Foundations of Relational Quantum Field Theory I: Scalars

TL;DR

The paper develops a relational quantum-field-theory (RQFT) framework built on operational quantum reference frames (QRFs) for the relativistic Poincaré group, focusing on scalar fields in Minkowski spacetime. It derives relational local observables and relational fields as kernels tied to frame preparations, establishing a relational notion of covariance and several causality concepts that are compatible with (and extend) standard QFT formalisms. By constructing vacuum expectation values and time-ordered correlators, the work shows close ties to Wightman QFT and explores how relational local algebras extend Algebraic QFT, while noting key distinctions from pointwise covariant QFTs. The paper also lays out a comprehensive outlook for extending the framework to spinors, gauge theories, curved spacetimes, and Euclidean settings, and for developing relational dynamics, measurement schemes, and relativistic renormalization. Overall, it provides a rigorous operational and mathematical foundation for rethinking QFT from a relational perspective with potential implications for quantum gravity and beyond. $P_+^$-covariant relational structures and frame-dependent observables form the core toolkit, enabling a principled bridge between relational QRFs and conventional QFT axioms through both Wightman-type kernels and AQFT-inspired algebras. $${ ext{Key relations}}: \\\hat{\\Phi}^{\\mathcal{R}}(\\omega) = \\int_F \\hat{\\phi}_{\\lambda}(x) \\mathrm{d}\\mu^{\\mathrm{E}_R}_{\\omega}(x,\\lambda), \\\text{transforms as } (a,\\Lambda) \\cdot \\\hat{\\Phi}^{\\mathcal{R}}(\\omega) = \\\hat{\\Phi}^{\\mathcal{R}}(\\omega \\cdot (a,\\Lambda)^{-1}),$$ and ${W}_n^{(\\Omega,\\mathcal{R})}$ recovers Wightman-like structure under suitable frame choices.}

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 Poincaré-covariant (quantum) frame observables defined over the space of (classical) 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 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, 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, and show how the properties of algebras generated by relational local observables suitably extend the core axioms of Algebraic QFT. We finish with an extensive outlook describing a number of further research directions. This work is an early step in revisiting the mathematical foundations of QFT from a relational and operational perspective.

Figures (3)

• Figure 1: Space of inertial frames for which every point represent a different viewpoint from which physical systems can be described. A classical inertial reference frame can be thought of as a Dirac delta distribution over this space, while an oriented relativistic quantum reference frame gives a Born probability distribution over it. The measure projected to $\mathbb{M}$ is a Born probablility measure of a marginal POVM $\mathsf{F}_\mathcal{R}$ as defined in Sec. \ref{['sec: covariance']} below.
• Figure 2: A comparison of weak and strong microcausality: pointwise commutation is shown through the circles. On the left, $\phi$ is strongly $\mathfrak{S}_\mathcal{R}$-microcausal: for any two preparations, the relational local quantum fields commute over spacelike-separated points. On the right, $\phi$ is weakly $\mathfrak{S}_\mathcal{R}$-microcausal: relational local quantum fields commute (over spacelike-separated points) only if the preparations are $\mathcal{R}$-spacelike separated.
• Figure 3: Pictorial representations of $\mathfrak{S}_\mathcal{R}$-spacelike resolution of frames and $\sigma(\mathfrak{S}_\mathcal{R})$-spacelike separation of regions.

Theorems & Definitions (92)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Theorem 3.1
• proof
• Definition 3.2
• Lemma 3.3