[论文解读] Additional degrees of freedom associated with position measurements in non-commutative quantum mechanics

Christian M. Rohwer, Scholtz, FG|arXiv (Cornell University)|Jun 6, 2012

Quantum Mechanics and Applications参考文献 9被引用 5

一句话总结

本文表明，在非对易量子力学中，为了完整描述量子态，位置测量需要超出空间坐标的额外自由度。通过正算子值测度（POVMs）的弱测量，本文表明非局域的星积形式可被等价地替换为涉及扩展结构粒子或具有内禀角动量的点粒子的局域描述，揭示了这些额外自由度在物理上具有意义，并且在规范不变的表述中可能具有规范性质。

ABSTRACT

In this thesis we shall demonstrate that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the formalism used entails a description that is non-local in that it requires all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary to specify quantum states completely. The remainder of the thesis, will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended, structured objects emerges quite naturally and unavoidably there. This description will be shown to be equivalent to one describing a two-charge harmonically interacting composite in a strong magnetic field found by Susskind. It will be argued that these notions also extend naturally to the quantum level, and constraints will be shown to arise there. A further local formulation will be introduced, with an interpretation in terms of objects located at a point with a certain angular momentum about that point. This again enforces the idea of particles that are not point-like. Both local descriptions make explicit the additional structure which is encoded more subtly in the non-local description. Additional degrees of freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in a gauge-invariant formulation of the theory.

研究动机与目标

• 解决由于最小长度尺度导致在非对易空间中定义位置测量时的模糊性问题。
• 表明仅基于位置的测量无法在非对易系统中完全重构量子态。
• 开发非对易量子力学的局域等价表述，使额外的结构自由度显式化。
• 从物理上解释这些额外自由度，将其与扩展物体或内禀角动量联系起来。
• 探讨这些自由度是否可在规范不变表述中被理解为规范自由度。

提出的方法

• 通过正算子值测度（POVMs）的弱（非投影）测量形式化位置测量。
• 使用星积将算符乘法映射为函数乘法，揭示其对所有阶位置导数的非局域依赖。
• 通过引入额外自由度（如内部结构或角动量）推导等价的局域描述。
• 构造一个经典极限，其中扩展的、谐振耦合的两电荷系统自然涌现。
• 引入第二种局域表述，其中粒子为点状但具有关于其位置的固定角动量。
• 分析理论的规范结构，表明额外自由度可在规范不变框架中被解释为规范自由度。

实验结果

研究问题

• RQ1 非对易量子力学中的位置测量能否完全确定一个量子态？
• RQ2 当位置测量不完整时，所需的额外自由度具有何种物理意义？
• RQ3 如何用使额外结构显式化的局域描述替代非局域的星积形式？
• RQ4 在经典极限下，这些额外自由度是否对应于扩展物体或内禀角动量？
• RQ5 这些额外自由度是否可在非对易量子力学的规范不变表述中被解释为规范自由度？

主要发现

• 由于星积对所有阶位置导数的非局域依赖，仅靠位置测量无法在非对易量子力学中完全重构量子态。
• 存在多种等价的局域表述，每种均引入额外自由度，使隐藏结构显式化。
• 一种局域表述自然导出扩展结构物体的经典图像，等价于强磁场下Susskind的两电荷谐振系统。
• 第二种局域表述描述了关于其位置具有固定角动量的点状粒子，强化了非点状量子物体的概念。
• 额外自由度可在非对易量子力学的规范不变表述中被解释为规范自由度。
• 这些表述的等价性证实了额外结构在物理上是必需的，而非单纯的数学构造。