Foundations of Relational Quantum Field Theory I: Scalars

TL;DR

This paper develops a relational framework for scalar quantum field theory using operational quantum reference frames, connecting foundational QFT concepts with relational and operational principles.

Contribution

It introduces a novel relational approach to QFT based on quantum reference frames, providing new mathematical and conceptual foundations for scalar fields in Minkowski spacetime.

Findings

Derived relational local observables and fields from quantum reference frames.

Reproduced key properties of Wightman functions within the relational framework.

Extended algebraic QFT axioms to relational local observables.

Abstract

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. Unlike other efforts in combining QFT with QRFs, we use the latter to provide novel mathematical and conceptual foundations for the former. We focus on scalar fields in Minkowski spacetime and discuss the emergence of relational local (bounded) observables and (pointwise) fields from the consideration of Poincar\'e-covariant (quantum) frame observables defined over the space of (classical) inertial reference frames. We recover a relational notion of Poincar\'e covariance, with transformations on the system directly linked to the state preparations of the QRF. We introduce and analyse various causality conditions, and construct an explicit example of a covariant scalar relational quantum field which is causal relative to operationally meaningful preparations of a relativistic QRF. The theory makes direct contact with established foundational approaches to QFT. We demonstrate that the vacuum expectation values derived within our framework reproduce many of the essential properties of Wightman functions and carry out a detailed comparison of 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 also show how the properties of algebras generated by relational local observables suitably extend 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.