Invitation: Gravity, Point Particles, and Group-Valued Momenta

Generalized quantum phase spaces for the κ-deformed extended Snyder model

“…we see that the internal and Lorentzian generators do not commute with each other. It can be checked that the relations ( 29)-( 32) are self dual under the generalized Born map (10), extended by (14). Moreover, it can be shown that the relations ( 29)-( 32) can be derived as the general solutions of the Jacobi identities for generators xµ , qµ and r.…”

“…[8][9][10]) that in the passage from classical to quantum gravity the algebraic framework of the noncommutative geometry with explicit dependence on the Planck constant is quite useful (see e.g. [10][11][12][13][14]). Snyder model describes, by means of particular algebraic relations, the NC geometry of quantum-deformed relativistic space-time but does not provide a dynamical system with the equations determining the time evolution.…”

An Introduction to Noncommutative Physics

“…It can be verified that [ L, Ĥ] = 0, which implies that rotational symmetry is broken in the noncommutative phase space. However, it should be noted that in some approaches to noncommutative geometry, such as the κ−Minkowski model of noncommutative spacetime, generalized operators are taken, by definition, as the generators of deformed-symmetry transformations [57,58]. In this approach, the Lie algebras of canonical quantum mechanics are replaced by Hopf algebras, and the dynamics of the canonical theory are generalized to include additional couplings between matter and gravity, as required by the structure of the κ−Poincare Hopf algebra [57,58].…”

“…However, in this pedagogical introduction to noncommutative physics we do not discuss Conne's approach to noncommutative geometry, K theory, noncommutative field theory, the Moyal star product technique, or κ−deformed symmetries of spacetime. For details of these (more mathematical) approaches, readers are referred to [14][15][16][56][57][58][59] and references therein. We summarise our conclusions, and offer a few opinions on the outlook for noncommutative geometry in physics research in Section 5.…”

Realizations of the Extended Snyder Model

“…The length scale enters the theory through commutators of spacetime coordinates in [1,2,8,9]. Deformations of spacetime symmetries-gravity, group-valued momenta, and noncommutative fields were presented in [3].…”