Graphical method in loop quantum gravity: I. Derivation of the closed formula for the matrix element of the volume operator

Abstract

To adopt a practical method to calculate the action of geometrical operators on quantum states is a crucial task in loop quantum gravity. In the series of papers, we will introduce a graphical method, developed by Yutsis and Brink, to loop quantum gravity along the line of previous works. The graphical method provides a very powerful technique for simplifying complicated calculations. In this first paper, the closed formula of volume operator is derived via the graphical method. By employing suitable and non-ambiguous graphs to represent the acting of operators as well as the spin network states, we use the simple rules of transforming graphs to yield the resulting formula. Comparing with the complicated algebraic derivation in some literatures, our procedure is more concise, intuitive and visual. The resulting matrix elements of volume operator is compact and uniform, fitting for both gauge-invariant and gauge-variant spin network states.

42 pages; a few modifications

References in corpus (8)

• The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
• Spectral Analysis of the Volume Operator in Loop Quantum Gravity
• The length operator in Loop Quantum Gravity
• New length operator for loop quantum gravity
• Hamiltonian operator for loop quantum gravity coupled to a scalar field
• New Hamiltonian constraint operator for loop quantum gravity
• Quantum spin dynamics
• Graphical method in loop quantum gravity: II. The Hamiltonian constraint and inverse volume operators

Cited by in corpus (6)

• New volume and inverse volume operators for loop quantum gravity
• Towards the self-adjointness of a Hamiltonian operator in loop quantum gravity
• Introduction to SU(2) recoupling theory and graphical methods for loop quantum gravity
• Graphical method in loop quantum gravity: II. The Hamiltonian constraint and inverse volume operators
• Note on quantum entanglement and quantum geometry
• Hermiticity of the Volume Operators in Loop Quantum Gravity