路径积分量子化与黎曼-辛流形

Symplectic quantization of multi-field Generalized Proca electrodynamics

We explicitly carry out the symplectic quantization of a family of multi-field Generalized Proca (GP) electrodynamics theories. In the process, we provide an independent derivation of the so-called secondary constraint enforcing relations -- consistency conditions that significantly restrict the allowed interactions in multi-field settings already at the classical level. Additionally, we unveil the existence of quantum consistency conditions, which apply in both single- and multi-field GP scenarios. Our newly found conditions imply that not all classically well-defined (multi-)GP theories are …

非紧辛流形的二维量子化的不存在性

Nonexistence of Finite-dimensional Quantizations of a Noncompact Symplectic Manifold

本文证明了在非紧辛流形M上不存在一个“观测值的基本代数”B的忠实的斜厄米矩阵有限维表示。因此，包含B的泊松代数C^ \infty（M）的任何李子代数都不存在有限维量子化。

Quantization of symplectic vector spaces over finite fields

In this paper, we construct a quantization functor, associating a complex vector space H(V) to a finite dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp(V). The main new technical result is a proof of a stronger form of the Stone-von Neumann property for the Heisenberg group. Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical vector space attached to a coadjoint orbit of a general unipotent group…

Quantum Polar Duality and the Symplectic Camel: a Geometric Approach to Quantization

We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our a…

电磁场中的辛区域、量子化和动力学

Symplectic areas, quantization, and dynamics in electromagnetic fields

在具有非均匀法拉第电磁张量的线性相空间中，发展了封闭积分形式的规范不变量子化。通过一个膜磁区得到了Groenewold乘积公式（对应于Weyl序）的一个类似形式，并推广到N个符号的乘积.量子化中的排序问题与膜的不同配置有关：配置的选择决定了固定排序的相位因子，并控制次级相空间上的辛广群结构。本文给出了电磁场中带电粒子量子演化问题的规范不变解的精确连续形式和通过动力膜面积的半经典近似。

辛量子化、不等价量子理论和海森堡测不准原理

Symplectic quantization, inequivalent quantum theories, and Heisenberg's principle of uncertainty

我们分析了海森堡图中非相对论性二维各向同性谐振子的量子动力学。这样的系统被作为玩具模型来分析一些各种各样的量子理论，可以建立从狄拉克的量子化规则的各种辛结构最近报道的这个经典系统的应用。指出这些量子理论是不等价的，因为与同一个物理经典可观测量相联系的算符（可观测量）的平均值彼此不一致。这种不等价性并不是因为算子排序的模糊性，而是因为在同一组坐标上定义了几个辛结构。它还表明，基本观测量之间的不确定性关系依赖于特定的量子理论选择。需要强调的是，这些（有些自相矛盾的）结果来自两种范式的结合：狄拉克量子化规则和通常的量子力学哥本哈根解释。

Berezin-Toeplitz quantization and the least unsharpness principle

We show that compatible almost-complex structures on symplectic manifolds correspond to optimal quantizations.

关于相空间量子化中“量子团”的优度

On the goodness of "quantum blobs" in phase space quantization

我们用“量子团”的概念取代了通常来自统计力学的量子胞的概念。一个量子团是一个相空间球的线性辛变换，其半径等于h-杆的平方根。量子团与任何辛平面的交点都是面积为h的一半的椭圆。这个非常特殊的性质，这是密切相关的原则，辛骆驼，导致一个辛不变陈述的不确定原理。此外，我们证明了相空间高斯在量子斑点的平均是最小不确定性高斯的维格纳变换。

辛Stiefel流形：易处理度量、二阶几何与牛顿法

Symplectic Stiefel manifold: tractable metrics, second-order geometry and Newton's methods

该研究针对辛约束优化问题，在辛Stiefel流形上构建易处理度量的黎曼优化几何要素，开发显式二阶几何与多种黎曼牛顿法，提出兼具全局与局部快速收敛性的混合算法，数值实验验证了方法有效性。

量化低秩量子态层析：双曲量化与黎曼最小二乘恢复

Quantized Low-Rank Quantum State Tomography: Hyperbolic Quantization and Riemannian Least-Squares Recovery

该研究针对有限比特泡利批量响应的低秩量子态层析问题，提出HyperQuant量化器与QuantRGD方法，建立相关恢复保证与比特- shots权衡，通过实验验证其有效性。

玻色子的第三量子化：辛对角化、非厄米哈密顿量与对称性

Third quantization for bosons: symplectic diagonalization, non-Hermitian Hamiltonian, and symmetries

本文提出一种基于单一辛变换的第三量子化方法，用于求解二次玻色子开放系统的刘维算符，并利用非厄米有效哈密顿量分析系统对称性及推导光子电流等可观测量的累积量生成函数。