Quantum Field Theory, Causal Structures and Weyl Transformations

We suggest that in the proper definition, Quantum Field Theories are quantum mechanical system which 'live' on the space of causal structures ${\cal C}$ of spacetime. That is, for any QFT a Hilbert space ${\cal H}$ on which local operators live is assigned not for each Lorentzian metric $g$, but for each causal structure ${\cal C}$. In practice one uses 'conformal frames' which all provide equivalent descriptions of the same QFT. To put it differently, Quantum Field Theories only know about causal structure of spacetime, and not its full Lorentzian metric. The Weyl group and the local RG flow naturally arise when one compares equivalent descriptions in different conformal frames. This is reduced to the usual RG flow of coupling constants when one only compares descriptions in conformal frames related by spacetime-independent Weyl rescalings. We point out that in this picture minimal coupling of a QFT to the metric is inconsistent and comment on the necessary violation of the equivalence principle in the presence of scalars.

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