Noncommutative spacetime realized in AdS_n+1 space: Nonlocal field theory out of noncommutative spacetime

In κ-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 κ^-1, which is a universal constant other than the velocity of light, under the ?κ-PoincarÉ 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.