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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 $y > 0$).

I've seen some papers about the subject, and it seems to be that a small group of people study conformal QM, probably because it's relatively easy to obtain exact results, such as closed-form expressions for conformal blocks. (This is a hard problem in general dimensions - I can give references if you're interested.) Mathematically there's no hope of extending these results to higher dimensions, but maybe these people try to get some intuition from this simple case.

You can also informally rephrase the subject by thinking of conformal QM as the study of quantum systems with no characteristic timescales, where all temporal correlation functions behave as power laws. (Of course, this is very crude, since scale invariance is less restrictive than conformal invariance.)

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