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摘要翻译：
分析了由Efetov引起的拉格朗日坐标系的鞍点。这个拉格朗日函数最初是作为计算对Bethe近似的系统修正的工具提出的，Bethe近似是一种在统计力学、眼镜、编码理论和组合优化中很重要的平均场近似。详细分析表明，平凡鞍点产生的几何和使人联想到动态三角化量子引力，这为设计几何和提供了新的可能性，以获得改进的d$维理论的平均场近似。在Efetov理论的情况下，主要几何是局部树状的，几何上的和在包含所有拓扑时以类似于量子引力发散的方式发散。来自动态三角化量子引力领域关于几何和的专门知识可能能够弥补这些缺陷，并实现Efetov理论最初的承诺。分析了Efetov-Lagrangian的其他鞍点；在这些点上的黑森是非正常的，是伪厄米特的，这在波速理论中是不寻常的。将高斯积分的标准公式推广到非正规核。
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英文标题：
《Sums over geometries and improvements on the mean field approximation》
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作者：
Vincent E. Sacksteder IV
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最新提交年份：
2007
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分类信息：
一级分类：Physics 物理学
二级分类：High Energy Physics - Theory 高能物理-理论
分类描述：Formal aspects of quantum field theory. String theory, supersymmetry and supergravity.
量子场论的形式方面。弦理论，超对称性和超引力。
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一级分类：Physics 物理学
二级分类：Disordered Systems and Neural Networks 无序系统与神经网络
分类描述：Glasses and spin glasses; properties of random, aperiodic and quasiperiodic systems; transport in disordered media; localization; phenomena mediated by defects and disorder; neural networks
眼镜和旋转眼镜；随机、非周期和准周期系统的性质；无序介质中的传输；本地化；由缺陷和无序介导的现象；神经网络
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一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
The saddle points of a Lagrangian due to Efetov are analyzed. This Lagrangian was originally proposed as a tool for calculating systematic corrections to the Bethe approximation, a mean-field approximation which is important in statistical mechanics, glasses, coding theory, and combinatorial optimization. Detailed analysis shows that the trivial saddle point generates a sum over geometries reminiscent of dynamically triangulated quantum gravity, which suggests new possibilities to design sums over geometries for the specific purpose of obtaining improved mean field approximations to $D$-dimensional theories. In the case of the Efetov theory, the dominant geometries are locally tree-like, and the sum over geometries diverges in a way that is similar to quantum gravity's divergence when all topologies are included. Expertise from the field of dynamically triangulated quantum gravity about sums over geometries may be able to remedy these defects and fulfill the Efetov theory's original promise. The other saddle points of the Efetov Lagrangian are also analyzed; the Hessian at these points is nonnormal and pseudo-Hermitian, which is unusual for bosonic theories. The standard formula for Gaussian integrals is generalized to nonnormal kernels.
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PDF链接：
https://arxiv.org/pdf/704.3129
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