Fractal Dimensions: Exploring the Unification of Physical Laws via Hausdorff Dimensions

This paper investigates the potential role of the Hausdorff dimension (DH)as a tuning parameter in the unification of physical laws. By integrating fractalgeometry with physical theories, we propose a mathematical framework to studykey physical phenomena across different Hausdorff dimensions, including:• At DH = 3, the unification of gauge fields and gravitational fields, with ananalysis of its implications for particle physics and cosmology;• At DH = 4, the mathematical mechanism of time-reversal symmetry breakingand its experimental manifestations;• At DH = 6, the construction of a fractal membrane model to investigate massspectrum corrections in higher-dimensional theories.Experimental validations include the observation of dimension-dependent phe?nomena in cold atomic optical lattice experiments and the statistical analysis ofgamma-ray burst afterglows. These results suggest that the Hausdorff dimensionmay act as a tuning parameter influencing the manifestation of physical laws, pro?viding new perspectives for the study of quantum gravity and higher-dimensionaltheories.Keywords: Hausdorff dimension, fractal geometry, quantum gravity, time?reversal symmetry breaking, topological phase transitions