Are there any known potentially useful nontrivial irreducible representations of the Lorentz Group $O(3,1)$ of dimension bigger than 4? Examples? Are there any known potentially useful, nontrivial, irreducible representations of the Lorentz Group $O(3,1)$ of dimension more than $4$? Examples? A $5$-dimensional representation? EDIT: Is there some deep reason why higher-dimensional representations (other than infinite-dimensional representations) are less useful?
 A: Irreducible representations of the Lorentz group are uniquely described by $(j_L,j_R)$ where both numbers belong to the set $\{0,1/2,1,3/2,2,\dots\}$. The dimension of the representation is simply
$$ d = (2j_L+1) (2j_R+1) $$
It's not hard to see that $d=5$ only occurs for $(j_L,j_R)=(2,0)$ or $(0,2)$. I haven't encountered such particles or fields in practice but it's possible to construct them.
A: In quantum physics, we are interested in unitary representations, because they preserve the Hilbert space norm. Most of the representations of the Lorentz group of interest in quantum
physics are infinite dimenional. The reason for that is that in the case of noncomapct groups, unitarity implies infinite dimensionality. Examples of such representations are the actions of the Lorentz group on the Hilbert spaces of solutions of the Klein-Gordon and Dirac equations, which are both infinite dimensional. For the case of the Klein gordon
equation, please see equation (2.59) in the following lecture notes by Arthur Jaffe.
A: All unitary irreducible representations of $SO(3,1)$ are necessarily infinite-dimensional.  Indeed, the classification in terms of $(j_R,j_L)$ is really a classification of the complexification of $so(3,1)$, which is isomorphic to $su(2)\oplus su(2)$: as finite-dimensional matrices, the generators of boost, when unwrapped back from $su(2)\oplus su(2)$, cannot be made into hermitian simultaneously with the generators of $so(3)$.
While finite dimensional non-unitary representations of the complexification of $so(3,1)$ are all labelled by two real numbers $(j_R,j_L)$, infinite dimensional unitary irreps are not.  While irreps are always labelled by two integers, some representations with a ground state are labelled by a real integer labelling the smallest representation of $so(3)$ in the irrep, and a purely imaginary number not necessarily integer.
