如何用量子环面的模糊环面逼近平坦谱三元组：一个扭曲的故事
How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale
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中文总结 AI 辅助
本文通过引入扭曲谱三元组和广义谱邻近性，证明了经典和量子平坦环面可以在微分层次上被有限维模糊环面严格逼近，克服了离散微积分非局域性导致的莱布尼茨法则失效问题。
中文摘要 AI 辅助
我们证明，在谱邻近性框架下，经典和量子平坦环面可以在微分层次上被有限维模糊环面严格逼近。标准的逼近尝试通常受到离散微积分内在的非局域性以及随之而来的莱布尼茨法则失效的阻碍。虽然当代的替代方案如谱截断通过放弃$C^*$-代数转而使用算子系统来规避这个问题，但我们通过推广交换子公式来保留$C^*$-代数范畴。为此，我们引入了一个宽松的扭曲谱三元组概念，其中扭曲是一个线性映射，充当离散化Riesz变换，封装了离散世界的非局域性。通过将谱邻近性推广到可能具有无界扭曲的扭曲谱三元组的广义设置中，我们证明了配备自然离散微积分的模糊环面收敛到环面上的标准平坦Dirac三元组，同时底层的扭曲收敛到恒等映射。
英文摘要
We prove that the classical and the quantum flat torus can be rigorously approximated at a differential level by finite-dimensional fuzzy tori within the framework of the spectral propinquity. Standard attempts to establish this convergence are traditionally obstructed by the intrinsic non-locality of discrete calculus and the subsequent failure of the Leibniz rule. While contemporary alternatives such as spectral truncations circumvent this issue by abandoning $C^*$-algebras in favor of operator systems, we instead preserve the $C^*$-algebraic category by generalizing the commutator formula. To this end, we introduce a relaxed notion of a twisted spectral triple where the twist is a linear map acting as a discretized Riesz transform that encapsulates the non-locality of the discrete world. Self-adjointness is obtained by adjoining the backward difference to the forward difference and representing this doubled calculus with the creation and annihilation operators of the CAR algebra. We also require the L-seminorm to control both the twisted commutator and the displacement of the twist from the identity. By extending the spectral propinquity to this generalized setting of twisted spectral triples with possibly unbounded twists, we prove that fuzzy tori equipped with their natural two-sided discrete calculus converge to an amplification of the standard flat Dirac triple, while the underlying twists converge to the identity.
发表机构
- University of Denver（丹佛大学）
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