  We have examined quantum theories of electric magnetic duality invariant
vector fields enjoying classical conformal invariance in 4-dimensional flat
spacetime. We extend Dirac's argument about "the conditions for a quantum field
theory to be relativistic" to "those for a quantum theory to be conformal". We
realize that electric magnetic duality invariant vector theories together with
classical conformal invariance defined in 4-$d$ flat spacetime are still
conformally invariant theories when they are quantized in a way that electric
magnetic duality is manifest.
