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

TL;DR

This paper develops an $bar$-perturbative algebraic approach to solve quantum Snyder and Yang models, revealing how non-zero zero-order terms indicate spontaneous symmetry breaking and how explicit calculations involve dual momenta and Heisenberg algebra.

Contribution

It introduces a novel $bar$-power series method for algebraic solutions of quantum Snyder and Yang models, linking zero-order terms to SSB parameters and providing explicit calculations with dual momenta.

Findings

Zero-order terms describe SSB parameters in the models.

Explicit $bar$-series terms can be calculated using dual momenta.

Models exhibit relativistic quantum space-times with Lorentz-covariance.

Abstract

We introduce the perturbative $\hbar$-power series ($\hbar$ = 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 ($\hbar $-independent) terms are non-vanishing they describe the spontaneous symmetry breaking (SSB) parameters of Lie-algebraic symmetries which characterize the considered models ($D=4$ dS symmetry in Snyder and $D=5$ dS symmetry in Yang cases). The consecutive terms in $\hbar$-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.