of the complexification of obtained via isomorphisms in , stand in one-to-one correspondence with the real linear representations of The set of all real linear irreducible representations of are thus indexed by a pair . The complex linear ones, corresponding precisely to the complexification of the real linear representations, are of the form , while the conjugate linear ones are the . All others are real linear only. The linearity properties follow from the canonical injection, the far right in , of into its complexification. Representations on the form or are given by real matrices (the latter are not irreducible). Explicitly, the real linear -representations of are where are the complex linear irreducible representations of and their complex conjugate representations. (The labeling is usually in the mathematics literature , but half-integers are chosen here to conform with the labeling for the Lie algebra.) Here the tensor product is interpreted in the former sense of . These representations are concretely realized below. The (m, n)-representations of so(3; 1) Via the displayed isomorphisms in and knowledge of the complex linear irreducible representations of upon solving for and , all irreducible representations of and, by restriction, those of are obtained. The representations of obtained this way are real linear (and not complex or conjugate linear) because the algebra is not closed upon conjugation, but they are still irreducible. Since is semisimple, all its representations can be built up as direct sums of the irreducible ones. Thus the finite dimensional irreducible representations of the Lorentz algebra are classified by an ordered pair of half-integers and , conventionally written as one of where is a finite-dimensional vector space. These are, up to a similarity transformation, uniquely given by where is the -dimensional unit matrix and are the -dimensional irreducible representations of also termed spin matrices or angular momentum matrices. These are explicitly given as where denotes the Kronecker delta. In components, with , , the representations are given by Common representations The representation is the one-dimensional trivial representation and is carried by relativistic scalar field theories. Fermionic supersymmetry generators transform under one of the or representations (Weyl spinors). The four-momentum of a particle (either massless or massive) transforms under the representation, a four-vector. A physical example of a (1,1) traceless symmetric tensor field is the traceless part of the energy–momentum tensor . Off-diagonal direct sums Since for any irreducible representation for which it is essential