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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?

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Any dot product of two vectors is a scalar, which is to say: the dot product of two objects that rotate like vectors, behaves like a scalar and is invariant under rotations. This is a geometric property.

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.

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