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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708248
• 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基础的数学模型大不相同。的确，经典力学描述了大规模的物理定律（例如，摇摆摆，行星轨道等的运动定律），并在数学上植根于差异几何形状，将计算扩展到允许曲率和更高维度的更通用空间。相比之下，量子力学以亚原子量表描述了物理定律，并植根于希尔伯特空间的语言。了解这两个模型之间的联系非常感兴趣。 直觉上，人们期望在亚原子水平上描述物理学的法律大规模确定的行为。从数学上讲，这种直觉是通过称为半古典限制的过程实现的，该过程从量子系统中产生经典系统。沿相反方向进行（即找到与经典的量子系统相对应的量子系统）被称为量化，尽管并非出于物理和数学原因总是可能的。 几何量化是对经典物理和量子物理学之间这种关系的数学理解。作为几何形状，它突出显示了每个框架中存在的对称性。该研究建议研究了量化几何方法可以适应某些类型的广义对称性（称为群体素的行为）的方法，这是对对称性的现代观点，即几何，拓扑和数学物理学的十字路口。 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转换，这些参数不会改变所描述的物理定律。与经典环境相反，不能总是构成群体素的对称性：并非总是有可能遵循另一个转换。 The particular groupoid symmetries investigated in this proposal, called quasi-symmetric 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 symmetric geometry, algebraic geometry, and conformal and quantum field theories. 尚不可用对群体固定动作的量化的完全理解。该研究建议中的目标通过（i）推进适应广义对称性的理论框架来填补这一空白； （ii）在一个这样一个框架中工作（准哈米尔顿行动）最终阐明了物理学的积极研究（例如，通过提供保形田间理论中的猜想公式）。