量子引力模拟：基于广义不确定性原理的最小长度量子模拟
Quantum Gravity Simulation: Quantum simulation with a minimum length based on the generalised uncertainty principle
浏览论文内容
中文总结 AI 辅助
该研究提出用\(L\)个量子比特模拟一维量子系统的方法，推导广义不确定性原理，探讨其在低能和高能情况的性质，给出高能动量新表达式及相关特定情况分析，为用量子模拟研究高能量子引力现象提供新途径。
中文摘要 AI 辅助
我们提出了一种在一次量子化框架内用\(L\)个量子比特模拟一维低能和高能量子系统的方法。在有限差分法中假设最小网格间距\(\Delta L\)，解析推导广义不确定性原理（GUP），其在低能和高能量子系统中有不同性质。低能时，\(\Delta L \ll \hbar\)时GUP趋近标准海森堡不确定性原理。有限\(\Delta L \neq 0\)时，GUP依平均动量有三个不同区域，波函数在有限动量不确定性下可单点局域。还推导了高能动量新表达式并关注与相对论相关的两个特定情况。一是放松HUP且高能动量匹配狭义相对论情况，GUP预测高能存在非零质量的最小长度；二是与HUP一致则恢复无最小长度的正则高能表示，但与狭义相对论动量不兼容，暗示修正的能量 - 动量方程。我们认为所提出的动量公式为用量子模拟工具研究高能量子引力现象提供了新途径。
英文摘要
We present a recipe for simulating one-dimensional quantum systems for low and high energies with $L$ qubits within the framework of first quantisation. Assuming a minimum grid spacing $ΔL$ in the finite-difference method, the generalised uncertainty principle (GUP) is derived analytically and shows distinct properties for low- and high-energy quantum systems. In the low-energy regime, the GUP approaches to the standard Heisenberg uncertainty principle (HUP) for $ ΔL \ll \hbar$. However, for finite $ΔL \neq 0$, the GUP mathematically provides three different regions dependent on the average momentum and suggests that a wavefunction can exhibit a single-point localisation at the finite momentum uncertainty due to lack of grid resolution. We then derive a new expression for the high-energy momentum and focus on two specific cases relevant to relativistic aspects. First, if the HUP is relaxed and the high-energy momentum is matched to the special-relativistic one, the resulting GUP predicts the existence of a minimum length with non-zero mass for high energy despite the continuous limit $ΔL = 0$. Second, enforcing consistency with the HUP allows the recovery of a canonical high-energy representation with no minimum length. But this requirement brings incompatibility with the special-relativistic momentum and suggests a modified energy-momentum equation. Therefore, we believe that the proposed momentum formulation provides a novel pathway to investigate quantum gravity phenomena for high energy using quantum simulation tools.
