Foundations of Relational Quantum Field Theory — scalars

We develop foundations for a relational approach to quantum field theory (RQFT) based on the operational quantum reference frames (QRFs) framework considered in a relativistic setting. We focus on scalar fields in Minkowski spacetime and discuss the emergence of relational local observables and pointwise fields from the consideration of Poincaré-covariant frame observables defined over the space of inertial reference frames. We recover a relational notion of Poincaré covariance, with transformations on the system directly linked to the state preparations of the QRF. We introduce various causality conditions which mirror standard Einstein causality and microcausality, now seen in a relational context. The theory makes direct contact with established foundational approaches to QFT: the vacuum expectation values derived within our framework reproduce many of the essential properties of Wightman functions, and we compare the proposed formalism with Wightman QFT with the frame smearing functions describing the QRF's localisation uncertainty playing the role of the Wightmanian test functions. We show how the algebras generated by relational local observables satisfy all of the core axioms of Algebraic QFT. This work is an early step in revisiting the mathematical foundations of QFT from a relational and operational perspective.