Looking at Quantum Phenomena in terms of "Classical Glasses" -Analysis of Quantum (Field) Theory by means of Path Integration-

在前两年，我们研究了具有非零背景经典分量的非阿贝尔规范场耦合的Nambu-Lona-Lasinio（NJL）模型。引入辅助场和积分费米场，分析了非阿贝尔规范场中包含量子效应的有效作用。这些计算只是双环阶的计算，所以非常麻烦。我们得到：(1)规范场的量子效应增强了手性对称性破缺（χsb）；(2)背景磁场对动力学质量的依赖性在4维单调增加，在3维单调增加，在χsb较小的区域单调增加，在χsb较大的区域单调增加。在第二年和最后一年，我们对辅助场在4费米系统中的作用进行了更深入的研究：将辅助场的量子效应包含在NJL的间隙方程中，发现χsb得到了增强，证明了辅助场方法的有效性。我们还检查了高阶辅助场总是改善0维4-费米系统（Grassmann积分模型）的结果，在这种情况下，我们遇到了作用的二阶导数在鞍点（即焦散点）消失的情况。

In the first two years, we have investigated the Nambu-Lona-Lasinio(NJL) model coupled to non-abelian gauge fields with non-zero background classical components. Introducing auxiliary fields and integrating fermi fields, we analize the effective action with being included quantum effects of non-abelian gauge fields. The calculations are nothing but those of two-loop orders so are very cumbersome. We have obtained results that (1) quantum effects of gauge fields enhance chiral symmetry breaking(χsb) , (2) dependence of the background magnetic fields to the dynamical mass is monotonically increasing in 4-dimension while in 3-dimension it is monotonically increasing in a region where χsb is small but decreasing where χsb is large. In the second and final years, we proceed to a more thorough investigation for the role of auxiliary fields in 4-fermi systems : inclusion of quantum effects of auxiliary fields to the gap equation in NJL to find that χsb is enhanced, justifying the auxiliary fields method. We also check that the higher order of auxiliary fields always improves results in 0-dimensional 4-fermi system (the Grassmann integration model) in which we encounter the case that the second derivative of the action vanishes at the saddle point, that is, of caustic.