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arXiv 0912.0450hep-latcond-mat.stat-mechhep-th
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格点Landau规范与代数几何

Lattice Landau Gauge and Algebraic Geometry

Dhagash Mehta, Andre Sternbeck, Lorenz von Smekal, Anthony G Williams

备注：
Talk given at International Workshop on QCD Green's Functions, Confinement, and Phenomenology - QCD-TNT09, Trento, Italy, September 07 - 11 2009
发表信息：
PoS QCD-TNT09:025,2009
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摘要：

在理论物理和化学的许多分支中，有效地求多元函数的全局极小值是一个基本而又困难的问题。然而，我们观察到，有许多物理系统的极值方程具有多项式样的非线性。这允许使用代数几何技术来完全解决这些方程。然后，可以通过二阶导数测试直接找到全局最小值。作为一个暖场例子，我们研究了紧致U（1）的格点Landau规范，并提出了两种求解相应规范固定方程的方法。在第一步中，我们得到所有的Gribov副本的一维和二维格。对于简单的3x3系统，它们的数量已经可以达到数千的数量级。我们预计，所采用的计算和数值代数几何方法具有深远的影响，超出了简单的，但说明这里讨论的例子。

英文摘要：

Finding the global minimum of a multivariate function efficiently is a fundamental yet difficult problem in many branches of theoretical physics and chemistry. However, we observe that there are many physical systems for which the extremizing equations have polynomial-like non-linearity. This allows the use of Algebraic Geometry techniques to solve these equations completely. The global minimum can then straightforwardly be found by the second derivative test. As a warm-up example, here we study lattice Landau gauge for compact U(1) and propose two methods to solve the corresponding gauge-fixing equations. In a first step, we obtain all Gribov copies on one and two dimensional lattices. For simple 3x3 systems their number can already be of the order of thousands. We anticipate that the computational and numerical algebraic geometry methods employed have far-reaching implications beyond the simple but illustrating examples discussed here.

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