标题： 规范构型空间上的度量 显示英文标题

标题： The Metric on the Space of Yang-Mills Configurations

摘要： 研究了费曼以及阿蒂亚、希钦和辛格所考虑的杨-米尔斯构型物理等价类集合上的距离函数，针对 $2+1$ 和 $3+1$- 维哈密顿量进行了探讨。 赋予此距离函数后，该集合成为一个度量空间，并且实际上如辛格观察到的那样，是一个黎曼流形。 此外，这个流形是完备的。 规范构型可以用来参数化这个流形。 未进行规范固定的度量张量具有零特征值，但在整个流形上没有歧义。 在 $2+1$ 维时，找到任意构型到纯规范构型的距离的问题是一个可积的二维微分方程系统。 发展了一种带有奇异度量张量的流形微积分，并利用这种微积分计算了黎曼曲率。 杨-米尔斯波函数算子上的拉普拉斯算子形式与之前声称的不同。 在 $3+1$- 维时，存在与纯规范构型有任意大距离但势能任意小的场构型。 这些构型类似于长波长胶子。 提出了尽管如此量子理论中仍可能存在质量间隙的原因。 显示英文摘要

摘要： A distance function on the set of physical equivalence classes of Yang-Mills configurations considered by Feynman and by Atiyah, Hitchin and Singer is studied for both the $2+1$ and $3+1$-dimensional Hamiltonians. This set equipped with this distance function is a metric space, and in fact a Riemannian manifold as Singer observed. Furthermore, this manifold is complete. Gauge configurations can be used to parametrize the manifold. The metric tensor without gauge fixing has zero eigenvalues, but is free of ambiguities on the entire manifold. In $2+1$ dimensions the problem of finding the distance from any configuration to a pure gauge configuration is an integrable system of two-dimensional differential equations. A calculus of manifolds with singular metric tensors is developed and the Riemann curvature is calculated using this calculus. The Laplacian on Yang-Mills wave functionals has a slightly different form from that claimed earlier. In $3+1$-dimensions there are field configurations an arbitrarily large distance from a pure gauge configuration with arbitrarily small potential energy. These configurations resemble long-wavelength gluons. Reasons why there nevertheless can be a mass gap in the quantum theory are proposed.