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arXiv:2503.02944

Extensions from within

Shadi Ali Ahmad, Marc S. Klinger

Published Mar 4, 2025• 0 citations

Abstract

Inclusions and extensions lie at the heart of physics and mathematics. The most relevant kind of inclusion in quantum systems is that of a von Neumann subalgebra, which is the focus of this work. We propose an object intrinsic to a given algebra that indexes its potential extensions into larger algebras. We refer to this object as a spatial Q-system, since it is inspired by the Q-system construction of Longo and others that has been used to categorify finite index inclusions for properly infinite algebras. The spatial Q-system broadens the usefulness of the Q-system by extending its applicability to the context of possibly infinite index inclusions admitting only operator valued weights rather than conditional expectations. An immediate physical motivation for the spatial Q-system arises in the form of the crossed product, which is a method for extending a given algebra by the generators of a locally compact group which acts upon it. If the group in question is not finite, the inclusion of the original algebra into its crossed product will be of infinite index and admit only an operator valued weight. Such an extension cannot be described by a standard Q-system. The spatial Q-system covers this case and, in fact, may be regarded as a far reaching generalization of the crossed product in which an algebra is extended by a collection of operators that possess a closed product structure compatible with the original algebra, but need not generate a symmetry object. We comment upon the categorical interpretation of the spatial Q-system, and describe how it may be interpreted as instantiating a protocol for approximate, or non-isometric, quantum error correction or in terms of quantum reference frames. We conclude by identifying a series of physical applications including the study of generalized symmetries, non-perturbative quantum gravity, black hole evaporation and interiors, and holography.

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Inclusions and extensions lie at the heart of physics and mathematics. The most relevant kind of inclusion in quantum systems is that of a von Neumann subalgebra, which is the focus of this work. We propose an object intrinsic to a given algebra that indexes its potential extensions into larger algebras. We refer to this object as a spatial Q-system, since it is inspired by the Q-system construction of Longo and others that has been used to categorify finite index inclusions for properly infinite algebras. The spatial Q-system broadens the usefulness of the Q-system by extending its applicability to the context of possibly infinite index inclusions admitting only operator valued weights rather than conditional expectations. An immediate physical motivation for the spatial Q-system arises in the form of the crossed product, which is a method for extending a given algebra by the generators of a locally compact group which acts upon it. If the group in question is not finite, the inclusion of the original algebra into its crossed product will be of infinite index and admit only an operator valued weight. Such an extension cannot be described by a standard Q-system. The spatial Q-system covers this case and, in fact, may be regarded as a far reaching generalization of the crossed product in which an algebra is extended by a collection of operators that possess a closed product structure compatible with the original algebra, but need not generate a symmetry object. We comment upon the categorical interpretation of the spatial Q-system, and describe how it may be interpreted as instantiating a protocol for approximate, or non-isometric, quantum error correction or in terms of quantum reference frames. We conclude by identifying a series of physical applications including the study of generalized symmetries, non-perturbative quantum gravity, black hole evaporation and interiors, and holography.

The subfactor theory provides a tool to analyze and construct extensions of Quantum Field Theories, once the latter are formulated as local nets of von Neumann algebras. We generalize some of the results of [62] to the case of extensions with infinite Jones index. This case naturally arises in physics, the canonical examples are given by global gauge theories with respect to a compact (non-finite) group of internal symmetries. Building on the works of Izumi–Longo–Popa [44] and Fidaleo–Isola [30], we consider generalized Q-systems (of intertwiners) for a semidiscrete inclusion of properly infinite von Neumann algebras, which generalize ordinary Q-systems introduced by Longo [58] to the infinite index case. We characterize inclusions which admit generalized Q-systems of intertwiners and define a braided product among the latter, hence we construct examples of QFTs with defects (phase boundaries) of infinite index, extending the family of boundaries in the grasp of [7].

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.

Q-systems describe "extensions" of an infinite von Neumann factor

N

, i.e., finite-index unital inclusions of

N

into another von Neumann algebra

M

. They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of

N

. We review the relation between Q-systems, their modules and bimodules as structures in a category on one side, and homomorphisms between von Neumann algebras on the other side. We then elaborate basic operations with Q-systems (various decompositions in the general case, and the centre, the full centre, and the braided product in braided categories), and illuminate their meaning in the von Neumann algebra setting. The main applications are in local quantum field theory, where Q-systems in the subcategory of DHR endomorphisms of a local algebra encode extensions

A(O)\subset B(O)

of local nets. These applications, notably in conformal quantum field theories with boundaries, are briefly exposed, and are discussed in more detail in two separate papers [arXiv:1405.7863, 1410.8848].

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.

© 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.

We define the perspective of any quantum reference frame (QRF) and construct reversible transformations between different perspectives. Our construction is based on two principles motivated operationally by the change from relative to absolute coordinates and leads to an incoherent group averaging approach with general symmetry group. Thereby, it extends the framework of [arXiv:2110.13199] from ideal QRFs, which generally require infinite resources like energy or angular momentum, to non-ideal QRFs, with only finite resources. We find that the perspective of a non-ideal QRF deviates significantly from that of an ideal QRF: Firstly, systems described relative to a non-ideal QRF appear superselected. Secondly, the structure of the perspective of a non-ideal QRF attests that successive relational operations on a system lead to back-reaction on this QRF.