3 Two Dimensional Electrodynamics

``Electrodynamics in $(1+1)$ dimensions,'' also known in its early quantum formulation as ``the Schwinger model'' [14], means different things to different people. Formally, the action (or Lagrangian) of the model is the same as that of the familiar Maxwell electrodynamics in $(3+1)$ dimensions, hence the name. The physical content, however, is vastly different. This is because of the stringent kinematical constraints that exist in $(1+1)$ dimensions: since there is no ``transversality'' in one spatial dimension, the concept of spin is undefined, and the notion of `` vector field'', massless or massive, is purely formal. Thus, there is no radiation field associated with the Maxwell tensor. There is, however, the same background vacuum energy and long range static interaction that we shall discuss in the next section for the membrane theory in $(3+1)$ dimensions. This is because in one spatial dimension a ``bubble'' degenerates into a particle-anti particle pair, moving left and right respectively, and the volume within the bubble is the linear distance between them [15]. Indeed, the main reason for the following discussion is to make it evident that those very kinematical constraints that exist in $(1+1)$ dimensions are intertwined with the production of mass and can be induced just as well in $(3+1)$ dimensions simply by increasing the spatial dimensions of the object: from a 0-brane in $(1+1)$ dimensions to a 2-brane, or bubble, in $(3+1)$dimensions, indeed, to a generic $p$-brane embedded in a target space with $p+2$-dimensions. In other words, the familiar theory of electrodynamics in $(3+1)$ dimensions does not represent a unique generalization of the so called ``electrodynamics in $(1+1)$ dimensions''. A more natural extension, especially from a cosmological standpoint, is the theory of a relativistic membrane coupled to a three index gauge potential. It is in the framework of bubble-dynamics, regardless of the dimensionality of the target space, that the cosmological constant drives the creation of particles of matter, and the engine of that process, at least at the classical level, is the ``topological symmetry breaking'' due to the existence of a boundary in the world history of the membrane.