= Positive energy unitary representation of the circle
{title2=$U_{e^{it}}=e^{itL}$, $\inf\sigma(L)>-\infty$}
A strongly continuous <unitary representation> of the circle has integer spectral weights: $H$ is the orthogonal sum of spaces on which $U_z=z^m$. Positive energy means that these weights are bounded below, equivalently its self-adjoint generator has spectrum bounded below. Multiplication by an integer character shifts the lower bound to zero. A unitary operator with a nonzero rotation weight cannot exist in its operator algebra: repeated application of it or its inverse would move a nonzero vector to arbitrarily low energy.
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