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Suppose that there is given a Wightman quantum field theory (QFT) whose Euclidean Green functions are invariant under the Euclidean conformal group (5~SO e (5, 1). We show that its Hubert space of physical states carries then a unitary representation of the universal (oo-sheeted) covering group (5* of the Minkowskian conformal group SO e (4, 2)/Z 2 . The Wightman functions can be analytically continued to a domain of holomorphy which has as a real boundary an oo-sheeted covering M of Minkowski-space M 4 . It is known that 05* can act on this space M and that M admits a globally @*-invariant causal ordering; M is thus the natural space on which a globally (5*-invariant local QFT could live. We discuss some of the properties of such a theory, in particular the spectrum of the conformal Hamiltonian H = i(P° + K°).As a tool we use a generalized Hille-Yosida theorem for Lie semigroups. Such a theorem is stated and proven in Appendix C. It enables us to analytically continue contractive representations of a certain maximal subsemigroup 6 of (5 to unitary representations of (5*.

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