The "Symplectic Camel Principle" and Semiclassical Mechanics

Symplectic Semiclassical Wave Packet Dynamics II: Non-Gaussian States

We generalize our earlier work on the symplectic/Hamiltonian formulation of the dynamics of the Gaussian wave packet to non-Gaussian semiclassical wave packets. We find the symplectic forms and asymptotic expansions of the Hamiltonians associated with these semiclassical wave packets, and obtain Hamiltonian systems governing their dynamics. Numerical experiments demonstrate that the dynamics give a very good approximation to the short-time dynamics of the expectation values computed by a method based on Egorov's Theorem or the Initial Value Representation.

A Surprising Property of Multidimensional Hamiltonian Systems; Application to Semiclassical Quantization of Phase Space

Symplectic topology has become a thriving area of research in mathematics and physics since Gromov's discovery in 1985 of a surprising property of canonical transformations (and hence of hamiltonian flows),"the principle of the symplectic camel". We exploit this property to show that the usually EBK quantization is equivalent to the physical assumption that the only allowewed semiclassical states are those with symplectic area nh+h/2. We introduce the terminology "quantum blobs" for these states. Quantum blobs are more general and powerful tools than the usual phase space cells of quantum stat…

Symplectic Techniques for Semiclassical Completely Integrable Systems

This article is a survey of classical and quantum completely integrable systems from the viewpoint of local ``phase space'' analysis. It advocates the use of normal forms and shows how to get global information from glueing local pieces. Many crucial phenomena such as monodromy or eigenvalue concentration are shown to arise from the presence of non-degenerate critical points.

Gutzwiller's Semiclassical Trace Formula and Maslov-Type Index Theory for Symplectic Paths

Gutzwiller's famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with an view towards the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley-Zhender index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley-Zehnder's index theo…

Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization

This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how to construct a generating function associated to the germ of any local symplectic groupoid and we provide an explicit (smooth, non-formal) universal formula $S_π$ for integrating any Poisson structure $π$ on a coordinate space. The second result involves the relation to semiclassical quantization. We show that the formal Taylor expansion of $S_{tπ}$ around $t=0$ yields an extract of Kontsevich's star product formula based o…

Symplectic Coarse-Grained Classical and Semi-Classical Evolution of Subsystems: New Theoretical Aspects

We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller's thawed Gaussian approximation introduced by Littlejohn. The key tool in our study is an extension of Gromov's "principle of the symplectic camel" obtained in collaboration with N. Dias and J. Prata. This extension says that the orthogonal projection of a symplectic phase space ball on a phase space with a smaller dimension also contains a symplectic ball with the same radius. In the quantum case, the radii of these symplectic balls are taken equal to s…

Symplectic Quantum Mechanics and Chern-Simons Gauge Theory II: Mapping Tori of Tori

We compute the semiclassical formulas for the partition functions obtained using two different Lagrangians: the Chern-Simons functional and the symplectic action functional.

Semiclassical almost isometry

Let M be a complex projective manifold, and L an Hermitian ample line bundle on it. A fundamental theorem of Gang Tian, reproved and strengthened by Zelditch, implies that the Khaeler form of L can be recovered from the asymptotics of the projective embeddings associated to large tensor powers of L. More precisely, with the natural choice of metrics the projective embeddings associated to the full linear series |kL| are asymptotically symplectic, in the appropriate rescaled sense. In this article, we ask whether and how this result extends to the semiclassical setting. Specifically, we relate …

The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics

Recently Ohsawa has studied the Marsden-Weinstein-Meyer quotient of the manifold $T^*\mathbb{R}^n\times T^*\mathbb{R}^{2n^2}$ under a $\operatorname{O}(2n)$-symmetry, and has used this quotient to describe the relationship between two different parametrisations of Gaussian wave packet dynamics commonly used in semiclassical mechanics. In this paper we suggest a new interpretation of (a subset of) the unreduced space as being the frame bundle $\mathcal{F}(T^*\mathbb{R}^n)$ of $T^*\mathbb{R}^n$. We outline some advantages of this interpretation, and explain how it can be extended to more general…

Symplectic inverse spectral theory for pseudodifferential operators

We prove, under some generic assumptions, that the semiclassical spectrum modulo O(h^2) of a one dimensional pseudodifferential operator completely determines the symplectic geometry of the underlying classical system. In particular, the spectrum determines the hamiltonian dynamics of the principal symbol.

辛群体上的结合半密度与量子化

Associative half-densities on symplectic groupoids and quantization

本文研究了满足适当结合条件的半密度，用于增强辛群体上的乘法映射，以实现对庞加莱-诺维科夫结构的完全半经典-解析近似。文章展示了此类结合半密度的存在性和分类，并将其应用于理解康特赛维奇量子化公式中的半经典因子。在线性泊松结构的情况下，恢复了杜福同态及其卡西瓦拉-韦尔内扩展中出现的因子作为规范的结合增强。

Symplectic Quantum Mechanics and Chern-Simons Gauge Theory I

In this article we describe the relation between the Chern-Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional.

Liouville 域的参数化 Gromov 宽度

Parametric Gromov width of Liouville domains

受辛骆驼问题启发，将 Gromov 宽度推广至参数空间上的辛球族嵌入，利用 Floer 上同调持续模和过滤弦拓扑为 Liouville 域建立上界，揭示超越经典情形的参数化辛嵌入约束。

辛切割与开/闭弦 I

Symplectic cuts and open/closed strings I

本文借助等变 gauged linear sigma model 与辛切割构造量子 Lebesgue 测度，统一刻画 CY3 的闭弦及 Lagrangian A-brane 的开弦零亏格 GW 不变量。