Relational Local Observables in QFT

• Relational local observables are operator-valued quantities defined relative to quantum reference frames, offering a rigorous alternative to background-dependent localization.
• They leverage measure-theoretic and functional analytic tools to replace traditional test functions with frame-dependent smearing, preserving transformation and causality properties.
• This framework reinterprets key quantum field theory concepts, paving the way for extensions to gauge fields, curved spacetimes, and quantum gravity measurement theories.

Relational local observables are operator-valued quantities in quantum field theory, quantum gravity, and related areas, defined not with respect to a fixed background spacetime or coordinate system but rather in relation to quantum or classical reference frames. Such observables are constructed to ensure invariance under the relevant group of symmetries (such as the Poincaré group in relativistic settings or the diffeomorphism group in gravity) and to encode operationally meaningful information about localized physical processes. They are central to current foundational approaches in both quantum field theory and quantum gravity as they provide a mathematically rigorous, physically interpretable framework for the localization of information without reference to absolute background structures.

1. Mathematical Framework for Relational Local Observables

The mathematical basis for relational local observables in the context of relativistic quantum field theory (QFT) utilizes operational quantum reference frames (QRFs) prepared in particular states and associated with probability operator-valued measures (POVMs) on the classical frame space. For scalar fields on Minkowski spacetime, the construction proceeds as follows (Fedida et al., 29 Jul 2025):

• Given a bounded system operator $\varphi\in\mathcal{B}(\mathcal{H})$, a relational local observable is defined as

$$\Phi^R(\omega) = \int_{F} \varphi_\lambda(x)\, d\mu^{E_R}_\omega(x, \lambda)$$

where $F$ is the classical inertial frame space ($M \times L_+$, with $L_+$ the proper orthochronous Lorentz group), $\omega$ is a state preparation for the QRF, $\varphi_\lambda(x)$ denotes the field operator “dressed” by Lorentz orientation $\lambda$ at point $x$, and $\mu^{E_R}_\omega$ is the probability measure derived from the QRF POVM outcome for $\omega$.

• Using the disintegration of $\mu^{E_R}_\omega$, the kernel field representation is

$$\varphi^R_\omega(x) = \int_{L_+} d\nu^{E_R}_\omega(\lambda|x) \varphi_\lambda(x)$$

and

$$\Phi^R(\omega) = \int_M \varphi^R_\omega(x) d\mu^{F_R}_\omega(x)$$

where $d\mu^{F_R}_\omega$ is the marginal measure for spacetime localization.

• This framework requires an ultraweakly continuous (projective) unitary representation of the Poincaré group acting on the system Hilbert space, ensuring correct transformation properties under Poincaré transformations.

The use of analytic measure-theoretic and functional analytic tools, including Bochner integration and duality between bounded and trace-class operators, is required for rigorous definition of these relationally localized observables.

2. Operational Quantum Reference Frames and Relational Covariance

The theory specifies that a relativistic QRF is a triple $R = (U_R, E_R, \mathcal{H})$, where $U_R$ is a projective unitary representation of the Poincaré group, and $E_R$ is a P-covariant POVM on the frame space $F$ (Fedida et al., 29 Jul 2025):

• The operational role of the QRF is manifested in the dependence of system observables on the QRF state preparation $\omega$.
• Relational covariance is formalized by the transformation law:

$$(a,\Lambda)\cdot \Phi^R(\omega) = \Phi^R((a,\Lambda)\cdot\omega)$$

indicating that an active system transformation is exactly mirrored by an appropriate passive transformation acting on the QRF state.

This establishes that all system properties (expectation values, algebraic relations, localization) are defined relative to the state and localization properties of the QRF, not to an external absolute structure.

3. Comparison with Wightman and Algebraic QFT

Relational QFT closely parallels, and in key respects generalizes, traditional Wightman and Algebraic QFT frameworks:

• In Wightman QFT, operator-valued distributions are smeared with Schwartz test functions. In the relational framework, the test functions are replaced by the “smearing” induced by the frame’s probability measure or its Radon–Nikodym derivative (the “frame smearing function”).
• Whereas Wightman fields are typically unbounded operators on a dense domain, the relational kernel fields $\varphi^R_\omega(x)$ are bounded operators on $\mathcal{H}$ (Fedida et al., 29 Jul 2025).
• The n-point vacuum expectation values

$$W_n^{(\Omega,R)}[\omega_1,\ldots,\omega_n](\varphi_1, \ldots, \varphi_n) = \operatorname{Tr}[\Omega \prod_{i=1}^n \Phi^R(\omega_i)]$$

satisfy the analogues of the Wightman axioms: transformation law, positive spectrum condition, Hermiticity, local commutativity, and clustering, where the localization uncertainty of the QRF smearing replaces the role of Schwartz test functions.

This foundational shift provides a mathematically rigorous operator algebra constructed from operationally meaningful, QRF-relative local observables, while retaining all key spectral, transformation, and algebraic properties of standard QFT.

4. Causality, Commutation, and Local Algebras

A hierarchy of causality conditions emerges in the relational paradigm:

• Epistemic (Einstein) causality: If the support of QRF state preparations $\omega_1$ and $\omega_2$ are spacelike separated, then relational local observables commute,

$[\Phi^R(\omega_1), \Phi^R(\omega_2)] = 0$

(Fedida et al., 29 Jul 2025).

• R–microcausality: If $x_1, x_2$ are spacelike separated, then $$[\varphi^R_{\omega_1}(x_1), \varphi^R_{\omega_2}(x_2)] = 0$$. This is a pointwise strengthening.
• For any spacetime region $U$, the relational local algebra $\mathfrak{A}^R(U)$ is the double commutant of all $\Phi^R(\omega)$ with QRF localization in $U$. The net of these algebras satisfies isotony, covariance, local commutativity (Einstein causality), and a suitable time-slice property.

These results demonstrate that the operationally-defined net of algebras generated by relational local observables fulfills the core axioms of Algebraic QFT (Fedida et al., 29 Jul 2025).

5. Recovery of Standard QFT and Relation to Measurement Theory

The vacuum expectation values of relational local observables reproduce many familiar properties of conventional Wightman functions, with notable operational differences (Fedida et al., 29 Jul 2025):

• The QRF-induced frame smearing function is directly identified with the Wightman test function, encoding the localization uncertainty of the quantum frame.
• The transformation properties, spectrum condition, and Hermiticity of the $\Phi^R(\omega)$ match those of Wightman fields, but the operational meaning is grounded in relative (rather than absolute) localizations.
• Relational measurement theory can be formulated by specifying interactions between system and detector described via tensor products of relational local observables associated to different QRFs, yielding a natural operational approach to quantum measurements and scattering.

This approach allows reinterpretation of key objects (fields, Wightman functions, distributions) in language where all localization is fundamentally relative to prepared quantum frames.

6. Outlook: Generalizations, Extensions, and Open Questions

Extensive directions are proposed for developing relational local observables in broader quantum and gravitational contexts (Fedida et al., 29 Jul 2025):

• Extension to spinor and gauge fields: For spinor fields, the frame space generalizes to include universal cover of the Lorentz group; for gauge fields, internal frame bundles enlarge the reference space.
• Curved spacetimes and quantum gravity: The reference frames become sections of Lorentzian frame bundles over curved backgrounds, and dynamical frame covariance is made explicit. This connects to frameworks in quantum gravity where all observable content is relational.
• Dynamics and renormalization: There is scope for formulating relational Lagrangians and Hamiltonians, relational versions of the Euler–Lagrange equations, and exploring frame-dependent aspects of mass gap and renormalization, with the QRF localization properties providing new control parameters.
• Detector models, scattering theory, and external frame transformations: Measurement schemes and LSZ-like formulations become “frame–neutral” and relational, and composition of different QRF transformations (“channels”) may yield new insights into renormalization group structure.
• Euclideanization and reconstruction: Relational QFT can be adapted to Euclidean signature and may provide a natural setting for relational Osterwalder–Schrader reconstruction.

A key theme is the possibility that physical masses, mass gaps, and effective field parameters could themselves depend on the localization or state of the QRF, resulting in a relational notion of energy thresholds.

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Relational local observables, rigorously defined via the operational framework of quantum reference frames with covariant POVMs, offer a foundationally robust and operationally meaningful generalization of localization and locality in quantum field theory. The approach yields a family of bounded, frame-relative kernel fields and a net of relational local algebras satisfying the full set of AQFT axioms, with transformation, spectral, and causality properties directly inherited from—and operationally grounded in—the QRF formalism. This paradigm not only recasts the mathematical and conceptual foundation of QFT but also opens multiple research avenues in the algebraic structure, measurement theory, and the quantum-to-gravity interface.