Aspects of quantum theory in 1+1 and slightly more dimensions

Abstract

We consider four problems in (1+1)-dimensional physics. Each of these problems had important connections to the physical behavior of (3+ 1 )-dimensional systems. First, we consider problem of fermions interacting with multiple bosonic solitons. We describe a new approximation scheme for determining the fermion energy spectrum and apply it to (1 + 1 )-dimensional two-component fermions coupled to scalar field solitons. Second, we study (1+1)-dimensional behavior in particles falling toward a Schwarzchild black hole . Using a non-covariant choice for the momentum cutoff, we examine the photon self-energy integral. We find evidence that the photons acquire an effective mass with a nonzero imaginary part, so that the photons may decay. Third, we consider cold fermions trapped in a high aspect ratio potential, which confines the particles to move in only one direction. The purely (1 + 1 )-dimensional aspects of this problem have been extensively studied. We examine the corrections that arise because of the underlying (3+ 1 )-dimensional character of the situation and determine the zero-temperature shifts in the (1+1)-dimensional energy spectrum. Fourth, we present a toy model, which is related, by analogy to the problem of electron-inhabited bubbles in liquid helium. An analysis of the I-dimensional model suggests that the recent suggestion that the electron bubbles may split in two is incorrect.