Knots and Links in Physical Systems 1 Roman

Few physical systems with topologies more complicated than simple gaussian linking have been explored in detail. Here we focus on examples with higher topologies in non-relativistic quantum mechanics and in QCD. 1 Generalized Aharonov-Bohm and Josephson effects Topology has played an important role in physics in recent years, but unlike in biology, most work done so far has involved the simplest nontrivial topology—gaussian linking. Here we discuss more complex topologies and their implications for physical systems. Let us start by recalling two prominent examples of the use of gaussian linking in nonrelativistic quantum mechanics, the magnetic Aharonov-Bohm effect [1] and the Josephson effect [2], see also Ref. [3]. We can generalize both these systems to higher order linking and/or knotting [4]. The magnetic Aharonov-Bohm effect, see Fig. 1, results when a charged particle travels around a closed path in a region of vanishing magnetic field but nonvanishing vector potential. The wave function of the particle is affected by the vector potential and a vector potential dependent interference pattern proportional to magnetic flux occurs at a detection screen. The conclusion one draws is that the vector potential is more fundamental than the magnetic field. The definitive experiments were done here in Japan [5].