标题： 量子力学没有时空：非对易几何的案例 显示英文标题

标题： Quantum mechanics without spacetime: a case for noncommutative geometry

摘要： 量子力学在目前的表述中需要一个外部的经典时间来进行描述。 经典时空流形和经典时空度规是由经典物质场产生的。 在没有这种经典物质场的情况下，量子力学应该在不参考经典时间的情况下进行表述。 如果存在这样的新表述，那么标准线性量子力学就是一种更基本的非线性量子理论的极限情况。 一种可能的新表述方法是通过非交换微分几何中的非交换时空坐标来实现。 在这里，非线性理论由属于Doebner-Goldin方程类的非线性薛定谔方程来描述，这些方程几年前已被发现。 这种与质量相关的非线性在粒子质量与普朗克质量相当时是显著的，否则可以忽略。 这种非线性原则上可以通过对介观系统进行量子力学的实验测试来检测，并且是对量子引力理论的一种有价值的实证探测。 我们还简要地讨论了我们的方法可能与环量子引力和弦理论之间的联系。 显示英文摘要

摘要： Quantum mechanics in its presently known formulation requires an external classical time for its description. A classical spacetime manifold and a classical spacetime metric are produced by classical matter fields. In the absence of such classical matter fields, quantum mechanics should be formulated without reference to a classical time. If such a new formulation exists, it follows as a consequence that standard linear quantum mechanics is a limiting case of an underlying non-linear quantum theory. A possible approach to the new formulation is through the use of noncommuting spacetime coordinates in noncommutative differential geometry. Here, the non-linear theory is described by a non-linear Schrodinger equation which belongs to the Doebner-Goldin class of equations, discovered some years ago. This mass-dependent non-linearity is significant when particle masses are comparable to Planck mass, and negligible otherwise. Such a non-linearity is in principle detectable through experimental tests of quantum mechanics for mesoscopic systems, and is a valuable empirical probe of theories of quantum gravity. We also briefly remark on the possible connection our approach could have with loop quantum gravity and string theory.