# 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?

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Are you talking about a 5 dimensional representation, or a representation of the 5 dimensional Lorentz group? All the reps of the 4d Lorentz group are useful, as are most of their higher dimensional analogs up to 11 dimensions. – Ron Maimon Jun 28 '12 at 20:17
Ok--- you explained in a comment to Lubos's answer. the spin 2 representation is traceless symmetric space-time tensor with 5 independent real components. – Ron Maimon Jun 28 '12 at 20:20
Look at the 12-dimensional Rarita Schwinger representation for (massive) gravitini: en.wikipedia.org/wiki/… . You can't do s. broken supergravity without it. – Cosmas Zachos Feb 21 at 23:07

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.
Slighty more precisely, finite-dimensional complex irreps of the Lie algebra of the Lorentz group are parameterized by spins $j_L$ and $j_R$. Every one of these gives a different representation of the double cover of the Lorentz group, labelled by the left & right integers $j_L$ and $j_R$. The other finite dimensional reps of the Lorentz group come from C, P, & T\$. None of which detracts from the main point. – user1504 Jun 28 '12 at 21:57