Representation Theory of the Lorentz Group

Abstract

We now come to fulfill the program formulated in chap. 6: to find and classify all quantities that behave linearly under Lorentz transformations just as tensors door in other words, to construct all finite-dimensional representations of the Lorentz group. From the commutation relations one reads off the adjoint representation, which happens to be identical with the representation in the space of sixtors (antisymmetric tensors of degree two) considered in sect. 6.5. From it one deduces that its Lie algebra is semisimple in the sense of the definition given in sect. 7.4. (The point here is the semisimplicity of its complexification: for the real Lorentz group, we already demonstrated even simplicity on the group level in appendix 2 to sect. 6.3.) It is an important theorem of H. Weyl that the finite-dimensional representations of semisimple Lie groups are fully reducible,¹ so that for their classification it suffices to find all irreducible representations. There result two fundamental representations, from which all others may be obtained by reducing tensor products: they are 2-dimensional and 2-valued and are again called spinor representations. From them, we develop some spinor algebra and give the relation to tensors. Finally we consider representations of the full Lorentz group.