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，而不是研究一类特定的物理系统。我的研究兴趣集中在自然界的运动学对称性，它们在场论中的应用，以及这些对称性之间的联系。对称性是对物理系统或描述这种系统的方程进行的变换，使得该系统的某些特征保持不变。对称性可以帮助公式化或求解描述物理系统的方程，或者更好地理解这种系统的性质。运动学对称性，如庞加莱代数（相对论理论的基础）、伽利略代数（低能物理学的内在）、德西特代数或牛顿-胡克代数（两者都是宇宙学的兴趣），决定了物理理论的基本结构，这些理论通常表现出额外的动力学对称性。这些对称性可以通过称为李代数的“压缩”和“变形”的数学过程相互转化。例如，伽利略代数是庞加莱代数在低速和大类时区间极限下的收缩。从对称性的观点看，爱因斯坦狭义相对论的基础在于用庞加莱代数代替伽利略代数。因此，从那一刻起，伽利略对称性受到的关注远少于庞加莱对称性，庞加莱对称性是自然界更基本的对称性。这项计划的主要目的是研究伽利略对称性的物理应用，其中包含一些令人惊讶和复杂的特征。 拟议的研究计划主要涉及伽利略不变性的应用，以促进非相对论系统的治疗。我的目标是描述凝聚态物理（自旋系统，超流性，超导性）和伽利略对称性的核物理中的低能现象的新模型。我和我的合作者已经利用伽利略协方差来研究场量子化、阿贝尔规范理论、任意自旋场、自旋系统、自旋统计联系等。我计划在形式和实践两个方面扩展我以前的工作。正式的方面包括解决巴巴波动方程（狄拉克，达芬-凯默-佩蒂奥，列维-Leblond）与各种潜在的物理利益，在交换和非交换相空间，以及考虑伽利略非线性方程孤子解决方案。实际目标包括继续我们以前的工作磁化阻尼，自旋系统的研究，自旋转移和自旋电子学的潜在应用。

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.