Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.24445

Finite-Precision Algebraic Quantum Field Theory

Abbas Edalat

quant-phmath-phSubmitted Jul 27, 2026• Announced Jul 28, 2026• 0 citations
v1 · Jul 27, 2026v2 · Aug 29, 2026
arXivHTMLPDF

Abstract

Recent work introduced Interval Quantum Mechanics (IQM), a finite-precision framework in which physical information is represented by sets of possible states rather than exact states. We extend this approach to algebraic quantum field theory (AQFT) by introducing Interval Algebraic Quantum Field Theory (IAQFT), whose basic objects are quantum parcels: weak* open convex regions of state space encoding finite-precision information obtained from finitely many local observations. We develop parcel reduction and measurement update, establish finite-dimensional information-contraction results, and formulate locality through compatible parcel nets. Major structural features of AQFT are recast in parcel-theoretic terms, including the Reeh--Schlieder property, Haag's theorem, modular theory, the KMS condition, and the Unruh effect. We show that spacelike vacuum correlations and strict Bell violations persist under finite precision, and that lattice approximations are compatible with the parcel framework. Finally, we recover the Murray--von Neumann classification of factors together with the trace and projection equivalence in Type~$\mathrm{II}_1$ factors from the geometry of limiting parcels. IAQFT thus provides a unified finite-precision formulation of the operational, modular, and operator-algebraic structures of relativistic quantum theory.

Related papers

Finite-Precision Quantum Mechanics: Quantum Parcels, Information and Geometry

Abbas Edalat

Quantum mechanics represents states by exactly specified density operators, whereas experimentally available information is necessarily finite in precision. We develop \emph{Interval Quantum Mechanics (IQM)}, in which finite observational information is represented by a \emph{quantum parcel}, a convex open set of density operators. Experimental parcels are determined by finitely many expectation intervals and form a basis for the state-space topology. A single parcel represents the states compatible with the available information, while a double parcel $(O_1,O_2)$ consists of disjoint possible and excluded regions. Exact point states are recovered as ideal limits of successive parcel refinement. Unitary dynamics lifts to a reversible evolution of parcels. Finite-resolution measurements are represented by fuzzy POVMs and Kraus updates; single parcels are preserved, while double-parcel updates are order-compatible whenever they remain double parcels. We give a useful sufficient condition for such preservation, together with examples showing it is not necessary. Under suitable conditions measurement contracts ambient Hilbert--Schmidt volume and strictly increases the associated geometric information. This contrasts with von Neumann entropy, which can remain unchanged under a selective measurement even though information has been gained. In this formulation, several familiar foundational paradoxes no longer arise in standard form. Wave--particle duality becomes a continuous suppression of interference as which-path resolution increases. Schrödinger's cat is described by a finite parcel updated toward the observed outcome sector, rather than an exact superposition undergoing abrupt collapse. Entanglement remains genuinely nonclassical: CHSH violation persists throughout an open parcel around a Bell state.

quant-pharXiv:2605.19706•May 20, 2026•0 citations

Quantum field logic

Héctor Freytes

Algebraic quantum field theory, or AQFT for short, is a rigorous analysis of the structure of relativistic quantum mechanics. It is formulated in terms of a net of operator algebras indexed by regions of a Lorentzian manifold. In several cases the mentioned net is represented by a family of von Neumann algebras, concretely, type III factors. Local quantum field logic arises as a logical system that captures the propositional structure encoded in the algebras of the net. In this framework, this work contributes to the solution of a family of open problems, emerged since the 30s, about the characterization of those logical systems which can be identified with the lattice of projectors arising from the Murray-von Neumann classification of factors. More precisely, based on physical requirements formally described in AQFT, an equational theory able to characterizethe type III condition in a factor is provided. This equational system motivates the study of a variety of algebras having an underlying orthomodular lattice structure. A Hilbert style calculus, algebraizable in the mentioned variety, is also introduced and a corresponding completeness theorem is established.

arXiv:2211.02325•Nov 4, 2022•0 citations

Quantum reference frames, measurement schemes and the type of local algebras in quantum field theory

CJ Fewster, DW Janssen, LD Loveridge +2 more

We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed by the existence of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type $\textnormal{II}_1$ factor. Our results build upon recent work of Chandrasekaran, Longo, Penington and Witten [JHEP 2023, 82 (2023)], providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.

•Jan 1, 2025•0 citations

Quantum Ergodicity and Thermalization in Interval Quantum Mechanics

Abbas Edalat

We combine Reimann's spectral typicality theorem -- a modern formulation of quantum ergodicity -- with the framework of Interval Quantum Mechanics (IQM). In IQM, quantum states are represented not by points but by \emph{quantum parcels}: weak open convex sets of density matrices defined by finitely many expectation intervals. Such parcels are the exact mathematical representation of the epistemic knowledge obtained from finite-precision measurements of macroscopic observables. We prove that for a single parcel in which every state has large effective dimension (a condition that ensures thermalization), the expectation interval of any bounded observable becomes concentrated around the microcanonical value for most late times. The asymptotic bound depends only on the minimal effective dimension within the parcel, not on its detailed shape. For a double parcel \((O_1,O_2)\) with both components contained in an energy shell, separated by a conserved quantity \(Q^*\) that is supported on the range of the measurement projector, we show that the expectation intervals of both parcels become concentrated near the microcanonical values of bounded observables, the separation is preserved exactly, and the updated double parcel after a fuzzy measurement remains valid.

quant-pharXiv:2606.00749•Jun 2, 2026•0 citations

Quantum Reference Frames, Measurement Schemes and the Type of Local Algebras in Quantum Field Theory

Christopher J. Fewster, Daan W. Janssen, Leon Loveridge +2 more

Abstract We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type $$\text {II}_1$$ II 1 factor. Our results build upon recent work of Chandrasekaran et al. (J High Energy Phys 2023(2): 1–56, 2023. arXiv:2206.10780 ), providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.

arXiv:2403.11973•Communications in Mathematical Physics•Dec 18, 2024•43 citations

Quantum Reference Frames, Measurement Schemes and the Type of Local Algebras in Quantum Field Theory

Christopher J. Fewster, Daan W. Janssen, Leon Loveridge +2 more

© The Author(s) 2024. We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type II1 factor. Our results build upon recent work of Chandrasekaran et al. (J High Energy Phys 2023(2): 1–56, 2023. arXiv:2206.10780), providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.

•Dec 18, 2024•0 citations
Recent work introduced Interval Quantum Mechanics (IQM), a finite-precision framework in which physical information is represented by sets of possible states rather than exact states. We extend this approach to algebraic quantum field theory (AQFT) by introducing Interval Algebraic Quantum Field Theory (IAQFT), whose basic objects are quantum parcels: weak* open convex regions of state space encoding finite-precision information obtained from finitely many local observations.
We develop parcel reduction and measurement update, establish finite-dimensional information-contraction results, and formulate locality through compatible parcel nets. Major structural features of AQFT are recast in parcel-theoretic terms, including the Reeh--Schlieder property, Haag's theorem, modular theory, the KMS condition, and the Unruh effect. We show that spacelike vacuum correlations and strict Bell violations persist under finite precision, and that lattice approximations are compatible with the parcel framework. Finally, we recover the Murray--von Neumann classification of factors together with the trace and projection equivalence in Type~II1\mathrm{II}_1II1​ factors from the geometry of limiting parcels. IAQFT thus provides a unified finite-precision formulation of the operational, modular, and operator-algebraic structures of relativistic quantum theory.
Quantum mechanics represents states by exactly specified density operators, whereas experimentally available information is necessarily finite in precision. We develop \emph{Interval Quantum Mechanics (IQM)}, in which finite observational information is represented by a \emph{quantum parcel}, a convex open set of density operators. Experimental parcels are determined by finitely many expectation intervals and form a basis for the state-space topology. A single parcel represents the states compatible with the available information, while a double parcel (O1,O2)(O_1,O_2)(O1​,O2​) consists of disjoint possible and excluded regions. Exact point states are recovered as ideal limits of successive parcel refinement. Unitary dynamics lifts to a reversible evolution of parcels. Finite-resolution measurements are represented by fuzzy POVMs and Kraus updates; single parcels are preserved, while double-parcel updates are order-compatible whenever they remain double parcels. We give a useful sufficient condition for such preservation, together with examples showing it is not necessary. Under suitable conditions measurement contracts ambient Hilbert--Schmidt volume and strictly increases the associated geometric information. This contrasts with von Neumann entropy, which can remain unchanged under a selective measurement even though information has been gained. In this formulation, several familiar foundational paradoxes no longer arise in standard form. Wave--particle duality becomes a continuous suppression of interference as which-path resolution increases. Schrödinger's cat is described by a finite parcel updated toward the observed outcome sector, rather than an exact superposition undergoing abrupt collapse. Entanglement remains genuinely nonclassical: CHSH violation persists throughout an open parcel around a Bell state.
Algebraic quantum field theory, or AQFT for short, is a rigorous analysis of the structure of relativistic quantum mechanics. It is formulated in terms of a net of operator algebras indexed by regions of a Lorentzian manifold. In several cases the mentioned net is represented by a family of von Neumann algebras, concretely, type III factors. Local quantum field logic arises as a logical system that captures the propositional structure encoded in the algebras of the net. In this framework, this work contributes to the solution of a family of open problems, emerged since the 30s, about the characterization of those logical systems which can be identified with the lattice of projectors arising from the Murray-von Neumann classification of factors. More precisely, based on physical requirements formally described in AQFT, an equational theory able to characterizethe type III condition in a factor is provided. This equational system motivates the study of a variety of algebras having an underlying orthomodular lattice structure. A Hilbert style calculus, algebraizable in the mentioned variety, is also introduced and a corresponding completeness theorem is established.
We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed by the existence of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type II1\textnormal{II}_1II1​ factor. Our results build upon recent work of Chandrasekaran, Longo, Penington and Witten [JHEP 2023, 82 (2023)], providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.
We combine Reimann's spectral typicality theorem -- a modern formulation of quantum ergodicity -- with the framework of Interval Quantum Mechanics (IQM). In IQM, quantum states are represented not by points but by \emph{quantum parcels}: weak open convex sets of density matrices defined by finitely many expectation intervals. Such parcels are the exact mathematical representation of the epistemic knowledge obtained from finite-precision measurements of macroscopic observables. We prove that for a single parcel in which every state has large effective dimension (a condition that ensures thermalization), the expectation interval of any bounded observable becomes concentrated around the microcanonical value for most late times. The asymptotic bound depends only on the minimal effective dimension within the parcel, not on its detailed shape. For a double parcel (O1,O2)(O_1,O_2)(O1​,O2​) with both components contained in an energy shell, separated by a conserved quantity Q∗Q^*Q∗ that is supported on the range of the measurement projector, we show that the expectation intervals of both parcels become concentrated near the microcanonical values of bounded observables, the separation is preserved exactly, and the updated double parcel after a fuzzy measurement remains valid.
Abstract We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type II1\text {II}_1II1​ II 1 factor. Our results build upon recent work of Chandrasekaran et al. (J High Energy Phys 2023(2): 1–56, 2023. arXiv:2206.10780 ), providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.
© The Author(s) 2024. We develop an operational framework, combining relativistic quantum measurement theory with quantum reference frames (QRFs), in which local measurements of a quantum field on a background with symmetries are performed relative to a QRF. This yields a joint algebra of quantum-field and reference-frame observables that is invariant under the natural action of the group of spacetime isometries. For the appropriate class of quantum reference frames, this algebra is parameterised in terms of crossed products. Provided that the quantum field has good thermal properties (expressed by the existence of a KMS state at some nonzero temperature), one can use modular theory to show that the invariant algebra admits a semifinite trace. If furthermore the quantum reference frame has good thermal behaviour (expressed in terms of the properties of a KMS weight) at the same temperature, this trace is finite. We give precise conditions for the invariant algebra of physical observables to be a type II1 factor. Our results build upon recent work of Chandrasekaran et al. (J High Energy Phys 2023(2): 1–56, 2023. arXiv:2206.10780), providing both a significant mathematical generalisation of these findings and a refined operational understanding of their model.