and chugging, one obtains which is the (infinitesimal) form for above and yields the relation . To obtain the affine relabelling, write After properly antisymmetrizing, one obtains the generator of symmetries given earlier. Thus, both and can be said to be the "generators of Lorentz transformations", but with a subtle distinction: the first corresponds to a relabelling of points on the affine frame bundle, which forces a translation along the fiber of the spinor on the spin bundle, while the second corresponds to translations along the fiber of the spin bundle (taken as a movement along the frame bundle, as well as a movement along the fiber of the spin bundle.) Weinberg provides additional arguments for the physical interpretation of these as total and intrinsic angular momentum. Other formulations The Dirac equation can be formulated in a number of other ways. Curved spacetime This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the Dirac equation in curved spacetime. The algebra of physical space This article developed the Dirac equation using four vectors and Schrödinger operators. The Dirac equation in the algebra of physical space uses a Clifford algebra over the real numbers, a type of geometric algebra. U(1) symmetry Natural units are used in this section. The coupling constant is labelled by convention with : this parameter can also be viewed as modelling the electron charge. Vector symmetry The Dirac equation and action admits a symmetry where the fields transform as This is a global symmetry, known as the vector symmetry (as opposed to the axial symmetry: see below). By Noether's theorem there is a corresponding conserved current: this has been mentioned previously as Gauging the symmetry If we 'promote' the global symmetry, parametrised by the constant , to a local symmetry, parametrised by a function , or equivalently the Dirac equation is no longer invariant: there is a residual derivative of . The fix proceeds as in scalar electrodynamics: the partial derivative is promoted to a covariant derivative The covariant derivative depends on the field being acted on. The newly introduced is the 4-vector potential from electrodynamics, but also can be viewed as a gauge field, or a connection. The transformation law under gauge transformations for is then the usual but can also be derived by asking that covariant derivatives transform under a gauge transformation as We then obtain