低能连续统与格点有效场论
Low Energy Continuum and Lattice Effective Field Theories
摘要：
在论文的第一部分中，我们考虑了粒子通过严格有限程相互作用的因果性和么正性约束。我们推广维格纳的因果关系的情况下，非零分波混频。具体来说，我们分析的质子和中子之间的低能相互作用的系统。我们还分析了低能散射系统的任意短程相互作用加上一个有吸引力的$1/r^α$尾巴$α\geq2$。特别是，我们专注于$α=6$的情况下，我们推导出这些系统的因果关系和么正性的约束，并发现范德华长度尺度占主导地位的参数特征的短距离物理的相互作用。这种尺度的分离表明了一个单独的普适性类，用于表征具有吸引力的$1/r^{6}$尾部的相互作用。我们认为，一个类似的普适性类存在任何有吸引力的潜在$1/r^α$为$α\geq2$。在论文的第二部分中，我们用绝热投影方法对零程相互作用极限下的费米子-二聚体散射进行了格子蒙特卡罗计算。绝热投影方法利用一组初始团簇态和欧氏时间投影，对有限体积中的低散射团簇态给出了系统的可改进的描述。我们使用Lüscher的有限体积关系来确定$s$波、$p$波和$d$波的相移。为了比较，我们还计算精确的晶格结果使用Lanczos迭代和连续体的结果使用Skorniakov-Ter-Martirosian方程。对于我们的蒙特卡罗计算，我们使用一种新的晶格算法称为杂质晶格蒙特卡罗。该算法可以被看作是一种混合技术，它结合了世界线和虚拟场蒙特卡罗模拟的元素。
英文摘要：
In the first part of the thesis we consider the constraints of causality and unitarity for particles interacting via strictly finite-range interactions. We generalize Wigner's causality bound to the case of non-vanishing partial-wave mixing. Specifically we analyze the system of the low-energy interactions between protons and neutrons. We also analyze low-energy scattering for systems with arbitrary short-range interactions plus an attractive $1/r^α$ tail for $α\geq2$. In particular, we focus on the case of $α=6$ and we derive the constraints of causality and unitarity also for these systems and find that the van der Waals length scale dominates over parameters characterizing the short-distance physics of the interaction. This separation of scales suggests a separate universality class for physics characterizing interactions with an attractive $1/r^{6}$ tail. We argue that a similar universality class exists for any attractive potential $1/r^α$ for $α\geq2$. In the second part of the thesis we present lattice Monte Carlo calculations of fermion-dimer scattering in the limit of zero-range interactions using the adiabatic projection method. The adiabatic projection method uses a set of initial cluster states and Euclidean time projection to give a systematically improvable description of the low-lying scattering cluster states in a finite volume. We use Lüscher's finite-volume relations to determine the $s$-wave, $p$-wave, and $d$-wave phase shifts. For comparison, we also compute exact lattice results using Lanczos iteration and continuum results using the Skorniakov-Ter-Martirosian equation. For our Monte Carlo calculations we use a new lattice algorithm called impurity lattice Monte Carlo. This algorithm can be viewed as a hybrid technique which incorporates elements of both worldline and auxiliary-field Monte Carlo simulations.
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