======Conformal quantum mechanics======

{{tag>1d}}

Haven't gotten to this yet. Sources:
  - Qualls CFT lectures [[https://arxiv.org/pdf/1511.04074.pdf|here]]
  - Model appears to have been introduced [[https://inspirehep.net/literature/108211|here]]
  - discussion on the non-invariance of the ground state [[https://arxiv.org/pdf/1106.0726.pdf|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 $SO(2,1)$