[论文解读] Expectation values, experimental predictions, events and entropy in quantum gravitationally decohered quantum mechanics

Bernard S. Kay, Varqa Abyaneh|ArXiv.org|Oct 4, 2007

Quantum Mechanics and Applications参考文献 33被引用 14

一句话总结

本文提出一种量子引力退相干机制，其中物理密度矩阵为对引力的约化迹，物理熵定义为物质-引力纠缠熵。研究表明，尽管某些可观测量（如动量平方）的期望值发生改变，但所有真实测量均为位置测量，因此实验上无法检测到偏差——即使在彭罗斯提出的实验中亦如此——同时消除了薛定谔猫态，并提出一种新颖的‘事件’诠释。

ABSTRACT

We restate Kay's 1998 hypothesis which simultaneously offers an objective definition for the entropy of a closed system, a microscopic foundation for the Second Law, a resolution of the Information Loss (and other) Black-Hole Puzzle(s) and an objective mechanism for decoherence. Presupposing a conventional unitary theory of low-energy quantum gravity, it offers all this by taking the physical density operator of a closed system to be the partial trace of its total density operator (assumed pure) over gravity and by defining its physical entropy to be its `matter-gravity entanglement entropy'. We also recall Kay's 1998 modified non-relativistic (many-body) quantum mechanics based on Kay's hypothesis with a Newtonian approximation to quantum gravity. In this modification, we find formal expectation values for certain `observables' such as momentum-squared and Parity are altered but those for functions of positions are unaltered. However, by arguing that every real measurement can ultimately be taken to be a position measurement, we prove that, in practice, it is impossible to detect any alteration at all and, in particular, we predict no alteration for Roger Penrose's experiment. Nevertheless, Kay's modification contains no Schrödinger Cat-like states, and also allows an `events' interpretation which we tentatively propose and begin to explore. We also obtain a Second-Law type result for a non-relativistic toy-model closed system and argue that similar results will apply for a wide class of model Newtonian and post-Newtonian closed systems although we argue that ordinary actual lab-sized systems can never be treated as closed for the purpose of calculating their entropy. Compared with `collapse models' such as GRW, Kay's Newtonian theory does a similar job while being free from ad hoc assumptions.

研究动机与目标

• 通过量子引力框架解决量子力学中的基础性问题，包括热力学第二定律和黑洞信息悖论。
• 通过物质-引力纠缠，为封闭系统中熵增提供客观的微观基础。
• 通过将物理密度算符重新定义为对引力自由度的约化迹，消除如薛定谔猫态等非物理叠加态。
• 基于与物理密度矩阵可交换的可造变量（beables）和谱投影算符，提出一种新颖的‘事件’诠释。
• 证明尽管存在理论修正，实验中对标准量子力学的偏差在实践中不可检测。

提出的方法

• 假设一个低能量子引力框架，总希尔伯特空间为物质与引力子空间的张量积。
• 将物理密度算符定义为总纯态密度算符对引力自由度的约化迹。
• 将物理熵识别为物质密度矩阵的冯诺依曼熵，即物质-引力纠缠熵。
• 推导出退相干指数 D(a)，抑制位置表象中非对角元，从而修正非位置可观测量的期望值。
• 将该框架应用于量子引力的牛顿近似，显示对动量平方和宇称期望值的修正。
• 引入‘可造变量’作为与物理密度矩阵可交换的自伴算符，定义每次事件的取值，从而实现基于事件的诠释。

实验结果

研究问题

• RQ1 幺正量子引力理论能否产生物理熵的单调增加，从而为热力学第二定律提供微观基础？
• RQ2 物质-引力纠缠熵是否能在封闭系统中解决黑洞信息丢失悖论？
• RQ3 该框架中对期望值的修正是否可在真实实验中检测到，特别是涉及叠加态的实验？
• RQ4 能否基于物理密度矩阵和可交换的可造变量，构建一个一致的量子力学‘事件’诠释？
• RQ5 为何尽管理论上存在修正（如动量平方期望值改变），这些偏离在实验中仍不可检测？

主要发现

• 物理密度算符被定义为对引力的约化迹，得到物质的混合态，其演化为幺正但熵持续增加。
• 物理熵被识别为物质密度矩阵的冯诺依曼熵，在幺正演化下单调增加，从而为热力学第二定律提供微观起源。
• 在牛顿近似下，动量平方和宇称的期望值发生修正，但位置函数的期望值保持不变。
• 由于所有真实测量最终均为位置测量，理论偏差在实验中不可检测，包括彭罗斯所提议的实验。
• 该理论消除了类似薛定谔猫态的叠加态，因为此类叠加态在相关区域并非物理密度矩阵的本征态。
• 提出一种新颖的‘事件’诠释，其中可造变量（与密度矩阵可交换的算符）为每次事件赋予确定值，其期望值与常规迹一致。