The QM/NCG Correspondence

The correspondence between quantum mechanics and noncommutative geometry is illustrated in the context of the noncommutative ${\mathrm{A}\mathrm{d}\mathrm{S}}_{\theta }^2\mathrm{/}\mathrm{C}\mathrm{F}{\mathrm{T}}_1$ duality where ${\mathrm{C}\mathrm{F}\mathrm{T}}_1$ is identified as conformal quantum mechanics. This model is conjectured to describe the gauge/gravity correspondence in one dimension. Applications to quantum black holes and emergent geometry/gravity are also discussed.