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Item BTZ black hole entropy from a Chern-Simons matrix model(IOP Publishing, 2013-10-30) Chaney, A.; Lu, Lei; Stern, A.; University of Alabama TuscaloosaShow more 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 x 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 2 pi-rotations.Show more Item Lorentzian Fuzzy Spheres(2015-09-16) Chaney, A.; Lu, Lei; Stern, Allen; University of Alabama TuscaloosaShow more We show that fuzzy spheres are solutions of Lorentzian Ishibashi-Kawai-Kitazawa-Tsuchiya-type 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.Show more Item Matrix model approach to cosmology(American Physical Society, 2016-03) Chaney, A.; Lu, Lei; Stern, A.; University of Alabama TuscaloosaShow more We perform a systematic search for rotationally invariant cosmological solutions to toy matrix models. These models correspond to the bosonic sector of Lorentzian Ishibashi, Kawai, Kitazawa and Tsuchiya (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 Poisson manifolds. The manifolds have a Lorentzian induced metric which can be associated with closed, open, or static space-times. 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 resolution of cosmological singularities, at least within the context of the toy matrix models. 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 particular limit where the lowest order noncommutativity vanishes at the boundary, but not in the interior.Show more Item Noncommutative spaces from matrix models(University of Alabama Libraries, 2016) Lu, Lei; Stern, Allen B.; University of Alabama TuscaloosaShow more Noncommutative (NC) spaces commonly arise as solutions to matrix model equations of motion. They are natural generalizations of the ordinary commutative spacetime. Such spaces may provide insights into physics close to the Planck scale, where quantum gravity becomes relevant. Although there has been much research in the literature, aspects of these NC spaces need further investigation. In this dissertation, we focus on properties of NC spaces in several different contexts. In particular, we study exact NC spaces which result from solutions to matrix model equations of motion. These spaces are associated with finite-dimensional Lie-algebras. More specifically, they are two-dimensional fuzzy spaces that arise from a three-dimensional Yang-Mills type matrix model, four-dimensional tensor-product fuzzy spaces from a tensorial matrix model, and Snyder algebra from a five-dimensional tensorial matrix model. In the first part of this dissertation, we study two-dimensional NC solutions to matrix equations of motion of extended IKKT-type matrix models in three-space-time dimensions. Perturbations around the NC solutions lead to NC field theories living on a two-dimensional space-time. The commutative limit of the solutions are smooth manifolds which can be associated with closed, open and static two-dimensional cosmologies. One particular solution is a Lorentzian fuzzy sphere, which leads to essentially a fuzzy sphere in the Minkowski space-time. In the commutative limit, this solution leads to an induced metric that does not have a fixed signature, and have a non-constant negative scalar curvature, along with singularities at two fixed latitudes. The singularities are absent in the matrix solution which provides a toy model for resolving the singularities of General relativity. We also discussed the two-dimensional fuzzy de Sitter space-time, which has irreducible representations of su(1,1) Lie-algebra in terms of principal, complementary and discrete series. Field theories on such backgrounds result from perturbations about the solutions. The perturbative analysis requires non-standard Seiberg-Witten maps which depend on the embeddings in the ambient space and the symplectic 2-form. We find interesting properties of the field theories in the commutative limit. For example, stability of the action may require adding symmetry breaking terms to the matrix action, along with a selected range for the matrix coefficients. In the second part of this dissertation, we study higher dimensional fuzzy spaces in a tensorial matrix model, which is a natural generalization to the three-dimensional actions and is valid in any number of space-time dimensions. Four-dimensional tensor product NC spaces can be constructed from two-dimensional NC spaces and may provide a setting for doing four-dimensional NC cosmology. Another solution to the tensorial matrix model equations of motion is the Snyder algebra. A crucial step in exploring NC physics is to understand the structure of the quantized space-time in terms of the group representations of the NC algebra. We therefore study the representation theory of the Snyder algebra and implementation of symmetry transformations on the resulted discrete lattices. We find the three-dimensional Snyder space to be associated with two distinct Hilbert spaces, which define two reducible representations of the su(2)*su(2) algebra. This implies the existence of two distinct lattice structures of Snyder space. The difference between the two representations is evident in the spectra of the position operators, which could only be integers in one case and half integers in the other case. We also show that despite the discrete nature of the Snyder space, continuous translations and rotations can be unitarily implemented on the lattices.Show more Item Particle dynamics on Snyder space(Elsevier, 2012-07-01) Lu, Lei; Stern, A.; University of Alabama TuscaloosaShow more 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 Poincare (Galilei) group on Snyder space. (C) 2012 Elsevier B.V. All rights reserved.Show more Item Snyder space revisited(Elsevier, 2012-01-21) Lu, Lei; Stern, A.; University of Alabama TuscaloosaShow more 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. Continuous rotations and translations are unitarily implementable on these lattices. (C) 2011 Elsevier B.V. All rights reserved.Show more