Groupoids and Geometric Quantization

群曲面和几何量化

• 批准号: RGPIN-2015-05833
• 负责人: Krepski, Derek
• 金额: $ 0.95万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613778
• 关键词: Groupoids; Geometric; Quantization

The mathematical models underlying classical mechanics and quantum mechanics are very different. Indeed, classical mechanics describes physical laws on a large scale (e.g. laws of motion for a swinging pendulum, planetary orbits, etc.) and is mathematically rooted in differential geometry, an extension of calculus to more general spaces that allow for curvature and higher dimensions. In contrast, quantum mechanics describes the physical laws on a subatomic scale and is rooted in the language of Hilbert spaces. There is great interest in understanding the connection between these two models. Intuitively, one expects the laws describing the physics at the subatomic level determine behaviour on the large scale. Mathematically, this intuition is realized by a process known as the semi-classical limit, which produces a classical system from a quantum one. Proceeding in the reverse direction (i.e. finding a quantum system that corresponds to a classical one) is known as quantization, although it is not always possible for physical and mathematical reasons. Geometric quantization is a mathematical understanding of such a relationship between classical and quantum physics. Being geometric, it highlights symmetries that are present in each framework. This research proposal investigates ways in which the geometric approach to quantization can be adapted to certain kinds of generalized symmetries, known as groupoid actions, a modern perspective on symmetry at the crossroads of geometry, topology, and mathematical physics. Like their classical counterparts, these symmetries can be viewed as transformations of the parameters in the mathematical models that do not change the physical laws being described. In contrast to the classical setting, groupoid symmetries cannot always be composed: it is not always possible to follow a given transformation by another one. The particular groupoid symmetries investigated in this proposal, called quasi-symplectic groupoid actions, describe examples of interest, such as moduli spaces of connections (which can be viewed as spaces of solutions of a differential equation), from various active areas of mathematics research in Canada and abroad, including symplectic geometry, algebraic geometry, and conformal and quantum field theories. A complete understanding of quantization for groupoid actions is not yet available.  The objectives in this research proposal contribute significantly towards filling this gap by (i) advancing the theoretical framework that accommodates generalized symmetry; and (ii) working within one such a framework (quasi-Hamiltonian actions) to ultimately clarify active research in physics (e.g. by proving conjectured formulas in conformal field theory).

经典力学和量子力学的数学模型是非常不同的。事实上，经典力学描述的是大尺度的物理定律（例如摆动的钟摆的运动定律，行星轨道等）。它在数学上植根于微分几何，这是微积分在更一般的空间中的延伸，允许曲率和更高的维度。相比之下，量子力学描述的是亚原子尺度上的物理定律，它植根于希尔伯特空间的语言，人们对理解这两个模型之间的联系非常感兴趣。 直觉上，人们认为描述亚原子水平物理学的定律决定了大尺度上的行为。在数学上，这种直觉是通过一个被称为半经典极限的过程实现的，这个过程从量子系统产生经典系统。反向进行（即找到一个对应于经典系统的量子系统）被称为量子化，尽管由于物理和数学原因并不总是可能的。 几何量子化是对经典物理和量子物理之间关系的数学理解。作为几何，它突出了每个框架中存在的对称性。本研究计划探讨量子化的几何方法如何适用于某些广义对称性，称为群胚作用，这是几何，拓扑和数学物理交叉点上对称性的现代观点。 像它们的经典对应物一样，这些对称性可以被视为数学模型中参数的变换，而不会改变所描述的物理定律。与经典的设定相反，群胚对称不可能总是被合成的：它不可能总是遵循一个给定的变换由另一个。在这个建议中研究的特殊群胚对称性，称为准辛群胚作用，描述了感兴趣的例子，例如连接的模空间（可以被视为微分方程的解的空间），来自加拿大和国外数学研究的各个活跃领域，包括辛几何，代数几何，共形和量子场论。 对广群作用的量子化还没有一个完整的理解。 本研究提案的目标通过以下方式为填补这一空白做出了重大贡献：（i）推进适应广义对称性的理论框架;（ii）在一个这样的框架（准哈密顿作用量）内工作，以最终澄清物理学中的积极研究（例如，通过证明共形场论中的固定公式）。