We examine basis functions on momentum space for the three dimensional Euclidean Snyder algebra. We argue that the momentum space is isomorphic to the SO(3) group manifold, and that the basis functions span either one of two Hilbert spaces. This implies the existence of two distinct lattice structures of space, on which continuous rotations and translations are unitarily implementable.Comment: 22 page

We perform a systematic search for rotationally invariant cosmological solutions to matrix models, or more specifically the bosonic sector of Lorentzian IKKT-type matrix models, in dimensions d less than ten, specifically d = 3 and d = 5. After taking a continuum (or commutative) limit they yield d − 1 dimensional space-time surfaces, with an attached Poisson structure, which can be associated with closed, open or static cosmologies. For d = 3, we obtain recursion relations from which it is possible to generate rotationally invariant matrix solutions which yield open universes in the continuum limit. Specific examples of matrix solutions have also been found which are associated with closed and static two-dimensional space-times in the continuum limit. The solutions provide for a matrix resolution of cosmological singularities. The commutative limit reveals other desirable features, such as a solution describing a smooth transition from an initial inflation to a noninflationary era. Many of the d = 3 solutions have analogues in higher dimensions. The case of d = 5, in particular, has the potential for yielding realistic four-dimensional cosmologies in the continuum limit. We find four-dimensional de Sitter dS 4 or anti-de Sitter AdS 4 solutions when a totally antisymmetric term is included in the matrix action. A nontrivial Poisson structure is attached to these manifolds which represents the lowest order effect of noncommutativity. For the case of AdS 4 , we find one particlular limit where the lowest order noncommutativity vanishes at the boundary, but not in the interior.

We examine Hamiltonian formalism on Euclidean Snyder space. The latter corresponds to a lattice in the quantum theory. For any given dynamical system, it may not be possible to identify time with a real number parametrizing the evolution in the quantum theory. The alternative requires the introduction of a dynamical time operator. We obtain the dynamical time operator for the relativistic (nonrelativistic) particle, and use it to construct the generators of Poincaré (Galilei) group on Snyder space. *

We show that fuzzy spheres are solutions of Lorentzian IKKT matrix models. The solutions serve as toy models of closed noncommutative cosmologies where big bang/crunch singularities appear only after taking the commutative limit. The commutative limit of these solutions corresponds to a sphere embedded in Minkowski space. This 'sphere' has several novel features. The induced metric does not agree with the standard metric on the sphere, and moreover, it does not have a fixed signature. The curvature computed from the induced metric is not constant, has singularities at fixed latitudes (not corresponding to the poles) and is negative. Perturbations are made about the solutions, and are shown to yield a scalar field theory on the sphere in the commutative limit. The scalar field can become tachyonic for a range of the parameters of the theory.

We examine a Chern-Simons matrix model which we propose as a toy model for studying the quantum nature of black holes in 2+1 gravity. Its dynamics is described by two N ×N matrices, representing the two spatial coordinates. The model possesses an internal SU (N ) gauge symmetry, as well as an external rotation symmetry. The latter corresponds to the rotational isometry of the BTZ solution, and does not decouple from SU (N ) gauge transformations. The system contains an invariant which is quadratic in the spatial coordinates. We obtain its spectrum and degeneracy, and find that the degeneracy grows exponentially in the large N limit. The usual BTZ black hole entropy formula is recovered upon identifying the quadratic invariant with the square of the black hole horizon radius. The quantum system behaves collectively as an integer (half-integer) spin particle for even (odd) N under 2π-rotations.