# Hamiltonian invariant under rotation of spins

Two spin half particles interact via the Hamiltonian $$H=J \vec{S}_{1}\cdot\vec{S}_{2}$$ It is said that this Hamiltonian is invariant under uniform rotation of spins. I don't see how this is apparent. What exactly is being rotated? Is it the states which live in the Hilbert space of $$S_1\otimes S_2$$? Or is it the spin vectors $$S_1$$ and $$S_2$$? Also, are they being rotated by the same amount?

Here, each spin is a vector (that is - they rotate like vectors) under rotations in spin space, generated by a unitary transformation $$U_{1,2} = e^{-i\theta_j S^j_{1,2}}$$ and as they commute and live in different orthogonal subspaces, we can rotate them both in a similar manner with $$U = e^{-i\theta_j (S^j_{1}+S^{j}_2)}$$ The result of this unitary transformation changes each spin, but leaves the Hamiltonian unchanged, as it is a dot product. This means that each matrix element $$\langle \psi | H | \phi \rangle$$ is unchanged under $$\langle \psi |U^{\dagger} H U| \phi \rangle$$. In this formulation, the question of whether the states are rotated or the operator transformed is a matter of choice.