# SO(3) x SU(2) Symmetry of the Hamiltonian

I have a question converning multiplets when describing atoms.

Let $H=\sum\limits_{k=1}^{N} (p_{k}^2 - \frac{Z}{|x_{k}|} + \sum\limits_{i<k}^{1..N} \frac{1}{|x_{i} - x_{k}|}$ be a (self-adjoint) operator on the Hilbertspace of totally antisymmetrized wave functions of N particles H^(N)_{a}.

In my notes it is written that SO(3) x SU(2) is a symmetry group of this hamiltonian. The group SO(3) x SU(2) consists of pairs (R,A)$\in SO(3) x SU(2)$ and has a unitary representation U on H^(N)_{a} given by:

$(U(R,A)\phi)(x_{1} s_{1} ,...., x_{N} s_{n}) = As_{1}s'_{N} ... A s_{N} s'_{N}\phi(R^{-1}x_{1} s_{1} s'_{1} ... R^{-1} x_{N} s'_{N})$

Here comes my problem. I don't really understand this definition. It's not given in my notes what the s' are and I don't understand what $As_{1}s'_{N}$ means...A vague guess might be that you multiply the vector $s_{1}$ in $C^{2}$ is multiplied by the matrix A in SU(2) and then scalar mulitplied by $s'_{N}$.

Butit's not clear to me..Can anyone help me to understand this definition better?

In addition to that I thought that if a group G is a symmetry group of a Hamiltonian H then it means that [H,U]=0 for ANY representation U of the group G. But in the above example one only gives one particular representation.

I'd be really happy if someone could help me. Thanks in advance.

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Hopefully the $SO(3)$ part of the symmetry is clear: it's just a rotation that acts on the position part of the wavefunction.
The $SU(2)$ part comes from the fact that these are (supposedly) spin 1/2 particles. Since each of these is described by two spin states we act on them by using a special unitary matrix acting on a two-dimensional complex space ${\mathbb C}^2$. But this is precisely a matrix from $SU(2)$ (in other words, this is a tautological representation where the matrix is represented by itself). Now, when acting on $N$ particles we take a tensor product of these representations. Therefore we get a representation ${\mathbb C}^2 \otimes \cdots \otimes {\mathbb C}^2$. I think what you've written must contain some typo but generally it has a correct structure of containing the matrix $A$ $N$ times and each time it acts on a pair of two spin-vectors.
Perhaps you were confused by some standard example like Coulomb potential problem. In this case every representation of $SO(3)$ (indexed by a non-negative integer) is realized on the Hilbert space. But it's easy to construct a Hilbert space where this fails, e.g. the ${\mathbb C}^2$ above only carries spin 0 and spin 1/2 representations of $SU(2)$ and no other (since reps with higher spin are more than two-dimensional).