Narrow Results

We find a class of Hermitian generalized Jordan triple systems (HGJTSs) and Hermitian (ϵ, δ)-Freudenthal–Kantor triple systems (HFKTSs). We apply one of the most simple HGJTSs which we find to a field theory and obtain a typical u(N) Chern–Simons gauge theory with a fundamental matter.

We define Hermitian generalized Jordan triple systems and prove a structure theorem. We also give some examples of the systems and study mathematical properties. We apply a Hermitian generalized Jordan triple system to a field theory and obtain a Chern–Simons gauge theory.

Following to the Weil method, we generalize the Heisenberg–Robertson uncertainty relation for arbitrary two operators. Consideration is made in spherical coordinates, where the distant variable $r$ is restricted from one side, $0\le r<\infty$. By this reason, accounting of suitable boundary condition at the origin for radial wavefunctions and operators is necessary. Therefore, there arise extra surface terms in comparison with traditional approaches. These extra terms are calculated for various solvable potentials and their influence is investigated. At last, the time–energy uncertainty relations are also analyzed. Some differences between our approach and that, in which a direct product for separate variances were considered, are discussed.