标题： 连接空间的相干态变换 显示英文标题

标题： Coherent State Transforms for Spaces of Connections

摘要： 塞加尔-巴格曼变换在线性场的量子理论中起着重要作用。最近，霍尔得到了一种非线性变换，这是对李群上量子力学的塞加尔-巴格曼变换的类比。 给定一个紧致连通李群$G$，其具有归一化的哈尔测度$\mu_H$，霍尔变换是从$L^2(G, \mu_H)$到${\cal H}(G^{\Co})\cap L^2(G^{\Co}, \nu)$的等距同构，其中$G^{\Co}$是$G$的复化，${\cal H}(G^{\Co})$是$G^{\Co}$上的全纯函数空间，$\nu$是$G^{\Co}$上的一个适当的热核测度。 我们将霍尔变换扩展到非阿贝尔规范理论的无限维背景，方法是将李群$G$替换为（某种扩展的）模规范变换的连接空间${\cal A}/{\cal G}$。 由此产生的“相干态变换”提供了实规范场的环路代数$C^\star$的全纯表示。 这种表示预期将在四维时空的非微扰、正则量子引力方法中起到关键作用。 显示英文摘要

摘要： The Segal-Bargmann transform plays an important role in quantum theories of linear fields. Recently, Hall obtained a non-linear analog of this transform for quantum mechanics on Lie groups. Given a compact, connected Lie group $G$ with its normalized Haar measure $\mu_H$, the Hall transform is an isometric isomorphism from $L^2(G, \mu_H)$ to ${\cal H}(G^{\Co})\cap L^2(G^{\Co}, \nu)$, where $G^{\Co}$ the complexification of $G$, ${\cal H}(G^{\Co})$ the space of holomorphic functions on $G^{\Co}$, and $\nu$ an appropriate heat-kernel measure on $G^{\Co}$. We extend the Hall transform to the infinite dimensional context of non-Abelian gauge theories by replacing the Lie group $G$ by (a certain extension of) the space ${\cal A}/{\cal G}$ of connections modulo gauge transformations. The resulting ``coherent state transform'' provides a holomorphic representation of the holonomy $C^\star$ algebra of real gauge fields. This representation is expected to play a key role in a non-perturbative, canonical approach to quantum gravity in 4-dimensions.