SPECTRAL ANALYSIS OF VOLUME OPERATOR IN LOOP QUANTUM GRAVITY FOR KINEMATICAL CASE - SEBASTIAN (NIM : 10214042), IGNATIUS
Loop Quantum Gravity has become one of the alternative solutions to quantum gravity. In this approach, the quantum effects are included into the General Relativity formulation through the Einstein-Hilbert Action using tetrads, resulting in quantized formulation for General Relativity. This formulation introduced a pair of new variables called Ashtekar's variable, which brought the group representation of SU(2). These variables are successfully used to model that in the quantum size, the space is actually discretized in the order of Planck length. The results came from the geometric operators, area and volume, which is formed by Ashtekar's variables. These operators have discrete eigenvalues, which is caused by the dependence of eigenvalues on the spin j of the spin network. The regularization process of these operators came from the classical definition of area and volume. Thus, the eigenvalues of area operator and volume operator are respectively the area and volume of the measured space. However, generally, there are two types of volume operator, the Ashtekar-Lewandowski operator and the Rovelli-Smolin operator. The significant difference between these two operators is the fact that Ashtekar-Lewandowski operator is sensitive to the direction of the spin network's link, while Rovelli-Smolin operator is not. This difference will result in a different spectrum of eigenvalue between these operators if they are applied to the same spin network.