Quantum surface holonomies for loop quantum gravity and their application to black hole horizons

Abstract

In this work we define a new type of flux operators on the Hilbert space of loop quantum gravity. We use them to solve an equation of the form $F(A)=c\Sigma$ in loop quantum gravity. This equation, which relates the curvature of a connection $A$ with its canonical conjugate $\Sigma =\ast E$, plays an important role for spherically symmetric isolated horizons, and, more generally, for maximally symmetric geometries and for the Kodama state. If the equation holds, the new flux operators can be interpreted as a quantization of surface holonomies from higher gauge theory. Also, they represent a kind of quantum deformation of SU(2). We investigate their properties and discuss how they can be used to define states that satisfy the isolated horizon boundary condition in the quantum theory.

arXiv comment39 pages, 8 figures. v2: some improvements to introduction and bibliography

References in corpus (28)

• Quantum Geometry and Black Hole Entropy
• Real Ashtekar Variables for Lorentzian Signature Space-times
• Linking Topological Quantum Field Theory and Nonperturbative Quantum Gravity
• An Invitation to Higher Gauge Theory
• Black hole entropy and SU(2) Chern-Simons theory
• Quantum gravity with a positive cosmological constant
• Detailed black hole state counting in loop quantum gravity
• SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry
• Chern-Simons-Matter Theory
• The SU(2) Black Hole entropy revisited
• On area and entropy of a black hole
• Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics
• Quantization maps, algebra representation and non-commutative Fourier transform for Lie groups
• Counting black hole microscopic states in loop quantum gravity
• Higher gauge theory -- differential versus integral formulation
• Four-dimensional Quantum Gravity with a Cosmological Constant from Three-dimensional Holomorphic Blocks
• On two-Dimensional Holonomy
• Abelian Chern-Simons theory
• Black hole horizons from within loop quantum gravity
• Chern-Simons expectation values and quantum horizons from LQG and the Duflo map
• Chern-Simons theory, Stokes' Theorem, and the Duflo map
• Diffeomorphism invariant cosmological symmetry in full quantum gravity
• Non-commutative Fourier transform for the Lorentz group via the Duflo map
• A Cubical Set Approach to 2-Bundles with Connection and Wilson Surfaces
• Some notes on the Kodama state, maximal symmetry, and the isolated horizon boundary condition
• Abelian Chern-Simons theory, Stokes' theorem, and generalized connections
• Extensions of the Duflo map and Chern-Simons expectation values
• Interpreting the isolated horizon boundary condition in terms of higher gauge theory

Cited by in corpus (1)

• The Generalization of Faulhaber's Formula to Sums of Arbitrary Complex Powers