Conformal quantum mechanics

Haven't gotten to this yet. Sources:

1. Qualls CFT lectures here
2. Model appears to have been introduced here
3. discussion on the non-invariance of the ground state here

The most general Lagrangian consistent with time and scale invariance is given by % \begin{align} L = \frac{1}{2} \dot{Q}^2 - \frac{g^2}{2 Q^2} \end{align} % This theory exhibits an $SL(2, \mathbb{R} \sim SO(2,1)$ symmetry. However, it does not have a normalizable ground state which is annihilated by all three of the generators of $SO(2,1)$