相空间中相对论质量壳因式分解产生的旋量结构
Spinor Structure and Quantum Mechanics from Relativistic Mass-Shell Factorisation in Phase Space
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中文总结 AI 辅助
该研究从相对论相空间质量壳因式分解出发，提出旋量结构的非量子起源，通过矩阵莫约星对易子等推导，协调了相对论协变性、经典相空间输运与量子自旋。
中文摘要 AI 辅助
自旋通常被视为最具内禀性的量子现象之一，其在经典物理学中缺乏自然对应物，这为从相空间变形量子化角度解释自旋起源造成了障碍。我们表明，这一困难可能源于对相对论相空间施加了过于严格的标量哈密顿结构。要求完整的有质量质量壳约束由一个在动量的四个分量上均为线性的有限维表达式表示，这迫使它的系数矩阵满足克利福德代数。该代数的最小复表示是四维的。在质量壳因子选定的秩二子空间内要求统计完备性，会产生一个四乘四的矩阵值系综分布。在每个在壳动量处，线性质量壳算子选定一个二维子空间，因此一般经典系综在量子化前由一个2×2矩阵描述。将外尔序的刘维尔方程投影到该子空间，可得到相对论输运，同时保留任意布居数和相干性，剩余投影器分量决定一阶变形修正。矩阵莫约星对易子的展开在领头阶产生对称化的经典矩阵刘维尔算符。若施加更强的双边星约束，它们会重现左右狄拉克-维格纳方程，并引入约化普朗克常数ℏ作为将无量纲内部代数转换为物理角动量的尺度。这些结果表明旋量结构具有非量子起源，并为协调相对论协变性、经典相空间输运与量子自旋提供了一条途径。
英文摘要
Spinor structure is usually introduced as part of relativistic quantum theory. We show that its underlying algebra can instead emerge without quantisation from a covariant statistical description of a massive relativistic particle where we require the theory retain both sheets of the massive mass shell. A finite-dimensional factorisation of the complete mass-shell constraint, linear in all four momentum components, forces a Clifford algebra. In (3+1) dimensions its minimal complex representation is four-dimensional, while each on-shell factor selects a rank-two sector; statistical completeness therefore requires an arbitrary $2\times2$ matrix distribution within that sector before quantisation. Projecting the Weyl-ordered matrix Liouvillian gives relativistic transport while preserving all internal populations and coherences. For universal deformations on flat canonical phase space satisfying the stated covariance, matrix-neutrality, homogeneity and associativity assumptions, the resulting product is the matrix Weyl--Moyal product. Its parameter has dimensions of action and, when identified with $\hbar$, supplies the scale of physical spin-$\tfrac12$. Requiring all-order constraint preservation gives a vanishing star commutator, while preservation of the selected star-spectral sector yields the two-sided Dirac--Wigner equations. Our central conclusion is therefore that relativistic mass-shell factorisation supplies spinor structure, while completion of the constraint-preservation hierarchy leads to the Weyl--Moyal phase-space formulation of quantum mechanics.
发表机构
- Quantalytics
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