玻色场论的欧几里德签名半经典方法：相互作用标量场
Euclidean signature semi-classical methods for bosonic field theories: interacting scalar fields
摘要：
优雅的“微局部”方法早已被广泛发展为传统的薛定谔本征值问题的分析。但是由于技术原因，这些方法迄今还不能应用于量子场论。然而，在这篇文章中，我们发起了一个'欧几里德签名半经典'计划，以扩大这些分析技术的范围，包括研究自相互作用的标量场在1 + 1，2 + 1和3 + 1维。基本的微局域方法首先需要解决一个单一的，非线性的Hamilton-Jacobi型方程的积分（为基态和激发态）的线性“传输”方程的序列沿着“流”产生的“基本解决方案"上述Hamilton-Jacobi方程。使用变分法，椭圆正则性理论和Banach空间版本的隐函数定理的直接方法相结合，我们建立，在一个合适的函数空间设置，存在性，唯一性和全局正则性，这需要的“基本解”的相关，欧几里得签名的Hamilton-Jacobi方程的系统研究。我们的方法适用于具有可重整化型多项式自相互作用的（大质量）标量场。正如我们将在其他地方说明的那样，它们也可以应用于2 + 1和3 + 1维的杨-米尔斯场。
英文摘要：
Elegant 'microlocal' methods have long since been extensively developed for the analysis of conventional Schroedinger eigenvalue problems. For technical reasons though these methods have not heretofore been applicable to quantum field theories. In this article however we initiate a 'Euclidean signature semi-classical' program to extend the scope of these analytical techniques to encompass the study of self-interacting scalar fields in 1 + 1, 2 + 1 and 3 + 1 dimensions. The basic microlocal approach entails, first of all, the solution of a single, nonlinear equation of Hamilton-Jacobi type followed by the integration (for both ground and excited states) of a sequence of linear 'transport' equations along the 'flow' generated by the 'fundamental solution' to the aforementioned Hamilton-Jacobi equation. Using a combination of the direct method of the calculus of variations, elliptic regularity theory and the Banach space version of the implicit function theorem we establish, in a suitable function space setting, the existence, uniqueness and global regularity of this needed 'fundamental solution' to the relevant, Euclidean signature Hamilton-Jacobi equation for the systems under study. Our methods are applicable to (massive) scalar fields with polynomial self-interactions of renormalizable type. They can, as we shall show elsewhere, also be applied to Yang-Mills fields in 2 + 1 and 3 + 1 dimensions.
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