# Infinitesimal rotation of classical fields: why are rotation group representations important?

I understand that $SO(3)$ representations are important in quantum physics, because eigenspaces of the Hamiltonian are irreps of $SO(3)$ if it is part of the symmetry group. But I don't see the reason why the irreps should be essential for classical field theories.

My professor talked about the conservation of angular momentum because of Noether's theorem. The infinitesimal symmetry is

$$x^\mu \mapsto x^\mu + \omega^{\mu\nu}x_\nu$$

with an asymmetric matrix $\omega$, so the field transforms as

$$\phi_i(x) \mapsto {S_i}^j(\omega)\phi_j(x^\mu + \omega^{\mu\nu}x_\nu)$$

where $S$ is the representation of $SO(3)$, which is trivial for scalar fields.

• Why are representations necessary for such an infinitesimal symmetry? For translations, the $x^\mu$ transformation was sufficient, no talk about representations.
• If the trivial irrep is for scalars, are all the other (real) irreps for vectors, and their tensor products for higher tensors?

The lowest non-trivial irreducible representation (labelled as "spin-1" in the usual diction) of $\mathrm{SO}(3)$, which is its fundamental representation, is the vector representation (let's call its representation space $V$). The tensor representations of rank $(k,l)$ are now $V\otimes\dots\otimes V\otimes V^*\otimes\dots\otimes V^*$ with $k$ factors of $V$ and $l$ factors of $V^*$. They are not irreducible, and split at least into the trace, symmetric traceless tensor and antisymmetric tensor representations.
• Thanks! Quick and very helpful. So the index in $\phi_i$ in the second formula does not necessarily run through $\{1,2,3\}$, it could also run through $\{1\}$ (for a scalar) or through cartesian products of $\{1,2,3\}$ for tensors? But for tensors, this would be a strange notation, wouldn't it?
• Please note my question on the $SO(3) irreducible representations on MSE: math.stackexchange.com/questions/1425825/… The third bullet point was inspired by your last paragraph. – Bass Commented Sep 8, 2015 at 14:30 • @BastianTreichler: You might be interested in this post of mine, where I show how to abstractly describe the irreducible representations from the algebra of ladder operators alone. Your first bullet point over there is incorrect, the higher spin representations do not act on$\mathbb{R}^3$, but on$\mathbb{R}^{2n+1}$(my ladder argument for the other post shows that there must be eigenvectors for every integer from$-n$to$n\$). Commented Sep 8, 2015 at 14:39