仿紧流形上的泛函积分
Functional Integration on Paracompact Manifolds
摘要：
1948年，费曼提出了函数积分。很久以前的问题方面的措施，在空间的领域是克服了引进体积元素的概率空间，导致随机配方。最近卡地亚和德维特-莫雷特专注于定义一个适当的积分措施，并建立了严格的数学公式的功能集成。他们的中心观察涉及到领域的分布性质，因为它导致了识别的分布泛函与施瓦茨空间测试功能的密度措施。这只是作者在不久前提出的泰勒-拉格朗日格式的数学内容。在这个计划领域是生活在度量施瓦茨-索伯列夫空间，受开放覆盖与下属的划分统一的测试功能。实际上，这些PU，通过卷积运算，导致在扩展的施瓦茨空间上的平滑场函数。这样，微分几何的基本假设--场存在于可微流形上--就得到了验证。接下来，它表明，卷积理论的分布导致一个健全的定义拉普拉斯-Stieltjes变换，源于存在的等距不变的Hausdorff措施在空间的领域。转向规范理论的建设顺利向量场的弯曲流形上建立，所需的可微向量纤维丛。适当选择一个连接进一步分离的功能Hausdorff积分措施有限高斯积分的规范参数的因素，并没有发挥任何物理作用。对于非阿贝尔杨-米尔斯规范理论，Vilkowisky-DeWitt联络和Landau-DeWitt协变背景规范的性质导致非常简单的计算规则。
英文摘要：
In 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More recently Cartier and DeWitt-Morette focused on the definition of a proper integration measure and established a rigorous mathematical formulation of functional integration. Their central observation relates to the distributional nature of fields, for it leads to the identification of distribution functionals with Schwartz space test functions as density measures. This is just the mathematical content of the Taylor-Lagrange Scheme developed by the authors in a recent past. In this scheme fields are living in metric Schwartz-Sobolev spaces, subject to open coverings with subordinate partition-of-unity test functions. In effect these PU, through the convolution operation, lead to smooth field functions on an extended Schwartz space. In this way the basic assumption of differential geometry -- that fields live on differentiable manifolds -- is validated. Next it is shown that convolution in the theory of distributions leads to a sound definition of Laplace-Stieltjes transforms , stemming from the existence of an isometry invariant Hausdorff measure in the space of fields. Turning to gauge theories the construction of smooth vector fields on a curved manifold is established, as required for differentiable vector fibre bundles. The proper choice of a connection further separates in the functional Hausdorff integration measure a finite Gaussian integration over the gauge parameter which factors out and plays no physical role. For non-Abelian Yang-Mills gauge theories the properties of Vilkowisky-DeWitt's connection and of Landau-DeWitt's covariant back-ground gauge result in very simple calculation rules.
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