无时空的量子力学--非对易微分几何的一种可能情形？
Quantum mechanics without spacetime - a possible case for noncommutative differential geometry?
摘要：
量子力学的规则需要一个时间坐标来表述。然而，时间的概念一般只有在经典时空几何存在时才是可能的。这样的几何体本身是由经典物质源产生的。因此可以说，目前已知的量子力学公式预先假定了经典物质场的存在。量子力学应该有一个更基本的公式，它避免了使用时间的概念。在本文中，我们将讨论如何为单粒子、非相对论性量子力学构造这样一个基本公式。我们认为，有一个潜在的非线性理论的量子引力，这两个标准的量子力学和经典的广义相对论的近似。当粒子的质量远小于普朗克质量时，量子力学的无时间公式可以从基本理论中推导出来。另一方面，当粒子的质量比普朗克质量大得多时，时空就出现了，其基础理论应该简化为经典力学和广义相对论。我们还建议，非交换微分几何是一个可能的候选人描述这一基本理论。
英文摘要：
The rules of quantum mechanics require a time coordinate for their formulation. However, a notion of time is in general possible only when a classical spacetime geometry exists. Such a geometry is itself produced by classical matter sources. Thus it could be said that the currently known formulation of quantum mechanics pre-assumes the presence of classical matter fields. A more fundamental formulation of quantum mechanics should exist, which avoids having to use a notion of time. In this paper we discuss as to how such a fundamental formulation could be constructed for single particle, non-relativistic quantum mechanics. We argue that there is an underlying non-linear theory of quantum gravity, to which both standard quantum mechanics and classical general relativity are approximations. The timeless formulation of quantum mechanics follows from the underlying theory when the mass of the particle is much smaller than Planck mass. On the other hand, when the particle's mass is much larger than Planck mass, spacetime emerges and the underlying theory should reduce to classical mechanics and general relativity. We also suggest that noncommutative differential geometry is a possible candidate for describing this underlying theory.
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