AI 中文总结
本文研究带边界Faddeev--Jackiw矩阵的零模与规范生成元的关系，证明残余核的精确含义，给出闭合为精确规范对称性的链判据，并通过反例和机械实例验证该判据能识别虚假生成元。
AI 中文摘要
一般而言，带边界的Faddeev--Jackiw矩阵的零模并不是规范生成元。我们精确确定了残余核所编码的内容，并识别出实现真正规范对称性所需的附加条件。记$f^{(m)}=\begin{pmatrix}f^{(0)}&B\\\\-B^{\mathsf T}&0\end{pmatrix}$和$Γ=N^{\mathsf T}B$，我们证明$\dim\ker f^{(m)}=\dim\kerΓ^{\mathsf T}+\dim\kerΓ-\operatorname{rank}(ΠMΠ)$，其中$M=B^{\mathsf T}f^{(0)+}B$，$Π$投影到$\kerΓ$上。两个核贡献具有不同含义：第一个由$f^{(0)}$的零方向组成，且与约束曲面相切；第二个则计数独立的第一类组合。随后我们证明$Γ$并非Dirac约束矩阵：它是约化约束矩阵的主-次级块，而$M$提供次级-次级块。对于具有非零乘子分量的残余零模，我们推导出相关的两步链，并证明当链收缩在模约束理想的效应环中消失时，该链恰好闭合为精确的规范对称性。一个四变量反例表明，Faddeev--Jackiw停止条件可能成立，而系统却没有规范自由度，且链判据会拒绝虚假生成元。机械实例随后展示了该判据，并表明参数分层可以携带不同的规范和动力学内容，而这些内容会被过早的消去所抹除。
英文摘要
A null mode of the bordered Faddeev--Jackiw matrix is not, in general, a gauge generator. We determine exactly what the residual kernel encodes and identify the additional condition required for genuine gauge symmetry. Writing $f^{(m)}=\begin{pmatrix}f^{(0)}&B\\-B^{\mathsf T}&0\end{pmatrix}$ and $Γ=N^{\mathsf T}B$, we prove $\dim\ker f^{(m)}=\dim\kerΓ^{\mathsf T}+\dim\kerΓ-\operatorname{rank}(ΠMΠ)$, where $M=B^{\mathsf T}f^{(0)+}B$ and $Π$ projects onto $\kerΓ$. The two kernel contributions have distinct meanings: the first consists of null directions of $f^{(0)}$ tangent to the constraint surface, while the second counts independent first-class combinations. We then show that $Γ$ is not the Dirac constraint matrix: it is the primary--secondary block of the reduced constraint matrix, with $M$ providing the secondary--secondary block. For a residual null mode with nonzero multiplier component, we derive the associated two-step chain and prove that it closes to an exact gauge symmetry precisely when the chain contraction vanishes in the effective ring modulo the constraint ideal. A four-variable counterexample shows that the Faddeev--Jackiw stopping condition can hold while the system has no gauge freedom, and that the chain criterion rejects the false generators. Mechanical examples then exhibit the criterion and show that parameter strata can carry distinct gauge and dynamical content that is erased by premature cancellation.
Comments50 pages, 4 figures. Supplementary material: self-contained Wolfram Language notebook
