International Journal of Theoretical Physics

, Volume 46, Issue 5, pp 1334–1359 | Cite as

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

  • Rainer Dick


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.


spinors fermions in low-dimensional systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baugh, J., Finkelstein, D. R., Galiautdinov A., and Saller, H. (2001). Journal of Mathematical Physics 42, 1489.MATHCrossRefADSMathSciNetGoogle Scholar
  2. Baulieu, L. and Bellon, M. (1987). Physics Letters B 196, 142.CrossRefADSMathSciNetGoogle Scholar
  3. Borel, A. and Hirzebruch, F. (1959). American Journal of Mathematics 81, 315.CrossRefMathSciNetGoogle Scholar
  4. Cardy, J. L. (1988). In Fields, Strings and Critical Phenomena, E. Brezin and J. Zinn-Justin, eds. North-Holland, Amsterdam, p. 169.Google Scholar
  5. Chern, S. S. (1955). Proceedings of the American Mathematical Society 6, 771.Google Scholar
  6. Courant, R. and Hilbert, D. (1962). Methods of Mathematical Physics, Vol. 2, Interscience Publishers, New York.Google Scholar
  7. Dick, R. (1989). Letters in Mathematical Physics 18, 67.MATHCrossRefMathSciNetGoogle Scholar
  8. Dick, R. (1992). Fortschritte der Physik 40, 519.MathSciNetGoogle Scholar
  9. Dick, R. (2003). International Journal of Theoretical Physics 42, 569.MATHCrossRefGoogle Scholar
  10. Hawley, N. S. and Schiffer, M. (1966). Acta Mathematica 115, 199.MATHCrossRefMathSciNetGoogle Scholar
  11. Itzykson, C. and Drouffe, J.-M. (1989). Statistical Field Theory, Vol. 2, Cambridge University Press, Cambridge.MATHGoogle Scholar
  12. Lehto, O. (1977). In Discrete Groups and Automorphic Functions, W. J. Harveym, eds., Academic Press, London, p. 121.Google Scholar
  13. Lüth, H. (2001). Solid Surfaces, Interfaces and Thin Films, 4th ed., Springer-Verlag, Berlin.MATHGoogle Scholar
  14. Milnor, J. (1963). L’Enseignement Mathématique 9, 198.MATHMathSciNetGoogle Scholar
  15. Mönch, W. (2001). Semiconductor Surfaces and Interfaces, 3rd ed., Springer-Verlag, Berlin.MATHGoogle Scholar
  16. Nayak, C. and Wilczek, F. (1996). Nuclear Physics B 479, 529.MATHCrossRefADSMathSciNetGoogle Scholar
  17. Penrose, R. and Rindler, W. (1984). Spinors and Space-time, Vol. 1, Cambridge University Press, Cambridge.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Physics & Engineering PhysicsUniversity of SaskatchewanSaskatoonCanada

Personalised recommendations