= Solution
The <Born rule> depends only on transition probabilities
$$
|\langle\phi|\psi\rangle|^2
$$
between rays. A physical transformation must preserve these probabilities. By <Wigner theorem>, such a ray transformation is induced by a unitary or antiunitary operator. For ordinary continuously connected transformations one chooses the unitary branch, so
$$
\boxed{U(g)^\dagger U(g)=I_{\mathcal H}}.
$$
The states $U(g_1)U(g_2)|\psi\rangle$ and $U(g_1g_2)|\psi\rangle$ represent the same ray. They may therefore differ by a state-independent <quantum phase>:
$$
\boxed{
U(g_1)U(g_2)
=e^{i\phi(g_1,g_2)}U(g_1g_2)}.
$$
Thus quantum transformations form a <projective unitary representation>.
If $g$ is a symmetry of the <Hamiltonian operator> $H$, it also obeys
$$
\boxed{U(g)^\dagger H U(g)=H},
$$
equivalently $[U(g),H]=0$. For a differentiable one-parameter subgroup, write
$$
U(s)=e^{-isQ/\hbar}
$$
with a self-adjoint generator $Q$. Differentiating the symmetry equation at $s=0$ yields
$$
[Q,H]=0.
$$
For a time-independent $Q$, the <Heisenberg picture> equation is
$$
\frac{dQ}{dt}=\frac i\hbar[H,Q]=0.
$$
\b[Hence the generator is a conserved observable. Conversely, a self-adjoint conserved $Q$ commutes with $H$ and its unitary group generates symmetries. This is the <conserved generator of a continuous quantum symmetry>.]
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