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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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Yes, such theories exist. They're known as "conformal quantum mechanics". See

http://scholar.google.com/scholar?q=%22conformal+quantum+mechanics%22&hl=en&lr=&btnG=Search
http://scholar.google.com/scholar?q=ads2-cft1&hl=en&lr=&btnG=Search

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.

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