[论文解读] Coherent States for Fractional Powers of the Harmonic Oscillator Hamiltonian

Kristina Giesel|arXiv (Cornell University)|Nov 16, 2021

Noncommutative and Quantum Gravity Theories参考文献 85被引用 4

一句话总结

本文通过两种方法——广义狄拉克量子化与群平均法，以及一种新颖的分数阶标签相干态构造——为哈密顿量包含谐振子分数幂的量子系统构建了相干态。核心贡献是提出了一类新型相干态，其能精确近似分数阶算符的半经典行为，并满足恒等分解关系，从而在量子引力与相对论性模型中实现可靠的半经典分析。

ABSTRACT

Inspired by special and general relativistic systems that can have Hamiltonians involving square roots or more general fractional powers, in this article, we address the question of how a suitable set of coherent states for such systems can be obtained. This becomes a relevant topic if the semiclassical sector of a given quantum theory is to be analysed. As a simple setup, we consider the toy model of a deparametrised system with one constraint that involves a fractional power of the harmonic oscillator Hamiltonian operator, and we discuss two approaches to finding suitable coherent states for this system. In the first approach, we consider Dirac quantisation and group averaging, as have been used by Ashtekar et al., but only for integer powers of operators. Our generalisation to fractional powers yields in the case of the toy model a suitable set of coherent states. The second approach is inspired by coherent states based on a fractional Poisson distribution introduced by Laskin, which however turn out not to satisfy all properties to yield good semiclassical results for the operators considered here and in particular do not satisfy a resolution of identity as claimed. Therefore, we present a generalisation of the standard harmonic oscillator coherent states to states involving fractional labels, which approximate the fractional operators in our toy model semiclassically more accurately and satisfy a resolution of identity. In addition, motivated by the way the proof of the resolution of identity is performed, we consider these kind of coherent states also for the polymerised harmonic oscillator and discuss their semiclassical properties.

研究动机与目标

• 为哈密顿量包含谐振子分数幂的量子系统开发合适的相干态。
• 将先前仅用于整数幂的狄拉克量子化与群平均技术扩展至分数幂。
• 解决目前缺乏同时满足精确半经典近似与恒等分解关系的分数阶算符相干态的问题。
• 将标准谐振子相干态推广至包含分数阶标签，以改善其半经典行为。
• 探讨这些新态在凝聚态谐振子模型中的半经典性质。

提出的方法

• 将狄拉克量子化与群平均方法推广至谐振子哈密顿量的分数幂。
• 引入一类具有分数阶标签的新相干态，以在半经典极限下更精确地近似分数阶算符。
• 基于带有分数参数的修正泊松分布构造相干态，但此类构造无法满足恒等分解关系。
• 利用恒等分解关系的证明结构，将该方法推广至凝聚态谐振子模型。
• 将新相干态应用于标准与凝聚态模型中，分析其半经典性质。
• 将新分数阶标签态与拉斯金的基于分数阶泊松分布的态进行比较，表明后者不满足恒等分解关系。

实验结果

研究问题

• RQ1 狄拉克量子化与群平均方法能否推广至谐振子哈密顿量的分数幂？
• RQ2 基于拉斯金分数阶泊松分布的相干态是否满足恒等分解关系，并能提供良好的半经典近似？
• RQ3 能否构造一类具有分数阶标签的新相干态，使其既能精确近似分数阶算符，又能满足恒等分解关系？
• RQ4 在凝聚态谐振子模型中，新分数阶标签相干态在半经典极限下的表现如何？
• RQ5 这些相干态对量子引力与相对论性量子系统中半经典分析有何影响？

主要发现

• 广义群平均方法在模型中成功构造出分数阶哈密顿量的相干态。
• 拉斯金的基于分数阶泊松分布的相干态虽在文献中有宣称，但不满足恒等分解关系。
• 新分数阶标签相干态对分数阶算符的半经典近似优于标准态或拉斯金基态。
• 这些新态满足恒等分解关系，这是量子理论中物理一致性的关键要求。
• 该构造方法足够稳健，可扩展至凝聚态谐振子模型，同时保持关键的半经典性质。
• 该方法为具有非多项式哈密顿量的量子引力模型提供了可靠的半经典分析框架。