The above can be further sharpened by examining the situation in ( p , q ) spatial dimensions. In this case, the complexified spin group Spin C ⁡ ⁡ ( p , q ) has a double covering by SO ⁡ ⁡ ( p , q ) × × S 1 with S 1 ≅ ≅ U ( 1 ) the circle. The implication is that SO ⁡ ⁡ ( p , q ) encodes the generalized Lorentz transformations (of course), while the circle can be identified with the U ( 1 ) action of the gauge group on electric charges. That is, the gauge-group action of the complexified spin group on a Dirac spinor can be split into a purely-real Lorentzian part, and an electromagnetic part. This can be further elaborated on non-flat (non-Minkowski-flat) spin manifolds. In this case, the Dirac operator acts on the spinor bundle. Decomposed into distinct terms, it includes the usual covariant derivative d + A . The A field can be seen to arise directly from the curvature of the complexified part of the spin bundle, in that the gauge transformations couple to the complexified part, and not the real-spinor part. That the A field corresponds to the electromagnetic potential can be seen by noting that (for example) the square of the Dirac operator is the Laplacian plus the scalar curvature R (of the underlying manifold that the spinor field sits on) plus the (electromagnetic) field strength F = d A . For the Majorana case, one has only the Lorentz transformations acting on the Majorana spinor; the complexification plays no role. A detailed treatment of these topics can be found in Jost while the ( p , q ) = ( 1 , 3 ) case is articulated in Bleeker. Unfortunately, neither text explicitly articulates the Majorana spinor in direct form.