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arxiv: 2507.21601 · v3 · submitted 2025-07-29 · 🪐 quant-ph · hep-th· math-ph· math.MP

Foundations of Relational Quantum Field Theory I: Scalars

Samuel Fedida , Jan G{\l}owacki This is my paper

Pith reviewed 2026-05-19 02:45 UTC · model grok-4.3

classification 🪐 quant-ph hep-thmath-phmath.MP
keywords relational quantum field theoryquantum reference framesscalar fieldsPoincaré covarianceWightman functionsalgebraic quantum field theorycausality in QFT
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The pith

Relational scalar quantum fields arise from Poincaré-covariant quantum reference frames and satisfy causality for meaningful preparations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops foundations for relational quantum field theory by considering scalar fields in Minkowski spacetime using an operational framework of quantum reference frames. It constructs relational local observables and pointwise fields from Poincaré-covariant frame observables defined over classical inertial frames. This setup recovers a relational notion of Poincaré covariance where system transformations link directly to the quantum reference frame's state preparations. Various causality conditions are introduced and analyzed, leading to an explicit example of a covariant scalar relational quantum field that remains causal under operationally meaningful preparations. The approach connects to standard quantum field theory by showing that vacuum expectation values reproduce essential properties of Wightman functions, with frame smearing functions playing the role of test functions, and extends the axioms of algebraic quantum field theory.

Core claim

We construct an explicit example of a covariant scalar relational quantum field which is causal relative to operationally meaningful preparations of a relativistic QRF, recovering relational Poincaré covariance and reproducing properties of Wightman functions with frame localization uncertainty as test functions.

What carries the argument

Poincaré-covariant quantum frame observables defined over the space of classical inertial reference frames, from which relational local observables emerge by linking transformations on the system to the QRF state preparations.

If this is right

  • Relational Poincaré covariance follows directly from the state preparations of the quantum reference frame.
  • Causality conditions hold for the constructed scalar field relative to meaningful QRF preparations.
  • Vacuum expectation values match essential properties of Wightman functions when using frame smearing as test functions.
  • The algebras generated by relational local observables extend the core axioms of algebraic quantum field theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the relational framework holds, standard QFT could be recovered in the limit where the quantum reference frame's localization uncertainty approaches zero.
  • This suggests potential extensions to interacting fields or fermionic fields by similar relational constructions.

Load-bearing premise

The operational quantum reference frames framework extends to a relativistic setting allowing Poincaré-covariant quantum frame observables over classical inertial frames from which relational local observables emerge.

What would settle it

A calculation showing that the commutator of two relational local observables at spacelike separation does not vanish for the explicit covariant scalar field example would falsify the causality claim.

Figures

Figures reproduced from arXiv: 2507.21601 by Jan G{\l}owacki, Samuel Fedida.

Figure 1
Figure 1. Figure 1: Space of inertial frames for which every point represent a different viewpoint from which physical systems [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper develops foundations for a relational approach to quantum field theory (RQFT) for scalar fields in Minkowski spacetime, using the operational quantum reference frames (QRFs) framework in a relativistic setting. It claims that relational local (bounded) observables and pointwise fields emerge from Poincaré-covariant quantum frame observables defined over the space of classical inertial reference frames. The work recovers a relational notion of Poincaré covariance linked directly to QRF state preparations, introduces and analyzes various causality conditions, constructs an explicit example of a covariant scalar relational quantum field that is causal relative to operationally meaningful QRF preparations, shows that the derived vacuum expectation values reproduce essential properties of Wightman functions (with frame smearing functions playing the role of test functions), and demonstrates that algebras generated by the relational local observables extend the core axioms of algebraic QFT (AQFT).

Significance. If the central constructions and comparisons hold, this provides a novel operational and relational foundation for QFT that integrates QRFs to derive covariance, locality, and causality without absolute background structures. The explicit example, direct reproduction of Wightman properties, and extension of AQFT axioms are notable strengths that could offer new tools for addressing foundational questions in quantum field theory, particularly regarding the emergence of standard structures from relational setups.

major comments (1)
  1. [explicit example construction] The section constructing the explicit covariant scalar relational quantum field: while the abstract states that causality holds relative to operationally meaningful preparations of a relativistic QRF, the load-bearing step of verifying the introduced causality conditions against the field operators is not sufficiently detailed to confirm the claim without additional assumptions on the QRF state preparations or smearing functions.
minor comments (2)
  1. [Wightman comparison] The notation and definitions for frame smearing functions and their relation to Wightman test functions should be introduced with explicit equations early in the manuscript to improve readability of the comparison section.
  2. [introduction or conclusions] A brief discussion of the limitations of restricting to scalar fields and classical inertial frames would help contextualize the scope of the relational Poincaré covariance recovery.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for the recommendation of minor revision. We address the single major comment below in a point-by-point manner.

read point-by-point responses

  1. Referee: [explicit example construction] The section constructing the explicit covariant scalar relational quantum field: while the abstract states that causality holds relative to operationally meaningful preparations of a relativistic QRF, the load-bearing step of verifying the introduced causality conditions against the field operators is not sufficiently detailed to confirm the claim without additional assumptions on the QRF state preparations or smearing functions.

    Authors: We thank the referee for this observation. The explicit construction of the covariant scalar relational quantum field appears in the dedicated section of the manuscript, where the operators are defined via the Poincaré-covariant QRF observables and causality is established by verifying that the relevant commutators (or correlation functions) vanish for spacelike separations when the QRF is prepared in states whose smearing functions have appropriate support. These support properties are tied directly to the operational localization uncertainty of the frame. We acknowledge, however, that the intermediate algebraic steps in this verification are presented at a level of detail that may require the reader to fill in some intermediate calculations. In the revised manuscript we will expand this section with an explicit step-by-step derivation of the commutator vanishing, together with a clear statement of the minimal support conditions on the smearing functions that suffice for the result. No new assumptions beyond those already stated in the operational framework will be introduced. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends the operational QRF framework to relativistic Poincaré-covariant frame observables over inertial frames, from which relational local observables and pointwise fields are derived for scalar fields in Minkowski space. Relational covariance is obtained by linking system transformations directly to QRF state preparations; causality conditions are introduced and checked against operationally meaningful preparations; an explicit covariant scalar relational field is constructed and shown to reproduce key Wightman function properties (with frame smearing replacing test functions) and to extend AQFT axioms. All steps are presented as direct consequences of the operational setup and compared independently to established structures, with no reduction of predictions to fitted inputs, no load-bearing self-citation chains, and no self-definitional loops visible in the claimed derivation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on the prior operational QRF framework as a domain assumption and standard QFT structures for comparisons, with frame smearing functions introduced to match test functions.

free parameters (1)
  • frame smearing functions
    Describe the QRF's localisation uncertainty and are used to play the role of Wightmanian test functions in comparisons.
axioms (1)
  • domain assumption The operational quantum reference frames framework extends consistently to relativistic Poincaré-covariant settings.
    Invoked to define frame observables over inertial frames and derive relational QFT structures.

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Forward citations

Cited by 1 Pith paper

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