Kinematical Symmetries in Field Theory

拟议的研究计划涉及应用于各种物理系统的对称性方法，而不是研究特定类别的物理系统。我的研究兴趣集中在自然界的运动对称性，它们在场论中的应用，以及这些对称性之间的联系。对称是对一个物理系统或描述这样一个系统的方程进行的变换，使得这个系统的某些特征保持不变。对称性可能有助于表述或求解描述物理系统的方程，或更好地理解此类系统的性质。运动对称性，如Poincaré代数(构成相对论理论的基础)、伽利莱代数(低能物理学的内在)、de Sitter或牛顿-胡克代数(两者都是宇宙学中感兴趣的)，决定了物理理论的基本结构，这些结构通常表现出额外的动力学对称性。这些对称性可以通过被称为李代数的“压缩”和“变形”的数学过程相互转换。例如，伽利莱代数是Poincaré代数在低速和大时间间隔范围内的压缩。从对称性的角度看，爱因斯坦狭义相对论的基础在于用庞加莱代数代替伽利莱代数。因此，从那一刻起，伽利莱对称比庞加莱对称受到的关注要少得多，庞加莱对称是自然的一种更基本的对称。这个提议的主要目的是研究伽利莱对称性的物理应用，它包含了一些令人惊讶和复杂的特征。

The proposed research program deals with symmetry methods applied to various physical systems, rather than studying a specific class of physical systems. My research interests focus on the kinematical symmetries of nature, their applications in field theory, and the connections between these symmetries. Symmetries are transformations performed on a physical system, or the equations which describe such a system, such that some feature of this system is preserved or unchanged. Symmetries may help formulating or solving the equations which describe a physical system, or better understanding the nature of such systems. Kinematical symmetries, such as the Poincaré algebra (which underlies the relativistic theories), Galilei algebra (intrinsic to low-energy physics), the de Sitter or Newton-Hooke algebras (both of interest in cosmology), determine the basic structures of physical theories, which often exhibit additional dynamical symmetries. These symmetries can be transformed into one another through mathematical procedures called "contractions" and "deformations" of Lie algebras. For instance, the Galilei algebra is a contraction of the Poincaré algebra in the limit of low-velocity and large time-like intervals. From the symmetries viewpoint, the foundation of Einstein's special relativity rests on substituting the Galilei algebra with the Poincaré algebra. Thus, from that moment on, the Galilei symmetry has received much less attention than the Poincaré symmetry, which is a more fundamental symmetry of nature. This proposal's main objective is to study physical applications of the Galilei symmetry, which contains some surprising and intricate features.