标题： 非海森堡量子力学 显示英文标题

标题： Non-Heisenberg quantum mechanics

摘要： 放松公理理论的公设是寻找更一般理论的自然方法，历史上，非欧几里得几何的发现是这一过程的一个著名例子。 在这里，我们使用这种方法通过忽略海森堡量子力学的核心来扩展量子力学——我们不假设存在一个满足海森堡对易关系的位置算符，$[\hat x,\hat p]=i\hbar$。 除了伽利略对称性之外，量子理论的其余公设导致了一个具有长度维度自由参数$l_0$的更一般的量子理论，使得当$l_0 \to 0$时，该理论退化为标准量子理论。 令人惊讶的是，这种没有先验假设非对易关系的非海森堡量子理论导致了一个修改的海森堡不确定性关系，$\Delta x \Delta p\geq \sqrt{\hbar^2/4+l_0^2(\Delta p)^2}$，这确保了存在一个最小位置不确定性，$l_0$，正如各种量子引力研究中所预期的那样。 通过将该框架的结果与一些观测数据进行比较，其中包括棒状引力波探测器AURIGA的第一个纵向正常模式和氢原子中的$1S-2S$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 $l_0$.