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首页> 外文期刊>Reviews in Mathematical Physics >MINIMAL LENGTH IN QUANTUM SPACE AND INTEGRATIONS OF THE LINE ELEMENT IN NONCOMMUTATIVE GEOMETRY
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MINIMAL LENGTH IN QUANTUM SPACE AND INTEGRATIONS OF THE LINE ELEMENT IN NONCOMMUTATIVE GEOMETRY

机译：非交换几何中量子空间的最小长度和线元素的积分

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摘要

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime (θ-Minkowski); on the other side, Connes' spectral distance in noncommutative geometry. Although in the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular, in the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d′L, which coincides exactly with the spectral distance dD on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator — together with their translations — d′L and dD coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d′L and dD as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.
机译：我们比较了出现在文献中各个不同点的两个概念，对量子时空中最小长度的出现提出了质疑：一方面，量子长度是Doplicher Fredenhagen Roberts（DFR）量子时空中算子L的谱，以及规范的非交换时空（θ-Minkowski）；另一方面，康尼斯在非交换几何中的光谱距离。尽管在欧几里得空间中这两个概念合并为测地距离之一，但它们在非交换框架中产生了截然不同的结果。特别是在Moyal平面中，量子长度从零开始有界，而光谱距离可以取任何实际的正值，包括无穷大。我们展示了如何通过将频谱三倍加倍来解决这一差异。这导致我们引入一个修改后的量子长度d'L，该长度与最佳定位状态集上的光谱距离dD完全吻合。在高能和大平移极限中，在量子谐波振荡器的本征态集及其平移上，d'L和dD渐近一致。在小能量下，我们将d'L和dD之间的差异解释为将线元素集成在量子空间上的两种不同方式。这使我们提出了Moyal平面上测地线的方程。

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