标题： 非海森堡量子力学 显示英文标题

标题： Non-Heisenbergian quantum mechanics

摘要： 放宽一个公理理论的公设是一种自然的方式去发现更一般的理论，历史上，非欧几何的发现就是这一程序的一个著名例子。 在这里，我们用这种方法来扩展量子力学，通过忽略海森堡量子力学的核心——我们不假设存在满足海森堡对易关系的算符 $[\hat x,\hat p]=i\hbar$。 除了伽利略对称性之外，量子论的其余公设导致了一个具有自由参数 $l_0$（长度量纲）的更广义的量子理论，当 $l_0 \to 0$ 时，该理论退化为标准量子理论。 令人惊讶的是，这种非海森堡量子理论，在没有先验假设非对易关系的情况下，导出了修正的海森堡不确定性关系 $\Delta x \Delta p\geq \sqrt{\hbar^2/4+l_0^2(\Delta p)^2}$，这保证了最小位置不确定性 $l_0$ 的存在，正如各种量子引力研究所预期的那样。 通过将这个框架的结果与一些观测数据进行比较，这些数据包括条形引力波探测器 AURIGA 的第一纵向正则模以及氢原子中的 $1S-2S$ 跃迁，我们得到了 $l_0$ 的上限。 显示英文摘要

摘要： Relaxing the postulates of an axiomatic theory is a natural way to find more general theories, and historically, the discovery of non-Euclidean geometry is a famous example of this procedure. Here, we use this way to extend quantum mechanics by ignoring the heart of Heisenberg's quantum mechanics -- We do not assume the existence of a position operator that satisfies the Heisenberg commutation relation, $[\hat x,\hat p]=i\hbar$. The remaining axioms of quantum theory, besides Galilean symmetry, lead to a more general quantum theory with a free parameter $l_0$ of length dimension, such that as $l_0 \to 0$ the theory reduces to standard quantum theory. Perhaps surprisingly, this non-Heisenberg quantum theory, without a priori assumption of the non-commutation relation, leads to a modified Heisenberg uncertainty relation, $\Delta x \Delta p\geq \sqrt{\hbar^2/4+l_0^2(\Delta p)^2}$, which ensures the existence of a minimal position uncertainty, $l_0$, as expected from various quantum gravity studies. By comparing the results of this framework with some observed data, which includes the first longitudinal normal modes of the bar gravitational wave detector AURIGA and the $1S-2S$ transition in the hydrogen atom, we obtain upper bounds on the $l_0$.