Kinematical Symmetries in Field Theory

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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.