# Conformal Quantum Mechanics

I heard the term Conformal Quantum Mechanics used today.

1. What exactly does this mean?

2. Why would one want to study this?

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It's slang for conformal field theory for manifolds of $D=1$ dimensions. (If you take the dimension to be time, 1-D QFT describes the time evolution of a system living in zero spatial dimensions, i.e. at a single point, so it's not really field theory but QM.)
It is special in the sense that normally, CFT representations are specified by two labels (the operator dimension $\Delta$ and spin $l$), but in 1D there is no spin, so you only have scaling dimensions. Also, all 1-D manifolds are trivially conformally flat (= they can be rescaled to obtain a Euclidean line). This fails in $D \geq 3$ dimensions. In 2D, this point is a bit subtle (you can have conformal manifolds with a boundary, such as the half plane with $y > 0$).