Matrix-Formulated Noncommutative Gauge Theories of Gravity

Abstract

In the present article first we review the construction of three-and four-dimensional fuzzy spaces and subsequently we present the construction of fuzzy gravity theories on these noncommutative spaces following the gauge theoretic formulation of gravity.

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

“…The non-commutative quantum space-times and quantum-deformed Heisenberg algebra representing NC phase spaces, both considered as the tools for the description of quantum gravity, are naturally expressed in the form of NC algebras, with the use of the framework of NC geometry. We considered here simple and quite popular Snyder and Yang models, algebraically described by 𝑜(4, 1) and 𝑜(5, 1) Lie algebras, which have been often used in the current quantum gravity research, in particular exploiting NC quantum geometry (see [17,21,44]). To be more specific, 𝑜(4, 1) describes an extended Snyder model, in which Snyder quantum space-time is a subspace of a larger non-commutative algebra, which includes Lorentz symmetry generators.…”

“…𝑟, q𝜇 (1) + r (1) , 𝑞 𝜇 = − 𝑖 𝑅 2 𝑥 𝜇 . (44) For the extended Snyder model, in the first order, we obtained the formulas ( 23), (24). In Yang model, due to the presence of additional coordinates (𝑞 𝜇 , 𝑟) one should add their dual momenta (𝑘 𝜇 , 𝑠)) (see (32), (33)) and extend the formulae ( 23), ( 24) by terms which are linear in momenta (𝑘 𝜇 , 𝑠) (see (33)) as follows:…”

Noncommutative spaces and superspaces from Snyder and Yang type models

“…introducing additional five numerical constants ˜ , ˜ , ˜ , and . The equations ( 40)- (44) impose the following constraints on eight parameters in ( 46)-(48):…”

“…The non-commutative quantum space-times and quantum-deformed Heisenberg algebra representing NC phase spaces, both considered as the tools for the description of quantum gravity, are naturally expressed in the form of NC algebras, with the use of the framework of NC geometry. We considered here simple and quite popular Snyder and Yang models, algebraically described by (4, 1) and (5, 1) Lie algebras, which have been often used in the current quantum gravity research, in particular exploiting NC quantum geometry (see [17,21,44]). To be more specific, (4, 1) describes an extended Snyder model, in which Snyder quantum space-time is a subspace of a larger non-commutative algebra, which includes Lorentz symmetry generators.…”