最小长度：伪装的截断？
The minimal length: a cut-off in disguise?
AI总结：
证明最小长度范式等价于波数空间截断，推广至多维与非对易几何，并揭示其运动学本质，指出实际上仅存在一种最小长度量子力学模型。
摘要：
最小长度范式，作为低能下量子引力的一个可能推论，通常被理解为对海森堡不确定性关系的唯象修改。我们证明，这一修改等价于与位置表象共轭的空间（即波数空间，它未必对应动量空间）中的一个截断。在给出波数的合适定义后，该结果可推广至多维情形与非对易几何。此外，我们发现了波数空间中随之产生的界限与最小长度尺度之间的直接关系。对于无法显式验证最小长度存在的情形，所提出的框架可用于澄清状况。事实上，将其应用于常见模型时，我们发现其中一个模型出乎意料地允许在位置测量中达到任意精度。最后，我们评论了这些发现对该领域的一般性意义。特别地，我们指出最小长度纯粹是运动学的，因此实际上只存在一种最小长度量子力学模型。
英文摘要：
The minimal-length paradigm, a possible implication of quantum gravity at low energies, is commonly understood as a phenomenological modification of Heisenberg's uncertainty relation. We show that this modification is equivalent to a cut-off in the space conjugate to the position representation, i.e. the space of wave numbers, which does not necessarily correspond to momentum space. This result is generalized to several dimensions and noncommutative geometries once a suitable definition of the wave number is provided. Furthermore, we find a direct relation between the ensuing bound in wave-number space and the minimal-length scale. For scenarios in which the existence of the minimal length cannot be explicitly verified, the proposed framework can be used to clarify the situation. Indeed, applying it to common models, we find that one of them does, against all expectations, allow for arbitrary precision in position measurements. In closing, we comment on general implications of our findings for the field. In particular, we point out that the minimal length is purely kinematical such that, effectively, there is only one model of minimal-length quantum mechanics.
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