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。这个建议的主要目的是研究伽利莱对称性的物理应用，它包含一些令人惊讶和复杂的特征。*建议的研究计划主要涉及伽利略不变性的应用，以便于处理非相对论系统。我的目标是描述具有伽利略对称性的凝聚态物理(自旋系统、超流性、超导电性)和核物理中的低能量现象的新模型。我和我的合作者已经利用伽利略协方差研究了场量子化、阿贝尔规范理论、任意自旋场、自旋系统、自旋统计联系等。我计划在正式和实际两个方向上扩展我以前的工作。形式方面包括在对易和非对易相空间中求解具有各种物理意义的势的Bhabha波动方程(Dirac，Duffin-Kemmer-Petiau，Lévy-Leblond)，以及考虑具有孤子解的伽利略非线性方程。实际目标包括继续我们之前在磁化衰减方面的工作，研究自旋系统，以及在自旋转移和自旋电子学方面的潜在应用。

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.