Quantum perturbative solutions of extended Snyder and Yang models with spontaneous symmetry breaking

We propose h¯ -expansions as perturbative solutions of quantum extended Snyder and Yang models, with h¯ -independent classical zero-th order terms responsible for the spontaneous breaking of D = 4 and D = 5 de Sitter symmetries. In such models, with algebraic basis spanned by oˆ(D, 1) Lie algebra generators, we relate the vacuum expectation values (VEV) of the spontaneously broken generators with the Abelian set of ten (Snyder, D = 4) or fifteen (Yang, D = 5) antisymmetric tensorial generalized coordinates, which are also used as zero order input for obtaining the perturbative solutions of quantum extended Snyder and Yang models. In such a way we will attribute to these Abelian generalized coordinates the physical meaning of the order parameters describing spontaneous symmetry breaking (SSB). It appears that the consecutive terms in h¯ -power series can be calculated explicitly if we supplement the SSB order parameters by the dual set of tensorial commutative momenta.