Is ?4n + 2? a general quantum electromagnetic topological invariant?

Abstract

We begin with a review of past work using a “gauge model” to compare the phase (or gauge) similarities of Hückel's and the Woodward–Hoffmann rules, and the Aharonov–Bohm effect. The conjugated circuits model provides a clearer description of the connection of aromaticity with the band model than previously used. A common attribute is the effect of a circular path enclosing at least one singularity which creates a nonsimply connected manifold in the presence of a vector potential, Ā. This condition leads to Dirac's ambiguity in the resultant magnetic field . A solution is a Dirac-like monopole proposed by Wu and Yang obtained by coordinate patching around the singularity. Another model attribute is the conservation of angular momentum of the molecule plus field. This obtains by consideration of the return flux, which links the circle of atoms in the molecule with a circle of flux, and provides a “linking” of the two circles. The linking is described by one of the oldest topological invariants, the “Gausslinking integral.” By expanding the monopole solution we can describe the linking integral by means of the (S3 → S2) Hopf map, which necessitates adding a Chern–Simons term to describe this effect properly. Following a brief description of the Chern–Simons basis for the Jones-Witten topological knot theory, we conclude that there are three possible factors which could be responsible for the WH/Hückel 4n + 2 effect: curvature, torsion, and writhing. In this model the monopole (curvature) accounts for the 2, the torsion (orbital) effect for 4n, and the writhing (spin) for 0. Because a topological theory has no metric, it has no size dependence; hence, the model will support a “shell structure” of the periodic table based on 4n + 2. We close with a discussion of the integer quantum Hall effect (IQHE), where the commutivity of translation operators is combined with gauge transformations, thereby defining magnetic translation operators. The same selection rule for commutivity of the magnetic translation operators in the IQHE seems to apply in 4n and 4n + 2 ring compounds. © 1995 John Wiley & Sons, Inc.