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Is there a way to see the conformal symmetry in momentum space in a CFT? I mean if I can recover the conformal group in some way in momentum space.

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  • $\begingroup$ This paper should be helpful. $\endgroup$ – MannyC Apr 26 '19 at 3:48
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The generator of special conformal transformation act, schematically, as $$ K\sim x^2\partial_x+x\partial_x. $$ In momentum space this becomes $$ K\sim p\partial_p^2+p\partial_p. $$ Since $K$ is a second order differential operator, $e^{\lambda K}$ does not act in a local, geometrical way on functions.

So while it is possible to check infinitesimal conformal invariance in momentum space by acting with the second-order differential operator, there is no simple geometric meaning to conformal invariance in momentum space.

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