Integrated Study of Fractal Loop Quantum Gravity and Fractal CDT Theory

This paper, based on the Dynamic Rigid Fractal Spacetime Manifold Theory (DRFSMT), delves into the integration of Fractal Loop Quantum Gravity (LQG) and Fractal Causal Dynamical Triangulations (CDT) theories. Key formulas in fractal LQG are derived and explained in detail, including fractal connections and momentum operators, fractal spin networks and area spectra, fractal Hamiltonian constraints, fractal black hole entropy, fractal volume operators, and fractal constraint algebra, with their physical implications elucidated. Simultaneously, the mathematical foundations of fractal CDT theory, the coupling of fractal measures and causal structures, the fractal CDT action, and the dynamical equations of fractal dimensions are integrated and interpreted. This study demonstrates the unique value and application prospects of this theory in quantum gravity and cosmology research, providing new perspectives and theoretical foundations for exploring the fractal geometric properties of spacetime.