Casimir Interactions Between Scatterers in Carbon Nanotubes

In this thesis we calculate interactions between localized scatterers in metallic carbon nanotubes. Backscattering of electrons between localized scatterers mediates long range forces between them. These interactions are mapped to Casimir forces mediated by one-dimensional massless fermions and calculated using a force operator approach. We first study interactions between scatterers described by spinor polarized potentials relevant to the single-valley problem in carbon nanotubes. We obtain the force between two finite width square barriers, and take the limit of zero width and infinite potential strength to study the Casimir force mediated by the fermions. For the case of identical scatterers we recover the conventional attractive one dimensional Casimir force. For the general problem with inequivalent scatterers we find that the magnitude and sign of this force depend on the relative spinor polarizations of the two scattering potentials which can be tuned to give an attractive, a repulsive, or a compensated null Casimir interaction. Next, we generalize our work on the single-valley Casimir problem to study interactions between physically realizable scatterers in nanotubes. We model spatially localized scatterers by local and non-local potentials and treat simultaneously the effects of intravalley and intervalley backscattering. We find that the long range forces between scatterers exhibit the universal power law decay of the Casimir force in one dimension, with prefactors that control the sign and strength of the interaction. These prefactors are nonuniversal and depend on the symmetry and degree of localization of the scattering potentials. We find that local potentials inevitably lead to a coupled valley scattering problem, though by contrast non-local potentials lead to two decoupled single-valley problems. The Casimir effect due to two-valley scattering potentials is characterized by the appearance of spatially periodic modulations of the force