to operate over the field of complex numbers, the direct sum of representations and have particular relevance to physics, since it permits to use linear operators over real numbers. is the bispinor representation. See also Dirac spinor and Weyl spinors and bispinors below. is the Rarita–Schwinger field representation. would be the symmetry of the hypothesized gravitino. It can be obtained from the representation. is the representation of a parity-invariant 2-form field (a.k.a. curvature form). The electromagnetic field tensor transforms under this representation. The group The approach in this section is based on theorems that, in turn, are based on the fundamental Lie correspondence. The Lie correspondence is in essence a dictionary between connected Lie groups and Lie algebras. The link between them is the exponential mapping from the Lie algebra to the Lie group, denoted If for some vector space is a representation, a representation of the connected component of is defined by This definition applies whether the resulting representation is projective or not. Surjectiveness of exponential map for SO(3, 1) From a practical point of view, it is important whether the first formula in can be used for all elements of the group. It holds for all , however, in the general case, e.g. for , not all are in the image of . But is surjective. One way to show this is to make use of the isomorphism the latter being the Möbius group. It is a quotient of (see the linked article). The quotient map is denoted with The map is onto. Apply with being the differential of at the identity. Then Since the left hand side is surjective (both and are), the right hand side is surjective and hence is surjective. Finally, recycle the argument once more, but now with the known isomorphism between and to find that is onto for the connected component of the Lorentz group. Fundamental group The Lorentz group is doubly connected, i. e. is a group with two equivalence classes of loops as its elements. Projective representations Since has two elements, some representations of the Lie algebra will yield projective representations. Once it is known whether a representation is projective, formula applies to all group elements and all representations, including the projective ones — with the understanding that the representative of a group element will depend on which element in the Lie algebra (the in ) is used to represent the