On the relation between the connection and the loop representation of quantum gravity

Abstract

Using Penrose binor calculus for $SU(2)$ ($SL(2,C)$) tensor expressions, a graphical method for the connection representation of Euclidean Quantum Gravity (real connection) is constructed. It is explicitly shown that: {\it (i)} the recently proposed scalar product in the loop-representation coincide with the Ashtekar-Lewandoski cylindrical measure in the space of connections; {\it (ii)} it is possible to establish a correspondence between the operators in the connection representation and those in the loop representation. The construction is based on embedded spin network, the Penrose graphical method of $SU(2)$ calculus, and the existence of a generalized measure on the space of connections modulo gauge transformations.

19 pages, ioplppt.sty and epsfig.sty

References in corpus (9)

• Quantum Theory of Gravity I: Area Operators
• Geometry Eigenvalues and Scalar Product from Recoupling Theory in Loop Quantum Gravity
• Volume and Quantizations
• Reality conditions inducing transforms for quantum gauge field theory and quantum gravity
• The complete spectrum of the area from recoupling theory in loop quantum gravity
• Closed formula for Wilson loops for $SU(N)$ Quantum Yang-Mills Theory in two dimensions
• A real alternative to quantum gravity in loop space
• Perturbation study on the spin and charge susceptibilities of the two-dimensional Hubbard model
• Quantizing Regge calculus

Cited by in corpus (40)

• Background Independent Quantum Gravity: A Status Report
• Loop Quantum Gravity
• ``Sum over Surfaces'' form of Loop Quantum Gravity
• Decomposition of entanglement entropy in lattice gauge theory
• Gauge Field Theory Coherent States (GCS) : I. General Properties
• Gauge Field Theory Coherent States (GCS) : II. Peakedness Properties
• Geometry Eigenvalues and Scalar Product from Recoupling Theory in Loop Quantum Gravity
• The Area Operator in the Spherically Symmetric Sector of Loop Quantum Gravity
• Quantum Symmetry Reduction for Diffeomorphism Invariant Theories of Connections
• Closed formula for the matrix elements of the volume operator in canonical quantum gravity
• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Gauge Field Theory Coherent States (GCS) : III. Ehrenfest Theorems
• Gauge Field Theory Coherent States (GCS) : IV. Infinite Tensor Product and Thermodynamical Limit
• Quantum Theory of Geometry III: Non-commutativity of Riemannian Structures
• Quantum Spin Dynamics (QSD) : VII. Symplectic Structures and Continuum Lattice Formulations of Gauge Field Theories
• The projector on physical states in loop quantum gravity
• 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
• Photons from quantized electric flux representations
• Strings, loops and others: a critical survey of the present approaches to quantum gravity
• Operators for quantized directions
• A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements
• Loop quantum gravity and quanta of space: a primer
• A Spin Network Primer
• Matrix Elements of Thiemann's Hamiltonian Constraint in Loop Quantum Gravity
• The basis of the physical Hilbert space of lattice gauge theories
• The future of spin networks
• Combinatorial space from loop quantum gravity
• Background independent quantizations: the scalar field I
• New volume and inverse volume operators for loop quantum gravity
• Background independent quantizations: the scalar field II
• Spin Networks and Recoupling in Loop Quantum Gravity
• Grasping rules and semiclassical limit of the geometry in the Ponzano-Regge model
• Fermion spins in loop quantum gravity
• SU(N) Glueball Masses in 2+1 Dimensions
• Quasilocal Energy for Spin-net Gravity
• Canonical ``Loop'' Quantum Gravity and Spin Foam Models
• Integrals over SU(N)
• Bibliography of publications related to Classical and Quantum Gravity in terms of Connection and Loop Variables
• No invariant perfect qubit codes