Real-space renormalization from a geometric fractal perspective

An introduction to real-space renormalization transformation (RT) is presented with geometric fractal aspects as focus. Critical exponents are equivalent to fractal dimensions. Details of the construction rule of deterministic fractals and the definition of their fractal dimensions illustrate differences in approach between field theory and condensed matter physics. The appearance of fractal structures is typically less obvious within the context of partition functions (equilibrium statistical mechanics and field theory) and master equations (dynamic processes). An RT amounts to a reformulation of those construction rules into a form similar to the conventional ones for deterministic fractals, but on an algebraic level without ever explicitly referring to the geometric structure. This is illustrated explicitly in the context of the Sierpinsky Gasket. Its construction rule can be reformulated as a growth process and also as a partition function.