Implications of conformal symmetry in quantum mechanics

In conformal quantum mechanics with the vacuum of a real scaling dimension and with a complete orthonormal set of energy eigenstates, which is preferable under the unitary evolution, the dilatation expectation value between energy eigenstates monotonically decreases along the flow from the UV to the IR. In such conformal quantum mechanics, there exist bounds on scaling dimensions of the physical states and the gauge operators.

Figure 1

The $D$ function. It is defined in the range $0\le E^2\le {\mathcal{C}}_2+\frac{1}{4}$ and decreases monotonically from the UV to the IR.

Figure 2

The bound on $(d,\mathrm{\Delta })$ in conformal quantum mechanics. In the orange region, the primary operators with the dimension $\mathrm{\Delta }$ and the vacuum states with the dimension $d$ are allowed, and the red line characterizes the charged physical states coupled to the gauge operators of ${\delta }_Q$. Without the favored condition (5.19) for the unitary evolution of the states, the green region is also allowed.