Probing the Shape of Quantum Surfaces: the Quadrupole Moment Operator
arXiv:1805.08257 · doi:10.1088/1361-6382/aae327
Abstract
The standard toolkit of operators to probe quanta of geometry in loop quantum gravity consists in area and volume operators as well as holonomy operators. New operators have been defined, in the U(N) framework for intertwiners, which allow to explore the finer structure of quanta of geometry. However these operators do not carry information on the global shape of the intertwiners. Here we introduce dual multipole moments for continuous and discrete surfaces, defined through the normal vector to the surface, taking special care to maintain parametrization invariance. These are raised to multipole operators probing the shape of quantum surfaces. Further focusing on the quadrupole moment, we show that it appears as the Hessian matrix of the large spin Gaussian approximation of coherent intertwiners, which is the standard method for extracting the semi-classical regime of spinfoam transition amplitudes. This offers an improvement on the usual loop quantum gravity techniques, which mostly focus on the volume operator, in the perspective of modeling (quantum) gravitational waves as shape fluctuations waves propagating on spin network states.
arXiv comment24 pages
References in corpus (29)
- Discreteness of area and volume in quantum gravity
- Quantum Spin Dynamics (QSD)
- A new spinfoam vertex for quantum gravity
- Twisted geometries: A geometric parametrisation of SU(2) phase space
- Lorentzian spin foam amplitudes: graphical calculus and asymptotics
- Quantum Spin Dynamics (QSD) II
- Polyhedra in loop quantum gravity
- Consistently Solving the Simplicity Constraints for Spinfoam Quantum Gravity
- Simplification of the Spectral Analysis of the Volume Operator in Loop Quantum Gravity
- Quantum tetrahedra and simplicial spin networks
- Spectral Analysis of the Volume Operator in Loop Quantum Gravity
- Spinor Representation for Loop Quantum Gravity
- U(N) Coherent States for Loop Quantum Gravity
- The Fine Structure of SU(2) Intertwiners from U(N) Representations
- Reconstructing Quantum Geometry from Quantum Information: Spin Networks as Harmonic Oscillators
- U(N) tools for Loop Quantum Gravity: The Return of the Spinor
- Consistency Check on Volume and Triad Operator Quantisation in Loop Quantum Gravity I
- Consistency Check on Volume and Triad Operator Quantisation in Loop Quantum Gravity II
- Holomorphic Factorization for a Quantum Tetrahedron
- The loop gravity string
- Bohr-Sommerfeld Quantization of Space
- A curvature operator for LQG
- Deformations of Polyhedra and Polygons by the Unitary Group
- A new Hamiltonian for the Topological BF phase with spinor networks
- Asymptotics of 4d spin foam models
- Quantum Surface and Intertwiner Dynamics in Loop Quantum Gravity
- Coherent States for 3d Deformed Special Relativity: semi-classical points in a quantum flat spacetime
- SO*(2N) coherent states for loop quantum gravity
- Coherent 3j-symbol representation for the loop quantum gravity intertwiner space