Chapter 6 Spinors

  • Mark J. D. Hamilton
Part of the Universitext book series (UTX)


We saw in the previous chapter that gauge fields correspond to gauge bosons and are described by 1-forms or, dually, vector fields. In physics, there exist of course also matter particles, like electrons, quarks and neutrinos. These particles are fermions and are described by spinor fields (spinors). Like vector fields or tensor fields, spinors have a specific transformation behaviour under rotations. However, spinors do not transform directly under the orthogonal group, but under a certain double covering, called the (orthochronous) spin group. In the case of Minkowski spacetime, rotations correspond to Lorentz transformations. The corresponding spin group is the Lorentz spin group.

In many mathematical expositions the discussion of spinors is restricted to the Riemannian case, because in most situations, manifolds in differential geometry carry a Riemannian metric. The pseudo-Riemannian case, like the case of Minkowski spacetime, is discussed less often, even though it is very important for physics. Since we are ultimately interested in applications of differential geometry and gauge theory to physics, it seems worthwhile to study orthogonal groups, Clifford algebras, spin groups and spinors from a mathematical point of view also in the Lorentzian and general pseudo-Riemannian case.


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Authors and Affiliations

  • Mark J. D. Hamilton
    • 1
  1. 1.Department of MathematicsLudwig-Maximilian University of MunichMunichGermany

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