量子弧长度量的变形
Quantum deformations of the arc length metric
AI总结：
本文研究了单位圆上弧长度量的q-变形，揭示了其在量子度量空间中的意义，并展示了该变形在经典度量下的恢复及连续依赖性。
摘要：
我们研究了单位圆上弧长度量的q-变形。这种q-变形自然地来自于Dirac算子，通过将标准整数替换为它们的q-变形类比来实现。然而，我们证明相应的度量结构仅在Marc Rieffel引入的量子度量空间水平上才有意义。这意味着我们在圆上连续函数上获得的量子度量并不来自于圆上的经典度量。在q等于1的特殊情况下，我们恢复了通常的弧长度量，并且证明我们的量子度量空间家族在David Kerr的完整Gromov-Hausdorff距离下连续地依赖于变形参数。
英文摘要：
We investigate a q-deformation of the arc length metric on the unit circle. This q-deformation arises naturally from the Dirac operator by replacing the standard integers by their q-deformed analogues. Nonetheless, we show that the corresponding metric structure only makes sense at the level of quantum metric spaces as introduced by Marc Rieffel. This means that the quantum metric we obtain on the continuous functions on the circle does not arise from a classical metric on the circle. In the special case where q equals one we recover the usual arc length metric and we show that our family of quantum metric spaces depend continuously on the deformation parameter with respect to David Kerr's complete Gromov-Hausdorff distance.
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