  S-matrix elements in flat space can be obtained from a large AdS-radius limit
of certain CFT correlators. We present a method for constructing CFT operators
which create incoming and outgoing scattering states in flat space. This is
done by taking the flat limit of bulk operator reconstruction techniques. Using
this method, we obtain explicit expressions for incoming and outgoing U(1)
gauge fields. Weinberg soft photon theorems then follow from Ward identites of
conserved CFT currents. In four bulk dimensions, gauge fields on AdS can be
quantized with standard and alternative boundary conditions. Changing the
quantization scheme corresponds to the S-transformation of SL(2,Z)
electric-magnetic duality in the bulk. This allows us to derive both, the
electric and magnetic soft photon theorems in flat space from CFT physics.
