finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group . Then where is the compact subgroup of consisting of unitary transformations of . The kernel of is a normal subgroup of . Since is simple, is either all of , in which case is trivial, or is trivial, in which case is faithful. In the latter case is a diffeomorphism onto its image, and is a Lie group. This would mean that is an embedded non-compact Lie subgroup of the compact group . This is impossible with the subspace topology on since all embedded Lie subgroups of a Lie group are closed If were closed, it would be compact, and then would be compact, contrary to assumption. In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of and used in the construction are Hermitian. This means that is Hermitian, but is anti-Hermitian. The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm. Restriction to SO(3) The representation is, however, unitary when restricted to the rotation subgroup , but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an representation have -invariant subspaces of highest weight (spin) , where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) is -dimensional. So for example, the (, ) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively. Since the angular momentum operator is given by , the highest spin in quantum mechanics of the rotation sub-representation will be and the "usual" rules of addition of angular momenta and the formalism of 3-j symbols, 6-j symbols, etc. applies. Spinors It is the -invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the representation has spin if is half-integral. The simplest are and , the Weyl-spinors of dimension . Then, for example, and are a spin representations of dimensions and respectively. According to the above paragraph, there are subspaces with spin both and in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under . It cannot be ruled out in general, however, that representations with multiple