Quantum Field Theory, Causal Structures and Weyl Transformations

TL;DR

This paper proposes a new perspective on quantum field theories as systems defined on the space of causal structures of spacetime, emphasizing their insensitivity to full metric details and exploring implications for Weyl transformations and RG flows.

Contribution

It introduces a framework where QFTs are associated with causal structures rather than metrics, highlighting the role of Weyl transformations and local RG flows in this context.

Findings

QFTs depend on causal structures, not full spacetime metrics

Weyl group and local RG flow emerge naturally in this framework

Minimal coupling to the metric is inconsistent, affecting the equivalence principle

Abstract

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.