Half-Differentials versus Spinor Formalism for Fermions in Low-Dimensional Systems

Abstract

The description of fermions on curved manifolds or in curvilinear coordinates usually requires a vielbein formalism to define Dirac γ-matrices or Pauli matrices on the manifold. Derivatives of the vielbein also enter equations of motion for fermions through the spin connection, which gauges local rotations or Lorentz transformations of tangent planes.

The present paper serves a dual purpose. First we will see how the zweibein formalism on surfaces emerges from constraining fermions to submanifolds of Minkowski space. In the second part, I will explain how in two dimensions the zweibein can be absorbed into the spinors to form half-order differentials. The interesting point about half-order differentials is that their derivative terms along a two-dimensional submanifold of Minkoski space look exactly like ordinary spinor derivatives in Cartesian coordinates on a planar surface, although there is a prize to pay in the form of a local mass term. The advantage of half-order differentials is that they allow for the use of the conformal field concept of conformal field theory even in low-dimensional fermion systems without conformal symmetry.