一维$O(N)$模型的启示
Lessons from $O(N)$ models in one dimension
摘要：
在一个时空维度（即普通的量子力学）的$O(N)$模型相关的各种主题被考虑。重点是教学介绍量子场论方法在一个简单的背景下，许多确切的结果是可用的，但某些微妙之处进行了讨论，这可能是感兴趣的积极研究人员在高维场理论以及。大$N$方法介绍的背景下，零维路径积分和斯特林系列的连接。整个谱的$O(N)$模型，其中包括熟悉的$l(l+1)$本征值的量子转子作为一个特殊的情况下，发现都图解通过大$N$方法和使用沃德身份。大的$N$方法已经准确的subleading阶和$\mathcal{O}\!\left(N^{-2}\right)$修正明确示出为消失。在$CP^{N-1}$ sigma模型的背景下讨论了$d=1$中规范理论的特殊性，找到了更一般的压缩球sigma模型的谱，并讨论了$O(N)$模型与具有$ϕ^4$相互作用的线性sigma模型之间的精确联系.给出了一种适用于硬约束$O(N)$模型的自洽屏蔽近似（SCSA）的有效形式。指出至少在$d=1$中，即使对于小的$N$，SCSA也可能比简单地将大的$N$扩展截断为子前导阶做得更差。在这两个超对称和非超对称版本的$O(N)$模型，天真的运动方程有关的真空期望值被证明是由正则化相关的有限的修正所产生的联系条款相关联的约束方程进行校正。
英文摘要：
Various topics related to the $O(N)$ model in one spacetime dimension (i.e. ordinary quantum mechanics) are considered. The focus is on a pedagogical presentation of quantum field theory methods in a simpler context where many exact results are available, but certain subtleties are discussed which may be of interest to active researchers in higher dimensional field theories as well. Large $N$ methods are introduced in the context of the zero-dimensional path integral and the connection to Stirling's series is shown. The entire spectrum of the $O(N)$ model, which includes the familiar $l(l+1)$ eigenvalues of the quantum rotor as a special case, is found both diagrammatically through large $N$ methods and by using Ward identities. The large $N$ methods are already exact at subleading order and the $\mathcal{O}\!\left(N^{-2}\right)$ corrections are explicitly shown to vanish. Peculiarities of gauge theories in $d=1$ are discussed in the context of the $CP^{N-1}$ sigma model, and the spectrum of a more general squashed sphere sigma model is found. The precise connection between the $O(N)$ model and the linear sigma model with a $ϕ^4$ interaction is discussed. A valid form of the self-consistent screening approximation (SCSA) applicable to $O(N)$ models with a hard constraint is presented. The point is made that at least in $d=1$ the SCSA may do worse than simply truncating the large $N$ expansion to subleading order even for small $N$. In both the supersymmetric and non-supersymmetric versions of the $O(N)$ model, naive equations of motion relating vacuum expectation values are shown to be corrected by regularization-dependent finite corrections arising from contact terms associated to the equation of constraint.
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