Spherical waves of spin 1 particle in anti de Sitter space-time

Abstract

Three possible techniques to deal with a vector particle in the anti de Sitter cosmological model are viewed: Duffin -Kemmer -Petiau matrix formalism based on the general tetrad recipe, group theory 5-dimensional approach based on the symmetry group SO(3.2), and a tetrad form of Maxwell equations in complex Riemann -Silberstein -Majorana -Oppenheimer representation. In the first part, a spin 1 massive field is considered in static coordinates of the anti de Sitter space-time in tetrad-based approach. The complete set of spherical wave solutions with quantum numbers (ǫ, j, m, l) is constructed; angular dependence in wave functions is described with the help of Wigner functions. The energy quantization rule has been found. Transition to massless case of electromagnetic field is specified, and electromagnetic solutions in Lorentz gauge have been constructed. In the second part, the problem of a particle with spin 1 is considered on the base of 5-dimensional wave equation specified in the same static coordinates. In the third part, a rarely used approach, based on tetrad form of Maxwell equations in complex representation is examined in the anti de Sitter model.

Normal modes of Proca fields in AdS spacetime

“…The solutions of Eqs. ( 9)- (11), i.e., of Dji q (i) = 0, i = 1, 2, 3, can be expressed analytically, as we shall see. For that we assume that the time dependence of the q (i) is e −iωt , where ω is the frequency.…”

“…The expression of the normal mode frequencies of the Proca field in AdS given in Eq. ( 17) differs from the expression given in [11] in the case of j 1 = 0, where their Eq. ( 24) has an additional term +1 compared with our Eq.…”

“…( 9) and ( 10), thus q (12) (r) will obey the same equation for q (3) with µ = 0, see Eq. (11). The equation for q (3) will be Eq.…”

“…For i = 1, 2, 3 we can write concisely Eqs. ( 9)- (11) as Dji q (i) = 0, where again j 1 = ℓ − 1, j 2 = ℓ + 1, j 3 = ℓ, and D ji is defined in Eq. (7).…”

“…In the AdS spacetime, one finds that when it is linearly perturbed with reflective boundary conditions, also called box boundary conditions, it oscillates normally, rather than quasinormally, as there are no losses of the field to infinity. For the AdS spacetime, the normal modes of scalar [8], electromagnetic and gravitational perturbations [9], and massive Proca perturbations [10,11] have been studied.…”