The volume operator for spin networks with planar or cylindrical symmetry

Abstract

This paper constructs a kinematic basis for spin networks with planar or cylindrical symmetry, by exploiting the fact that the basis elements are representations of an O(3) subgroup of O(4). The action of the volume operator on this basis gives a difference equation for the eigenvalues and eigenvectors of the volume operator. For basis elements of low spin, the difference equation can be solved readily on a computer. For higher spins, I solve for the eigenvalues using a WKBJ method. This paper considers only the case where the gravitational wave can have both polarizations. The single polarization case is considered in a spearate paper.

35 pages

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Cited by in corpus (10)

• Semiclassical Mechanics of the Wigner 6j-Symbol
• The Evolution of $\Lambda$ Black Holes in the Mini-Superspace Approximation of Loop Quantum Gravity
• Quantization of Midisuperspace Models
• Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials
• The screen representation of vector coupling coefficients or Wigner 3j symbols: exact computation and illustration of the asymptotic behavior
• Symmetric angular momentum coupling, the quantum volume operator and the 7-spin network: a computational perspective
• Quadrilaterals on the square screen of their diagonals: Regge symmetries of quantum-mechanical spin-networks and Grashof classical mechanisms of four-bar linkages
• The volume operator for singly polarized gravity waves with planar or cylindrical symmetry
• Spin network coherent states for planar gravitational waves. I
• Plane wave holonomies in loop quantum gravity II: sine wave solution