Scale and Conformal Invariance in Quantum Field Theory

现代理论物理学的两大支柱是量子力学和广义相对论。虽然它们还不能结合在一起形成一个一致的框架（弦理论是一个明显的例外），但量子力学和广义相对论的基础——狭义相对论的结合，导致了一个一致的框架。这个框架被称为量子场论，它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20世纪物理学中最重要的发现之一，它是量子场论框架不可避免的结果。重整化群是一个迭代过程，其中高动量自由度被整合出来，在所有理论的空间中产生流，这些理论决定了物理系统在不同距离尺度上所经历的变化。它在现代物理学，特别是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类系统地研究远距离物理（突现现象）。

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. The proposed research program deals with the implications of scale and conformal invariance in quantum field theory. Scale-invariant and conformal-invariant quantum field theories exhibit interesting renormalization group flows and have applications in several areas of research including string theory and second-order phase transitions in condensed matter physics. A better understanding of scale and conformal invariance could have far-reaching consequences, leading to a complete classification of all quantum field theories in terms of relevant deformations of conformal field theories. More specifically, the proposal investigates two important issues: the connections between scale invariance, conformal invariance and the c-theorem (a long-lasting conjecture which has not been proved in all generality as of now) in quantum field theory; and the existence of scale-invariant but not conformal-invariant non-relativistic field theories. Over the past years, some remarkable progress has been made on the implications of scale invariance and conformal invariance for quantum field theory. The research program proposes to extend these results to the more technically-challenging realm of strongly-coupled quantum field theory as well as the non-relativistic limit relevant to condensed matter physics. For the non-relativistic case, charting the novel behaviours encountered in such theories could lead to interesting developments in condensed matter physics.