2015/05/01 by Jinsong Yang, Yongge Ma, Yang, Jinsong +1
Medicine · Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Neuroblastoma Research and Treatments #Noncommutative and Quantum Gravity Theories #gr-qc
paper · pdf · doi:10.48550/arxiv.1505.00223
42 pages; a few modifications
openalex publication_date 2015/05/01 · arxiv created 2016/02/24 · arxiv updated 2016/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.