$G_{\mathrm{NC}}$的中心字符、达布规范形以及NCQM与QM的运动学不等价性
From Bopp Shifts to Toroidal Shadows: K-Theoretic Gap Labels in Noncommutative Quantum Mechanics
AI总结：
本文从$G_{\mathrm{NC}}$的单位表示理论角度分析了非交换量子力学中的广义Bopp位移和达布规范形，证明了非退化的NCQM态与普通量子力学态在$G_{\mathrm{NC}}$表示下不等价。
摘要：
我们从$G_{\mathrm{NC}}$的运动学对称群的单位表示理论角度，分析了二维非交换量子力学（NCQM）中的广义Bopp位移和达布规范形。该群是一个二阶nilpotent李群，具有三维中心，其正规部分的单位双射$\widehat{G_{\mathrm{NC}}}$由中心字符$(\hbar,\vartheta,B_{\mathrm{in}})$标示。普通二维量子力学（QM）作为$\widehat{G_{\mathrm{NC}}}$中沿商映射$G_{\mathrm{NC}}\rightarrow G_{\mathrm{WH}}$膨胀的Weyl-Heisenberg表示族，其中心字符为$(\hbar,0,0)$。我们证明，一个一般的非退化NCQM态$(\hbar_0,\vartheta_0,B_0)$，其中$\hbar_0,\vartheta_0,B_0\neq 0$且$\hbar_0-B_0\vartheta_0\neq 0$，作为$G_{\mathrm{NC}}$表示与普通QM态$(\hbar_0,0,0)$不等价。因此，尽管广义Bopp位移和达布规范形可以产生满足规范交换关系的辅助算子四元组，但它们并不建立相应态的运动学等价性。我们进一步解释，这种看似的识别仅在通过达布规范形或粗略星积描述后才出现，其中原始的$G_{\mathrm{NC}}$中心字符标签不再作为数据的一部分。该结果澄清了NCQM中算子层面的达布规范形、变形量子化等价性和表示论等价性之间的关系。
英文摘要：
We study Bopp shifts in two-dimensional noncommutative quantum mechanics (NCQM) through a functorial lens. A nondegenerate NCQM sector with central character $(\hbar,\vartheta,B_{\rm in})$ determines a self-adjoint infinitesimal representation of the NCQM Lie algebra $\mathfrak{g}_{\rm NC}$. A Darboux normalization of its represented phase-space operators produces a self-adjoint infinitesimal representation of the Weyl-Heisenberg Lie algebra $\mathfrak{g}_{\rm WH}$ with central parameter $\hbar$, and hence defines a Bopp-shift functor collapsing $(\hbar,\vartheta,B_{\rm in}) \mapsto \hbar$. In particular, a generic NCQM sector is not equivalent, as a $\mathfrak{g}_{\rm NC}$-sector, to the ordinary QM sector $(\hbar,0,0)$, even though their Bopp-shift images have the same Weyl-Heisenberg parameter. To measure what this collapse forgets, we construct a toroidal shadow functor assigning to each periodic datum $L=(a_x,a_y)$ and each NCQM sector $χ$ a phase-space noncommutative four-torus $A^4_{Θ_{χ,L}}$. Its $K_0$-trace pairing yields sector-sensitive gap labels whose top-degree coefficient is $(\vartheta B_{\rm in}-\hbar)/((2π)^2\hbar)$. This coefficient is independent of the spatial cell area $a_xa_y$ and equals the Pfaffian ${\rm Pf}(Θ_{χ,L})$, the top-degree generator of the $K_0$-trace range. Since strong Morita equivalence preserves this trace range up to positive scaling, non-proportionality of the trace ranges obstructs Morita equivalence of the shadows, separating equal-$\hbar$ sectors that the Bopp-shift functor identifies. In the arithmetic subfamily where $\vartheta/A$ and $B_{\rm in}A/\hbar$ are algebraic, the trace-range scale is forced to be trivial and $|{\rm Pf}|$ becomes a computable separation criterion: $|\vartheta B_{\rm in}-\hbar| \neq |\vartheta' B'_{\rm in}-\hbar|$ already implies inequivalence.
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