Scale and Conformal Invariance in Quantum Field Theory

现代理论物理的两大支柱是量子力学和广义相对论。尽管它们还不能结合在一起形成一个一致的框架(一个明显的例外是弦理论)，但量子力学和狭义相对论的结合--广义相对论的基础--导致了一个一致的框架。这个被称为量子场论的框架产生了迄今为止发现的最精确的基本粒子理论。重整化群是二十世纪物理学中最重要的发现之一，它是量子场论框架下不可避免的结果。重整化群是一个迭代过程，其中高动量自由度被积分出来，它在所有理论的空间中产生流，该流决定了物理系统在不同距离尺度下所经历的变化。它在现代物理学，特别是在量子场论和凝聚态物理中具有极其重要的意义，因为它允许通过普适性类系统地研究远程物理(浮现现象)。

The two pillars of modern theoretical physics are quantum mechanics and general relativity. Although they cannot be combined together into a consistent framework yet (a notable exception being string theory), the union of quantum mechanics and special relativity, the basis for general relativity, leads to a consistent framework. This framework, called quantum field theory, gives rise to the most precise theory of elementary particles found thus far. One of the most important discoveries in physics of the twentieth century, the renormalization group, is an unavoidable consequence of the framework of quantum field theory. The renormalization group, which is an iterative process where high-momentum degrees of freedom are integrated out, generates a flow in the space of all theories which determines the changes a physical system undergoes as viewed at different distance scales. It is of paramount importance in modern physics, especially in quantum field theory and condensed matter physics, since it allows the systematic study of long-distance physics (emergent phenomena) through universality classes.