# Do a 1-dimensional conformal theory exist?

can we have in physic or can we speak about 1-d conformal theory in physics ??

for example in this one dimensional theory what would be the generators $x \partial _{x}$ or $\partial _{x}$ ??

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There is an $SL(2,{\mathbb R})$ symmetry in them, or its (e.g. supersymmetric) extensions. The potential $1/r^2$ may occur in such QM models. Because there is only time dimension, there is no $\partial_x$, just $\partial_t$, and similarly there is only $t \partial_t$, formally speaking.