Title: Poincaré, Scale and Conformal Symmetries: Gauge Perspective and Cosmological Ramifications Show Chinese title

Title: 庞加莱、尺度和共形对称性：规范观点与宇宙学影响

Abstract: In the first part of the thesis we focus on local symmetries. We review a self-consistent framework that we employed in order to discuss the dynamics of the theories of interest. Its merit lies in that we can make the symmetry group act internally and thus be effectively separated from coordinate transformations. We investigate under which conditions it is not needed to introduce extra compensating fields to make relativistic as well as nonrelativistic theories invariant under local symmetries and more precisely under scale transformations. We clarify the role that torsion plays in this context. We highlight the difference between Weyl and conformal invariance and we demonstrate that not all conformal theories can be coupled to gravity in a Weyl invariant way. Once this minimalistic treatment for gauging symmetries is left aside, new possibilities appear. Namely, if we consider the Poincar\'e group, the presence of the extra modes leads to nontrivial particle dynamics. We derive constraints such that the theory is free from pathologies. In the second part we make clear that even when not gauged, the presence of scale invariance is appealing. First, it makes possible for the dimensionful parameters that appear in a theory to be generated dynamically and be sourced by the vacuum expectation value of the dilaton. If the Standard Model is embedded into a scale-invariant framework, a number of interesting implications for cosmology arise. The inflationary stage of our Universe and its present-day acceleration become linked, a connection that might give us insight into the dark energy dynamics. We show that in the context of gravitational theories which are invariant under restricted coordinate transformations, the dilaton instead of being introduced ad hoc, can emerge from the gravitational part of a theory. We discuss the consequences of the nontrivial way this field emerges in the action. Show Chinese abstract

Abstract: 在论文的第一部分中，我们关注局部对称性。我们回顾了一个自洽的框架，我们使用该框架来讨论感兴趣理论的动力学。其优点在于我们可以使对称群内部作用，从而有效地与坐标变换分离。我们研究在什么条件下不需要引入额外的补偿场，以使相对论和非相对论理论在局部对称性下，更精确地说在标度变换下保持不变。我们阐明了在这一背景下挠率所起的作用。我们强调了Weyl不变性和共形不变性的区别，并证明并非所有共形理论都可以以Weyl不变的方式与引力耦合。一旦摒弃这种极简的对称性规范处理方式，就会出现新的可能性。即，如果我们考虑庞加莱群，额外模式的存在会导致非平凡的粒子动力学。我们推导出约束条件，使得理论免受病理现象的影响。在第二部分中，我们明确指出，即使没有规范，尺度不变性的存在也是有吸引力的。首先，它使得理论中出现的具有维度的参数可以动态生成，并由标量场的真空期望值作为来源。如果将标准模型嵌入到一个尺度不变的框架中，会引出一些有趣的宇宙学后果。我们宇宙的暴胀阶段及其当前的加速膨胀变得相关联，这种联系可能为我们提供关于暗能量动力学的见解。我们表明，在遵循受限坐标变换不变性的引力理论背景下，标量场而不是被人为引入，而是可以从理论的引力部分中产生。我们讨论了该场以非平凡方式在作用量中出现所带来的后果。