任意维度与号型下的狄拉克、马约拉纳与外尔旋量
Dirac, Majorana, and Weyl Spinors in Arbitrary Dimension and Signature
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中文总结 AI 辅助
本文针对任意维度与号型，通过分别处理实克利福德代数及其偶子代数，给出明确约定的费米子场分类方法，确定各周期性扇区下的各类旋量场，区分不变配对与相关约束，完善了旋量结构的系统分类。
中文摘要 AI 辅助
任意号型下的费米子场类型常仅通过复旋量表示推导得出，这会掩盖与实结构、不连通时空反射以及作用量层面双线性约束相关的区别。我们提出一种明确约定的分类方法，分别处理完整的实克利福德代数及其偶子代数，从而区分旋子（pinors）与旋量（spinors），以及Pin等变的马约拉纳型结构与仅为Spin等变的结构。我们将伴随、复共轭与转置交织子，同基本及整体反射提升相结合，形成统一的符号演算。所得的局域克利福德分类按32个周期性扇区组织，由Bott类及总维度模8确定，不含依赖理论的规范、反常或整体约束。对每个扇区，我们明确了可存在的狄拉克、外尔、马约拉纳、辛马约拉纳、马约拉纳-外尔及辛马约拉纳-外尔场；并区分了不变配对的存在性与交织子兼容性、格拉斯曼及手征选择规则、协变指标收缩，以及局域双线性算子的厄米性。
英文摘要
Fermionic field types in arbitrary signature are often inferred from complex Spin representations alone, which obscures distinctions associated with real structures, disconnected spacetime reflections, and action-level bilinear constraints. We give a convention-explicit classification that treats the full real Clifford algebra and its even subalgebra separately, thereby distinguishing pinors from spinors and Pin-equivariant Majorana-type structures from structures that are only Spin-equivariant. Adjoint, complex-conjugation, and transposition intertwiners are combined with elementary and collective reflection lifts into a unified sign calculus. The resulting local Clifford classification is organised into thirty-two periodicity sectors determined by the Bott class and the total dimension modulo eight, without including theory-dependent gauge, anomaly, or global constraints. For each sector, we identify the available Dirac, Weyl, Majorana, symplectic Majorana, Majorana-Weyl, and symplectic Majorana-Weyl fields. The existence of an invariant pairing is distinguished from intertwiner compatibility, Grassmann and chiral selection rules, covariant index contraction, and Hermiticity of local bilinear operators.
