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].
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].
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.
A subtheory of a quantum field theory specifies von Neumann subalgebras [Formula: see text] (the ‘observables’ in the space-time region [Formula: see text]) of the von Neumann algebras [Formula: see text] (the 'field' localized in [Formula: see text]). Every local algebra being a (type III1) factor, the inclusion [Formula: see text] is a subfactor. The assignment of these local subfactors to the space-time regions is called a ‘net of subfactors’. The theory of subfactors is applied to such nets. In order to characterize the ‘relative position’ of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a single local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows one to characterize, and reconstruct, local extensions ℬ of a given theory [Formula: see text] in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.
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)⊂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].
A subtheory of a quantum field theory specifies von Neumann subalgebras A(phi) (the 'observables' in the space-time region phi) of the von Neumann algebras B(phi) (the 'fields' localized in phi). Every local algebra being a (type III1) factor, the inclusion A(phi) subset of B(phi) is a subfactor. The assignment of these local subfactors to the space-time regions is called a 'net of subfactors'. The theory of subfactors is applied to such nets. In order to characterize the 'relative position' of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a single local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows one to characterize, and reconstruct, local extensions B of a given theory A in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.
The low-energy excitations of non-Abelian topological order are described byanyon fusion categories. In the thermodynamic limit, the algebra of local observables may form a tower of von Neumann subfactor inclusions, and the correspondence between the Jones index and the total quantum dimension of anyons hasbeen supported by Longo’s theorem in algebraic quantum field theory and by explicit calculations in Kitaev’s quantum double model. Taking this as a startingpoint, this paper does not presuppose the universal validity of subfactor structures in generic strongly correlated lattice systems; instead, it sets them as explicittheoretical hypotheses and systematically constructs a derivation chain from thediscrete sequence of Jones indices to constraints on physical excitation gap ratios.The central link is the definition of a subfactorizable Hamiltonian– expressing thelow-energy effective Hamiltonian, within an energy cutoff, as a quadratic form inTemperley-Lieb projections– thereby using the representation theory of the TLalgebra to establish a mapping between anyon excitations and Bratteli diagramnodes, leading to the conclusion that the gap ratio depends only on the Jones parameter τ. The discrete valueing of the Jones index when it is less than 4 forcesτ to be locked to a countable set, yielding the rigid prediction that within thesame topological phase region the ratio of different excitation gaps cannot driftcontinuously with external parameters. This paper designs an experimental testscheme based on scanning tunneling spectroscopy and transport measurements,and systematically provides falsification conditions, so that each hypothesis in thetheoretical framework can be independently tested. The contribution of this paper lies in extending subfactor algebra theory from exactly solvable models to atheoretical framework with operational physical predictions, through a cautioushypothesis–deduction–test structure.
An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III _{1} 1 , in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II _∞ ∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.