Global conformal invariance in quantum field theory

Abstract

Suppose that there is given a Wightman quantum field theory (QFT) whose Euclidean Green functions are invariant under the Euclidean conformal group ⋍SO e (5,1). We show that its Hilbert space of physical states carries then a unitary representation of the universal (∞-sheeted) covering group * of the Minkowskian conformal group SO e (4, 2)ℤ2. The Wightman functions can be analytically continued to a domain of holomorphy which has as a real boundary an ∞-sheeted covering $$\tilde M$$ of Minkowski-spaceM 4. It is known that * can act on this space $$\tilde M$$ and that $$\tilde M$$ admits a globally *-invariant causal ordering; $$\tilde M$$ is thus the natural space on which a globally *-invariant local QFT could live. We discuss some of the properties of such a theory, in particular the spectrum of the conformal HamiltonianH=1/2(P 0+K 0). 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 $$\mathfrak{S}$$ of to unitary representations of *.