# 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

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.

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Thank you. I thought it was an odd coincidence that all the common representations have a dimension of 4 or less and that relativity is defined by a 4 dimensional metric. I thought, maybe, there is some deep reason why higher order representations aren't as useful. –  MadScientist Jun 28 '12 at 15:23
If I accept an answer can people continue to anwer? I think I'll wait for more answers before accepting. I've edited the question and added another too. :) –  MadScientist Jun 28 '12 at 15:32
@BB1, people will be able to answer even if an answer is accepted. Even you can add a new answer. People can even add bounties to your question, which they will award as they see fit. Of course, once the question seems to be fairly well answered, others will feel less motivated to compete with a new answer. Also, at any moment you can change the accepted answer. –  lurscher Jun 28 '12 at 17:44
@Lurscher Thank You. –  MadScientist Jun 28 '12 at 20:56
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

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.

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Thank you. I'm reading the notes now. I've been looking for good notes on Group Theory and related topics. I would like to be able to accept multiple answers. –  MadScientist Jun 28 '12 at 15:26