Non-chiral Bosonization of Fermions in One Dimension*

An alternative to the conventional approach to bosonization in one dimension that invokes the Dirac equation in 1+1 dimension with chiral ``right-movers'' and ``left-movers'' is proposed that works directly with the bounded parabolic energy bands relevant to Condensed Matter problems. This technique allows us to use a basis different from the plane wave basis that makes this non-chiral approach ideally suited to study Luttinger liquids that have boundary or impurities that break translational symmetry. We provide a simple solution to the electron Green function for the problem of Luttinger liquid (LL) with a boundary and also for a LL with a single impurity. The present method is significantly easier than the g-ology based standard bosonization and other methods that require a combination of RG along with bosonization/refermionization techniques. Our results are broadly consistent with these more established approaches.