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.