- Non-chiral Bosonization of Fermions in One Dimension
In this chapter, an alternative to the conventional approach to bosonization in one dimension that invokes the Dirac equation in 1+1 dimension with chiral ‘rightmovers’ and ‘left-movers’ is proposed (the conventional approach is discussed, e.g., in the text book by Giamarchi, Quantum Physics in One Dimension, Oxford). 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 a boundary(s) or impurities that break translational symmetry. We provide a simple solution to the electron Green function for the problem of a 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 (renormalization group) along with bosonization/refermionization techniques. This method is superior because it reproduces, as a limiting case, the exact closed formula for the (random phase approximation) RPA Green function of free fermions with the single impurity without mutual interactions obtainable trivially using Fermi algebra, whereas the conventional bosonization/RG approaches are not able to do so.