On the q-quantum gravity loop algebra

A class of deformations of the q-quantum gravity loop algebra is shown to be incompatible with the combinatorics of Temperley-Lieb recoupling theory with deformation parameter at a root of unity. This incompatibility appears to extend to more general deformation parameters.Comment: v2: text clarified, version to be publishe

Operators for quantized directions

Inspired by the spin geometry theorem, two operators are defined which measure angles in the quantum theory of geometry. One operator assigns a discrete angle to every pair of surfaces passing through a single vertex of a spin network. This operator, which is effectively the cosine of an angle, is defined via a scalar product density operator and the area operator. The second operator assigns an angle to two ``bundles'' of edges incident to a single vertex. While somewhat more complicated than the earlier geometric operators, there are a number of properties that are investigated including the full spectrum of several operators and, using results of the spin geometry theorem, conditions to ensure that semiclassical geometry states replicate classical angles.Comment: v1: 20 pages, 23 figures v2: changes in presentation and regularization (final results unchanged). This is an expanded version of the one to be published in Class. Quant. Gra

New Operators for Spin Net Gravity: Definitions and Consequences

Two operators for quantum gravity, angle and quasilocal energy, are briefly reviewed. The requirements to model semi-classical angles are discussed. To model semi-classical angles it is shown that the internal spins of the vertex must be very large, ~10^20.Comment: 7 pages, 2 figures, a talk at the MG9 Meeting, Rome, July 2-8, 200

A Spin Network Primer

Spin networks, essentially labeled graphs, are ``good quantum numbers'' for the quantum theory of geometry. These structures encompass a diverse range of techniques which may be used in the quantum mechanics of finite dimensional systems, gauge theory, and knot theory. Though accessible to undergraduates, spin network techniques are buried in more complicated formulations. In this paper a diagrammatic method, simple but rich, is introduced through an association of 2 by 2 matrices to diagrams. This spin network diagrammatic method offers new perspectives on the quantum mechanics of angular momentum, group theory, knot theory, and even quantum geometry. Examples in each of these areas are discussed.Comment: A review of spin networks suitable for students of advanced quantum mechanics (undergraduate). 16 pages, many eps figures, to be published in Am. J. Phys v2: Updated to include key referenc

Quantum geometry phenomenology: Angle and semiclassical states

The phenomenology for the deep spatial geometry of loop quantum gravity is discussed. In the context of a simple model of an atom of space, it is shown how purely combinatorial structures can affect observations. The angle operator is used to develop a model of angular corrections to local, continuum flat-space 3-geometries. The physical effects involve neither breaking of local Lorentz invariance nor Planck scale suppression, but rather only involve the combinatorics of SU(2) recoupling. Bhabha scattering is discussed as an example of how the effects might be observationally accessible

Plane gravitational waves and loop quantization

Starting from the polarized Gowdy model in Ashtekar variables, the Killing equations characteristic for plane-fronted parallel gravitational waves are introduced in part as a set of first-class constraints, in addition to the standard ones of General Relativity. These constraints are expressed in terms of quantities that have an operator equivalent in Loop Quantum Gravity, making plane wave space-times accessible to loop quantization techniques

Toward loop quantization of plane gravitational waves

The polarized Gowdy model in terms of Ashtekar–Barbero variables is reduced with an additional constraint derived from the Killing equations for plane gravitational waves with parallel rays. The new constraint is formulated in a diffeomorphism invariant manner and, when it is included in the model, the resulting constraint algebra is first class, in contrast to the prior work done in special coordinates. Using an earlier work by Banerjee and Date, the constraints are expressed in terms of classical quantities that have an operator equivalent in loop quantum gravity, making these plane gravitational wave spacetimes accessible to loop quantization techniques