Quantum Algebra Research Articles

Boundary algebras of the Kitaev quantum double model

The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group.

Novel non‐involutive solutions of the Yang–Baxter equation from (skew) braces

AbstractWe produce novel non‐involutive solutions of the Yang–Baxter equation coming from (skew) braces. These solutions are generalisations of the known ones coming from braces and skew braces, and surprisingly in the case of braces, they are not necessarily involutive. In the case of two‐sided (skew) braces, one can assign such solutions to every element of the set. Novel bijective maps associated to the inverse solutions are also introduced. Moreover, we show that the recently derived Drinfeld twists of the involutive case are still admissible in the non‐involutive frame, and we identify the twisted ‐matrices and twisted coproducts. We observe, as in the involutive case, that the underlying quantum algebra is not a quasi‐triangular bialgebra, as one would expect, but a quasi‐triangular quasi‐bialgebra. The same applies to the quantum algebra of the twisted ‐matrices, contrary to the involutive case.

Algebraic discrete quantum harmonic oscillator with dynamic resolution scaling

Algebraic discrete quantum harmonic oscillator with dynamic resolution scaling

Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups

AbstractLet be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient. For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

Quantization, dequantization, and distinguished states

Abstract Geometric quantization is a natural way to construct quantum models starting from classical data. In this work, we start from a symplectic vector space with an inner product and — using techniques of geometric quantization — construct the quantum algebra and equip it with a distinguished state. We compare our result with the construction due to Sorkin — which starts from the same input data — and show that our distinguished state coincides with the Sorkin-Johnson state. Sorkin’s construction was originally applied to the free scalar field over a causal set (locally finite, partially ordered set). Our perspective suggests a natural generalization to less linear examples, such as an interacting field.

Commutative Poisson algebras from deformations of noncommutative algebras

It is well-known that a formal deformation of a commutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} leads to a Poisson bracket on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document} and that the classical limit of a derivation on the deformation leads to a derivation on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalization of it for formal deformations of an arbitrary noncommutative algebra A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The deformation leads in this case to a Poisson algebra structure on Π(A):=Z(A)×(A/Z(A))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A}){:}{=}Z(\\mathcal {A})\ imes (\\mathcal {A}/Z(\\mathcal {A}))$$\\end{document} and to the structure of a Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}-Poisson module on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. The limiting derivations are then still derivations of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}, but with the Hamiltonian belong to Π(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi (\\mathcal {A})$$\\end{document}, rather than to A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantized Grassmann algebra.

A generalized dynamic asymmetric exclusion process: orthogonal dualities and degenerations

Abstract In this paper, a generalized version of dynamic asymmetric simple exclusion process (ASEP) is introduced, and it is shown that the process has a Markov duality property with the same process on the reversed lattice. The duality functions are multivariate q-Racah polynomials, and the corresponding orthogonality measure is the reversible measure of the process. By taking limits in the generator of dynamic ASEP, its reversible measure, and the duality functions, we obtain orthogonal and triangular dualities for several other interacting particle systems. In this sense, the duality of dynamic ASEP sits on top of a hierarchy of many dualities. For the construction of the process, we rely on representation theory of the quantum algebra U q ( sl 2 ) . In the standard representation, the generator of generalized ASEP can be constructed from the coproduct of the Casimir. After a suitable change of representation, we obtain the generator of dynamic ASEP. The corresponding intertwiner is constructed from q-Krawtchouk polynomials, which arise as eigenfunctions of twisted primitive elements. This gives a duality between dynamic ASEP and generalized ASEP with q-Krawtchouk polynomials as duality functions. Using this duality, we show the (almost) self-duality of dynamic ASEP.

Hamiltonians for the quantised Volterra hierarchy

This paper builds upon our recent work, published in Carpentier et al (2022 Lett. Math. Phys. 112 94), where we established that the integrable Volterra lattice on a free associative algebra and the whole hierarchy of its symmetries admit a quantisation dependent on a parameter ω. We also uncovered an intriguing aspect: all odd-degree symmetries of the hierarchy admit an alternative, non-deformation quantisation, resulting in a non-commutative algebra for any choice of the quantisation parameter ω. In this study, we demonstrate that each equation within the quantum Volterra hierarchy can be expressed in the Heisenberg form. We provide explicit expressions for all quantum Hamiltonians and establish their commutativity. In the classical limit, these quantum Hamiltonians yield explicit expressions for the classical ones of the commutative Volterra hierarchy. Furthermore, we present Heisenberg equations and their Hamiltonians in the case of non-deformation quantisation. Finally, we discuss commuting first integrals, central elements of the quantum algebra, and the integrability problem for periodic reductions of the Volterra lattice in the context of both quantisations.

Time delay in κ-anti-de Sitter spacetime

Based on deformed translations in the κ-anti-de Sitter algebra, we derive a delay in the time of detection between a soft and a hard photon, which are simultaneously emitted at a distant event, to first order in the quantum gravity parameter. In the basis analyzed, the trajectories are undeformed, and the effect depends exclusively on the symmetry properties of the quantum algebra. The time delay depends linearly on the energy of the hard particle and has a sinusoidal dependence on the redshift of the source.

Localization in quantum field theory

We review the issue of localization in quantum field theory and detail the nonrelativistic limit. Three distinct localization schemes are examined: the Newton–Wigner, the algebraic quantum field theory, and the modal scheme. Among these, the algebraic quantum field theory provides a fundamental concept of localization, rooted in its axiomatic formulation. In contrast, the Newton–Wigner scheme draws inspiration from the Born interpretation, applying mainly to the nonrelativistic regime. The modal scheme, relying on the representation of single particles as positive frequency modes of the Klein–Gordon equation, is found to be incompatible with the algebraic quantum field theory localization.This review delves into the distinctive features of each scheme, offering a comparative analysis. A specific focus is placed on the property of independence between state preparations and observable measurements in spacelike separated regions. Notably, the notion of localization in algebraic quantum field theory violates this independence due to the Reeh–Schlieder theorem. Drawing parallels with the quantum teleportation protocol, it is argued that causality remains unviolated. Additionally, we consider the nonrelativistic limit of quantum field theory, revealing the emergence of the Born scheme as the fundamental concept of localization. Consequently, the nonlocality associated with the Reeh–Schlieder theorem is shown to be suppressed under nonrelativistic conditions.

Crossed products, conditional expectations and constraint quantization

Recent work has highlighted the importance of crossed products in correctly elucidating the operator algebraic approach to quantum field theories. In the gravitational context, the crossed product simultaneously promotes von Neumann algebras associated with subregions in diffeomorphism covariant quantum field theories from type III to type II, and provides the necessary ingredients to gravitationally dress operators, thereby enforcing the constraints of the theory. In this note we enhance the crossed product construction to the context of general gauge theories with arbitrary combinations of internal and spacetime local symmetries. This is done by leveraging the correspondence between the crossed product and the extended phase space. We then undertake a detailed study of constraint quantization from the perspective of a generic crossed product algebra. We study and compare four distinct approaches to constraint quantization from this point of view: refined algebraic quantization, BRST quantization, path integral quantization, and the commutation theorem for crossed products. Far from simply reproducing existing analyses, the operator algebraic viewpoint sheds new light on old problems by reformulating the dressing of operators in terms of conditional expectations and other closely related projection maps. We conclude by applying our approach to the constraint quantization of three distinct gauge theories including a discussion of gravity on null hypersurfaces.

Geometric Hawking radiation of Schwarzschild Black Hole with novel quantum algebra

In the context of an extended General Relativity theory with boundary terms included, we introduce a new nonlinear quantum algebra involving a quantum differential operator, with the aim to calculate quantum geometric alterations when a particle is created in the vicinity of a Schwarzschild black-hole by the Hawking radiation mechanism. The boundary terms in the varied action give rise to modifications in the geometric background, which are investigated by analyzing the metric tensor and the Ricci curvature within the framework of a renormalized quantum theory of gravity.

Seminal electromagnetic fields from preinflation

We investigate the geometric dynamics of the primordial electric and magnetic fields during the early stages of the universe by extending a recently introduced quantum algebra (Bellini et al., 2023). We work on an extended model of gravity that considers the boundary terms from the Einstein–Hilbert action as geometric quantum fluctuations of the spacetime. We propose that the extended Riemann manifold is generated by a new connection δΓˆαβμ. This connection contains geometric information about the fluctuations of gravitational and electromagnetic fields in the vacuum, which could have been crucial during the primordial stages of the universe’s evolution. We revisit a preinflationary cosmological model (Bellini, 2023) with a variable time scale and negative spatial curvature, such that the universe begins with a null initial background energy density. We observed the emergence of large scale magnetic fields starting from small values during the early phases of the universe’s evolution. Subsequently, these fields decrease to reach present day values on the order of δBˆ≃10−12G on cosmological scales (between 1024 and 1026 meters). This significant deviation from inflationary models eliminates the need to impose excessively large initial values on these fields.

Holography on the quantum disk

Motivated by recent study of DSSYK and the non-commutative nature of its bulk dual, we review and analyze an example of a non-commutative spacetime known as the quantum disk proposed by L. Vaksman. The quantum disk is defined as the space whose isometries are generated by the quantum algebra Uqsu1,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {U}_q\\left(\\mathfrak{s}{\\mathfrak{u}}_{1,1}\\right) $$\\end{document}. We review how this algebra is defined and its associated group SUq(1, 1) that it generates, highlighting its non-trivial coproduct that sources bulk non-commutativity. We analyze the structure of holography on the quantum disk and study the imprint of non-commutativity on the putative boundary dual.

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

Renormalization on the DFR quantum spacetime

An approach to renormalization of scalar fields on the Doplicher–Fredenhagen–Roberts (DFR) quantum spacetime is presented. The effective nonlocal theory obtained through the use of states of optimal localization for the quantum spacetime is reformulated in the language of (perturbative) Algebraic Quantum Field Theory. The structure of the singularities associated to the nonlocal kernel that codifies the effects of non-commutativity is analyzed using the tools of microlocal analysis.

Quasi-free Isomorphisms of Second Quantization Algebras and Modular Theory

Using Araki–Yamagami’s characterization of quasi-equivalence for quasi-free representations of the CCRs, we provide an abstract criterion for the existence of isomorphisms of second quantization local von Neumann algebras induced by Bogolubov transformations in terms of the respective one particle modular operators. We discuss possible applications to the problem of local normality of vacua of Klein-Gordon fields with different masses.

Crossed product algebras and generalized entropy for subregions

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.

Integrability of rank-two web models

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [1,2]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [3]. These models are based on the A2, G2 and B2 Lie algebras, and their local vertex configurations are intertwiners of the corresponding q-deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of B2 also on the square lattice. For specific root-of-unity choices of q, we show the equivalence to a number of three- and four-state spin models on the dual lattice.The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For q on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix.In the A2 case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the G2 case we identify four regimes numerically. The B2 case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.

On the combinatorial rank of quantum groups of type F4

In this paper, we calculate the combinatorial rank of the positive part [Formula: see text] of the multiparameter version of the small Lusztig quantum group, where [Formula: see text] is a simple Lie algebra of type [Formula: see text]. For this purpose, we also present the basis and relations of the quantum algebra, calculate the coproduct of its PBW-generators and determine all skew-primitive elements of this quantum group. In conclusion, we obtain that the combinatorial rank of [Formula: see text] is 4.