The Validity of Dimensional Regularization Method on Fractal Spacetime

Figure 1

The Cantor ternary set is defined by repeatedly removing the middle thirds of line segments [8, 9]. (a) One starts by removing the middle third from the interval [𝑥₂⁽¹⁾,𝑥₁⁽¹⁾], leaving [𝑥₄⁽²⁾,𝑥₃⁽²⁾] and [𝑥₂⁽²⁾,𝑥₁⁽²⁾]. (b) Next, the “middle third” of all remaining intervals is removed. (c) This process is continued ad infinitum. Finally, the Cantor ternary set consists of all points in the interval [𝑥₂⁽¹⁾,𝑥₁⁽¹⁾] that are not removed at any step in this infinite process.