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
*.
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Lüscher, M., Mack, G. Global conformal invariance in quantum field theory. Commun.Math. Phys. 41, 203–234 (1975). https://doi.org/10.1007/BF01608988
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DOI: https://doi.org/10.1007/BF01608988