The Casimir Effect in Non-Abelian Gauge Theories on the Lattice

We present non-perturbative results of the Casimir potential in non-abelian gauge theories in (2+1)D and (3+1)D for SU(2) and SU(3) in the confined and deconfined phases. For the first time, geometries beyond parallel plates in (3+1)D SU($N_c)$ are explored, and we show that the Casimir effect for the symmetrical and asymmetrical tube and box is attractive. The result for the tube is contrary to the weakly coupled, massless non-interacting scalar field theory result where a repulsive Casimir force, described by a negative slope of the potential, is measured in this geometry. We propose various techniques that can be used to account for the energy contributions from creating the boundaries in the geometry of a tube and box where the size of the faces forming the walls of the geometries changes with separation distance. We show that increasing the temperature from a confined to a deconfined phase does not alter the measured potential and we motivate for this observation by showing that the masses of the particles in the Casimir interactions are lower than the lowest glueball mass, $M_{0^{++}}$ in the pure gauge ground state, thus suggesting that the region inside the geometries is a boundary-induced deconfined phase. Through the measured expectation value of the Polyakov loop, we propose that the Casimir effect for the asymmetrical tube in the large separation distance limit $R\to \infty$ should be equivalent to that of the parallel plate geometry at smallest separation distance, while the asymmetrical box in the same limit should be equivalent to the symmetrical tube at the smallest separation distance.