时空准晶体与通往量子引力之路中的计算复杂性理论
Spacetime Quasicrystals and Computational Complexity Theory in the Road Towards Quantum Gravity
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中文总结 AI 辅助
本文证明半经典引力可高效求解NP完全问题，暗示引力量子化；并构造洛伦兹准晶体，提出其与量子引力及跷跷板关系的联系，且在数-体积对应和时空超均匀性上优于随机集。
中文摘要 AI 辅助
第一部分：半经典引力高效求解NP完全问题。假设引力场是经典的，并且通过半经典爱因斯坦场方程（EFEs）与量子场耦合，我们证明了一个大质量且非相对论性量子比特的弱场动力学原则上可以用于在多项式时间内求解NP完全问题。我们将这种巨大的计算能力归因于半经典EFEs所提供的非线性量子比特动力学。因此，上述两个假设意味着对物理扩展丘奇-图灵论题的违反，我们将其视为引力量子化的证据。第二部分：时空准晶体。我们将自相似准晶体（如彭罗斯镶嵌）从欧几里得空间推广到闵可夫斯基时空。我们构造了洛伦兹准晶体的首批例子，并识别了它们的关键新特征。然后我们探讨了与基础物理的可能联系。首先，我们提出一个推测性场景，其中我们的（3+1）维宇宙嵌入在一个特别对称的（9+1）维环面T^{9,1}中，该环面先前被认为是超弦最对称的环面紧化。我们建议这一构造可能有助于阐明普朗克尺度、真空能量尺度和电弱尺度之间的跷跷板关系M_Pl M_vac ≈ M_EW^2。其次，我们将因果集理论中的随机因果集替换为有序的、高度对称的准晶因果集。我们发现暗示性证据表明，我们的准晶体在（1+1）维和（3+1）维中的数-体积对应性方面优于泊松随机集，而这种行为在之前被认为是不可能的。最后，将超均匀性从空间扩展到时空，我们证明我们的（1+1）维准晶体是时空超均匀结构的第一个实例。
英文摘要
Part I: Semiclassical Gravity Efficiently Solves $\mathsf{NP}$-Complete Problems Assuming the gravitational field is classical and that it couples to quantum fields via the semiclassical Einstein field equations (EFEs), we show that the weak-field dynamics of a massive and non-relativistic qubit can in principle be used to solve an $\mathsf{NP}$-complete problem in polynomial time. We attribute this vast computational power to the non-linear qubit dynamics afforded by the semiclassical EFEs. Consequently, the above two assumptions entail a violation of the Physical Extended Church--Turing Thesis, which we regard as evidence for the quantization of gravity. Part II: Spacetime Quasicrystals We generalize self-similar quasicrystals, such as the Penrose tilings, from Euclidean space to Minkowski spacetime. We construct the first examples of Lorentzian quasicrystals and identify their key novel features. We then explore possible connections to fundamental physics. First, we propose a speculative scenario in which our $(3+1)$D universe is embedded in a particularly symmetric $(9+1)$D torus $\mathbb{T}^{9,1}$, previously identified as the most symmetric toroidal compactification of the superstring. We suggest that this construction may shed light on the seesaw relation $M_{\rm Pl}M_{\rm vac} \approx M_{\rm EW}^{2}$ between the Planck, vacuum-energy, and electroweak scales. Second, we replace the random causal sets of causal set theory with ordered, highly symmetric quasi-crystalline causal sets. We find suggestive evidence that our quasicrystals outperform Poisson random sets in the number-volume correspondence in both $(1+1)$D and $(3+1)$D, where such behaviour has been conjectured to be impossible. Finally, extending hyperuniformity from space to spacetime, we show that our $(1+1)$D quasicrystal is the first demonstrated example of a spacetime-hyperuniform structure.
发表机构
- University of Waterloo（滑铁卢大学）
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