KINEMATICAL ANALYSIS FOR GEOMETRICAL SPECTRUM OF LENGTH, AREA AND ANGLE IN LOOP QUANTUM GRAVITY - HUSIN (NIM : 20216039), IDRUS
Loop Quantum Gravity(LQG) is one of the candidates of quantum theory for EinsteinÃ‚Â’s general relativity. This theory quantize general relativity using the Hamiltonian formulation for the Einstein-Hilbert action. Using the tetrad formulation it is found that the dynamical variables of this theory can be expressed in terms of triad and SU(2) connections known as Ashtekar connections. This formulation also provides an additional constraint equation namely the Gauss constraint, which is not obtained from the standard ADM formulation. This constraint is important in kinematical theory of LQG in order to construct the spin-network state as the basis of the Hilbert space in LQG. The discussion in this thesis is restricted only to the kinematical part of LQG, so we only work on kinematical Hilbert space where the basis for this space is spin-network state. For 3d there are three geometrical operators which can be discussed: length of segment, angle between segments and area, meanwhile for 4d there are five geometrical operators which can be discussed: length of segment, area, angle between segments, angle between faces and volume. In this thesis the discussion is given for both cases. For 3d case the three operators are discuss thoroughly. Explicit expression for matrix element of area operator is given for triangular and rectangular discretization. It was also shown that the classical relation in the form of triangular inequality still applies to any spin number. The 3d case ends with analysis of area spectrum for rectangle built in two ways: first, from coupling two triangles and second rectangle that built itself. Analysis is done in two simple cases: ground state and first excited monochromatic. Next for 4d case the discussion is given specifically for length operator and spectrum analysis is only given for ground state monochromatic with tetrahedron discretization. For this case we calculated the set of eigenvalues of length of every wedge alongside with the eigenvectors. At the end we given the expression for Heisenberg uncertainty between wedge length.