Fractal geometry in quantum mechanics, field theory and spin systems

The goal of this article is to review the role of fractal geometry in quantum physics. There are two aspects: (a) The geometry of underlying space (space–time in relativistic systems) is fractal and one studies the dynamics of the quantum system. Example: percolation. (b) The underlying space–time is regular, and fractal geometry which shows up in particular observables is generated by the dynamics of the quantum system. Example: Brownian motion (imaginary time quantum mechanics), zig-zag paths of propagation in quantum mechanics (Feynmanʹs path integral). Historically, the first example of fractal geometry in quantum mechanics was invoked by Feynman and Hibbs describing the self-similarity (fractal behavior) of paths occurring in the path integral. We discuss the geometry of such paths. We present analytical as well as numerical results, yielding Hausdorff dimension dH=2. Velocity-dependent interactions (propagation in a solid, Bruecknerʹs theory of nuclear matter) allow for dH<2. Next, we consider quantum field theory. We discuss the relation of self-similarity, the renormalization group equation, scaling laws and critical behavior, also violation of scale invariance, like logarithmic scaling corrections in hadron structure functions. We discuss the fractal geometry of paths of the path integral in field theory. We present numerical results for the length of propagation and fractal dimension for the free fermion propagator which is relevant for the geometry of quark propagation in QCD. Then we look at order parameters for the confinement phase in QCD. The fractal dimension of closed monopole current loops is such an order parameter. We discuss properties of a fractal Wilson loop. We look at critical phenomena, in particular at critical exponents and its relation to non-integer dimension of space–time by use of an underlying fractal geometry with the purpose to determine lower or upper critical dimensions. As an example we consider the U(1) model of lattice gauge theory. As another topic we discuss fractal geometry and Hausdorff dimension of quantum gravity and also for gravity coupled to matter, like to the Ising model or to the 3-state Potts model. Finally, we study the role that fractal geometry plays in spin physics, in particular for the purpose to describe critical clusters.