Electrons in curved low-dimensional systems: spinors or half-order differentials?

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. However, it is known e.g. in superstring theory, that so called half-order differentials can also be used to describe fermions in two dimensions. Therefore, 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, and the whole effect of the background geometry reduces to a universal factor multiplying orthogonal derivative terms and mass terms.