix expectation value $\langle \cdot \rangle_c$ and related constructs, with a spacetime example $[X^{\mu},X^{\nu},X^{\rho}] = -i l_P^2 \varepsilon^{\mu\nu\rho\sigma} X_{\sigma}$ yielding $\delta X^1 \delta X^2 \delta X^3 \geq \tfrac{l_P^2}{6}|\langle X^0\rangle|$. The work provides an algebraic framework for many-index objects and hints at connections to Nambu-like mechanics and quantum gravity phenomena, suggesting directions for interpreting cubic-matrix elements and linking to string/M-theory-inspired uncertainty concepts.

Abstract

We propose a generalization of spin algebra using three-index objects. There is a possibility that a triple commutation relation among three-index objects implies a kind of uncertainty relation among their expectation values.