Intrinsic Dissipation in Nonassociative Algebras: Structural Classification of a Three-Dimensional Dissipative Algebra and an Integrable Quantum Toy Model 非结合代数中的内禀耗散:三维耗散代数的结构分类与一个可积量子玩具模型

Abstract

Abstract This paper studies the complete structure of a three-dimensional commutative nonassociative algebra (Three-Dimensional Dissipative Algebra, TDDA) defined by three axioms Σ(e²=1, i²=−1, ei=0), and its relation to the framework of quantum mechanics. There are four main results: (1) Structural classification — TDDA is a simple, power-associative, non-alternative algebra whose idempotents form the hyperboloid δ²−t²=1/4, with a nondegenerate trace form of signature (+,+,−); (2) Representation classification — TDDA admits no one-dimensional representation, its multiplication algebra is the full matrix algebra M₃(ℝ), and its unique irreducible representation is the natural module ℝ³ carrying a (2+1)-dimensional Minkowski metric; (3) Quantum toy model — the condition “all left-multiplication operators are self-adjoint” uniquely determines the metric g=diag(1,1,−1); the Hamiltonian H=Lₑ reproduces, pointwise, the Rabi oscillations of the standard qubit H=σₓ, and the physical subspace automatically decouples from the ghost direction; (4) Intrinsic dissipation theorem — the complete classification of the associator operator Δ(a,b)=Lₐ·ₙ−LₐLₙ shows that Δ is always a rank-one operator acting only on the non-classical plane, and Δ is a nilpotent one-way pump if and only if the (e,i) components of a and b are Minkowski-orthogonal. For H=Lᵢ+λΔ(e,i), the Schrödinger equation is exactly solvable, with the physical-direction probability P_δ(t)=λ²(cosh t−1)² strictly monotonically increasing — irreversibility emerges as a theorem of the algebra, with no external heat bath required. All algebraic identities and numerical results are verified by symbolic computer computation. 摘要 本文研究由三条公理 Σ(e²=1, i²=-1, ei=0) 定义的三维交换非结合代数(三维耗散代数,TDDA)的完整结构及其与量子力学框架的关系。主要结果有四:(1)结构分类——TDDA 是单的、幂结合的、非交错的代数,其幂等元构成双曲面 δ²-t²=1/4,迹型非退化且符号差为 (+,+,-);(2)表示分类——TDDA 不存在一维表示,其乘法子代数为全矩阵代数 M₃(ℝ),唯一不可约表示为携带 (2+1) 维闵可夫斯基度规的自然模 ℝ³;(3)量子玩具模型——“所有左乘算子自伴”这一条件唯一确定度规 g=diag(1,1,-1);哈密顿量 H=Lₑ 给出与标准量子比特 H=σₓ 逐点一致的 Rabi 振荡,且物理子空间与鬼态方向自动解耦;(4)内禀耗散定理——结合子算子 Δ(a,b)=Lₐ·ₙ-LₐLₖ 的完全分类表明:Δ 恒为秩一算子且仅作用于非经典平面;Δ 为幂零单向泵当且仅当 a,b 的 (e,i) 分量在闵可夫斯基度规下正交。对 H=Lᵢ+λΔ(e,i),薛定谔方程精确可解,物理方向概率 P_δ(t)=λ²(cosh t-1)² 严格单调增长——不可逆性作为代数的定理出现,无需外部热浴。全部代数恒等式与数值结果经符号计算机器验证。