Two major electromagnetic phenomena, magneto-optical effects and the Casimir effect, have seen much theoretical and experimental use for many years.

On the other hand, recently there has been an explosion of theoretical and experimental work on so-called topological materials, and a natural question to ask is how such electromagnetic phenomena change with these novel materials.

Specifically, we will consider are topological insulators and Weyl semimetals.

When Dirac electrons on the surface of a topological insulator are gapped or Weyl fermions in the bulk of a Weyl semimetal appear due to time-reversal symmetry breaking, there is a resulting quantum anomalous Hall effect (2D in one case and bulk 3D in the other, respectively).

For topological insulators, we investigate the role of localized in-gap states which can leave their own fingerprints on the magneto-optics and can therefore be probed.

We have shown that these states resonantly contribute to the Hall conductivity and are magneto-optically active.

For Weyl semimetals we investigate the Casimir force and show that with thickness, chemical potential, and magnetic field, a \emph{repulsive and tunable} Casimir force can be obtained.

Additionally, various values of the parameters can give various combinations of traps and antitraps.

We additionally probe the topological transition called a Lifshitz transition in the band structure of a material and show that in a Casimir experiment, one can observe a non-analytic ``kink'' in the Casimir force across such a transition.

The material we propose is a spin-orbit coupled semiconductor with large $g$-factor that can be magnetically tuned through such a transition.

Additionally, we propose an experiment with a two-dimensional metal where weak localization is tuned with an applied field in order to definitively test the effect of diffusive electrons on the Casimir force---an issue that is surprisingly unresolved to this day.

Lastly, we show how the time-continuous coherent state path integral breaks down for both the single-site Bose-Hubbard model and the spin path integral.

Specifically, when the Hamiltonian is quadratic in a generator of the algebra used to construct coherent states, the path integral fails to produce correct results following from an operator approach.

We note that the problems do not arise in the time-discretized version of the path integral, as expected.