Divergence, Spacetime Dimension and Fractal Structure

Abstract

The dimension of spacetime 1) is one of the fundamental concepts in physics and should be determined 2) from a truly fundamental theory of everything. We take for granted that the dimension of our physical spacetime is four. However, in some physical theories, other dimensions have been considered. Kaluza-Klein theories 3) and superstring theories 4) involve integer dimensions larger than four. In the dimensional regularization procedure 5) and the expansion about 4 dimensions, 6) nonintegral dimensions are used as a useful mathematical device without physical significance. We are interested in theories given in fractal spacetime dimensions 7) or generalized Hausdorff dimensions, 8) because quantum electrodynamics (QED) in a spacetime with fewer than four dimensions would be free of ultraviolet divergences. 8) We may consider our spacetime to be of a fractal nature, 9) with a non-integer fractal dimension. The most basic property of a fractal is perhaps its fractal dimension n, 7) which is derived from n = limδ→0 ln(k(δ))/ ln(δ−1). The quantity k(δ) here is the minimum number of elemental cubes of side δ needed to cover the fractal. Let us consider the Cantor set as a concrete example of a fractal space. The twodimensional three-Cantor space has a fractal dimension n = ln(32−1)/ ln 3 = 1.89 · · ·. This result can be generalized to a D-dimensional N -Cantor space, 10) which has the fractal dimension n = ln(ND − 1)/ lnN D − 1/(ND lnN). With the above example in mind, in this paper, we assume the dimension of spacetime to be slightly smaller than four. 8) Within the framework of QED, this dimension of spacetime can be determined by calculating three Feynman diagrams, that of the electron self-energy, the photon self-energy, and the vertex graph. In addition, we investigate the validity of the perturbative expansion and the positivity of the fine-structure constant. Let us start with the self-energy of an electron to second order,