C³ 量子力学公理化框架 Axiomatic Framework of C³ Quantum Mechanics

Abstract

宣言 Manifesto 标准量子力学的数学基础是结合代数:算子乘法满足 (AB)C = A(BC),态空间是希尔伯特空间(正定内积空间),演化由幺正群 U(N) 描述。 The mathematical foundation of standard quantum mechanics is associative algebra: operator multiplication satisfies (AB)C = A(BC), the state space is a Hilbert space (positive-definite inner product space), and evolution is described by the unitary group U(N). 然而,自然界存在深刻的非结合结构:从八元数在弦论中的角色,到 Jordan 代数在量子逻辑中的起源,非结合性不是病态,而是更一般的对称性破缺。中道积分理论已经证明:非结合代数上的围道积分可以通过算术平均唯一地定义,结合子精确量化每条分支的偏差。 However, nature harbors profound non-associative structures: from octonions in string theory to Jordan algebras in quantum logic, non-associativity is not pathological but a more general symmetry breaking. The Mid-Path Integral theory has proven: contour integrals on non-associative algebras can be uniquely defined via arithmetic averaging, with the associator precisely quantifying each branch’s deviation. 本文提出一个根本性的转向:不再把非结合代数塞进希尔伯特空间的模子里,而是直接从复化三维数(ℂ³)的内生结构出发,重建量子力学的全部公理。标准量子力学将是本文框架在“结合性极限”(δ → 0)和“标量子空间限制”(ψ₁ = ψ₂ = 0)下的子理论。 This paper proposes a fundamental pivot: instead of forcing non-associative algebras into the Hilbert-space mold, we reconstruct all axioms of quantum mechanics directly from the endogenous structure of the complexified ternary numbers (ℂ³). Standard quantum mechanics will be a sub-theory of this framework in the “associative limit” (δ → 0) and the “scalar subspace restriction” (ψ₁ = ψ₂ = 0). 预备知识:ℂ³ 代数结构 Preliminaries: ℂ³ Algebraic Structure 乘法表 Multiplication Table 基为 {1, e₁, e₂},乘法由以下关系定义: Basis {1, e₁, e₂}, multiplication defined by: ✅ e₁² = −1 + δ e₂, e₁ e₂ = e₂ e₁ = α e₁, e₂² = β e₂ ✅ e₁² = −1 + δ e₂, e₁ e₂ = e₂ e₁ = α e₁, e₂² = β e₂ 基本性质 Basic Properties • 交换性:X·Y = Y·X(乘法表对称) Commutativity: X·Y = Y·X (symmetric multiplication table) • 分配律:X·(Y+Z) = X·Y + X·Z(严格成立) Distributivity: X·(Y+Z) = X·Y + X·Z (strictly holds) • 非结合性:一般地 (X·Y)·Z ≠ X·(Y·Z) Non-associativity: generally (X·Y)·Z ≠ X·(Y·Z) • 结合子:Δ(X,Y,Z) = (X·Y)·Z − X·(Y·Z) Associator: Δ(X,Y,Z) = (X·Y)·Z − X·(Y·Z) • 严格定理:Δ(X,X,X) ≡ 0(幂结合性在自乘时成立) Strict theorem: Δ(X,X,X) ≡ 0 (power-associativity holds for self-multiplication) • 严格定理:Δ(X,X,Y) ≠ 0 当 Y ≠ X(混合非结合性) Strict theorem: Δ(X,X,Y) ≠ 0 when Y ≠ X (mixed non-associativity) • 定理 2.5(非结合激活条件):当 x₀=y₀=z₀=0 时 Δ(X,Y,Z) ≡ 0。非结合性只在至少一个元素的标量分量非零时显现。 Theorem 2.5 (Non-associative activation): Δ(X,Y,Z) ≡ 0 when x₀=y₀=z₀=0. Non-associativity only manifests when at least one element has non-zero scalar component.