Kinematical Symmetries in Field Theory

提出的研究计划涉及应用于各种物理系统的对称方法，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该系统的方程进行的变换，使该系统的某些特征得以保留或不变。对称性可以帮助表述或解决描述物理系统的方程，或者更好地理解这些系统的本质。运动对称，如庞加莱代数（它是相对论理论的基础）、伽利莱代数（低能物理的内在特征）、德西特代数或牛顿-胡克代数（两者都对宇宙学感兴趣），决定了物理理论的基本结构，这些理论往往表现出额外的动态对称性。这些对称性可以通过李代数的“收缩”和“变形”的数学过程相互转换。例如，伽利莱代数是在低速和大类时区间的极限下对庞加莱代数的压缩。从对称的观点来看，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，伽利略对称比庞加莱对称受到的关注要少得多，庞加莱对称是自然界更基本的对称。这个提议的主要目的是研究伽利略对称的物理应用，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. The proposed research program deals mainly with applications of Galilean invariance in order to facilitate the treatment of non-relativistic systems. My aim is to describe new models of low-energy phenomena in condensed matter physics (spin systems, superfluidity, superconductivity) and nuclear physics with Galilean symmetry. My collaborators and I have already utilized Galilean covariance to study field quantization, abelian gauge theories, arbitrary spin fields, spin systems, the spin-statistics connection, and others. I plan to expand my previous work both in formal and practical directions. Formal aspects include solving the Bhabha wave equations (Dirac, Duffin-Kemmer-Petiau, Lévy-Leblond) with various potentials of physical interest, in both commutative and non-commutative phase spaces, as well as considering Galilean non-linear equations with soliton solutions. Practical objectives comprise the continuation of our previous work on magnetization damping, the study of spin systems, with potential applications to spin transfer and spintronics.