$\hbar$ -perturbative solutions of quantum Snyder and Yang models with parameters describing spontaneous symmetry breaking

Abstract

We introduce the perturbative ℏ-power series (ℏ -Planck constant) providing the algebraic solutions of 𝐷 = 4 quantum Snyder and Yang models which describe relativistic quantum space-times and Lorentz-covariant quantum phase spaces. We argue that if in these series the zero order (ℏindependent) terms are non-vanishing they describe the spontaneous symmetry breaking (SSB) parameters of Lie-algebraic symmetries which characterize the considered models (𝐷 = 4 dS symmetry in Snyder and 𝐷 = 5 dS symmetry in Yang cases). The consecutive terms in ℏ-power series can be calculated explicitly if we supplement the SSB order parameters (Nambu-Goldstone or NG modes) by dual set of commutative momenta, which together define the canonical tensorial Heisenberg algebra.