Noncommutative spacetime realized in Ad S n 1 space: Nonlocal field theory out of noncommutative spacetime

In $\kappa $ -Minkowski spacetime, the coordinates are Lie algebraic elements such that time and space coordinates do not commute, whereas space coordinates commute with each other. The noncommutativity is proportional to a Planck-length-scale constant $\kappa ^{-1}$ , which is a universal constant other than the velocity of light, under the $\kappa $ -Poincar transformation. In this sense, the spacetime has a structure called doubly special relativity. Such a noncommutative structure is known to be realized by $SO(1,4)$ generators in 4-dimensional de Sitter space. In this paper, we try to construct a noncommutative spacetime having a commutative $n$ -dimensional Minkowski spacetime based on $AdS_{n+1}$ space with $SO(2,n)$ symmetry. We also study an invariant wave equation corresponding to the first Casimir invariant of this symmetry as a nonlocal field equation expected to yield finite loop amplitudes