Quantum Algebra Research Articles

Lorentzian 2D CFT from the pAQFT Perspective

We provide a detailed construction of the quantum theory of the massless scalar field on two-dimensional, globally hyperbolic (in particular, Lorentzian) manifolds using the framework of perturbative algebraic quantum field theory. From this we obtain subalgebras of observables isomorphic to the Heisenberg and Virasoro algebras on the Einstein cylinder. We also show how the conformal version of general covariance, as first introduced by Pinamonti as an extension of the construction due to Brunetti, Fredenhagen and Verch, may be applied to natural Lagrangians, which allow one to specify a theory consistently across multiple spacetimes, in order to obtain a simple condition for the conformal covariance of classical dynamics, which is then shown to quantise in the case of a quadratic Lagrangian. We then compare the covariance condition for the stress-energy tensor in the classical and quantum theory in order to obtain a transformation law involving the Schwarzian derivative of the new coordinate, in accordance with a well-known result in the Euclidean literature.

The effective Dirac algebra by gauge field interaction in relativistic electrodynamics

Conventional relativistic electrodynamics is set on flat Minkowski spacetime, where all computable quantities are calculated from the flat metric [Formula: see text]. We can redefine the metric of spacetime from the Dirac algebra. In this paper, we study how an electrodynamic interaction can alter the normal gamma matrix to an effective one and result in a shift in the metric perturbatively. The curvature properties inferred from the curved metric are also investigated. We also study how the spin operator is changed under the interaction that contributes to an effective spin operator and how the spin of an electron will be slightly deviated from [Formula: see text]. Then we perform canonical quantization of the effective Dirac algebra. Finally, we apply our results to the relativistic hydrogen case and demonstrate how such system curves the spacetime metric.

The classical limit and spontaneous symmetry breaking in algebraic quantum theory

In this paper an overview of some recent developments on the classical limit and spontaneous symmetry breaking (SSB) in algebraic quantum theory is given. In such works, based on the theory of C∗-algebras, the concept of the classical limit has been formalized in a complete algebraic manner. Additionally, since this setting allows for commutative as well as non-commutative C∗-algebras, and hence for classical and quantum theories, it provides an excellent framework to study SBB as an emergent phenomenon when transitioning from the quantum to the classical world by turning off a semi-classical parameter. We summarize the main results and show that this algebraic approach sheds new light on the connection between the classical and the quantum realm, where particular emphasis is placed on the role of SSB in Theory versus Nature. To this end a detailed analysis is carried out and illustrated with three different physical models: Schrödinger operators, mean-field quantum spin systems and the Bose–Hubbard model.

K-Theoretic Hall Algebras of Quivers with Potential as Hopf Algebras

Abstract Preprojective K-theoretic Hall algebras (KHAs), particular cases of KHAs of quivers with potential, are conjecturally positive halves of the Okounkov–Smirnov affine quantum algebras. It is thus natural to ask whether KHAs of quivers with potential are halves of a quantum group. For a symmetric quiver with potential satisfying a Künneth-type condition, we construct (positive and negative) extensions of its KHA, which are bialgebras. In particular, there are bialgebra extensions of preprojective KHAs and one can construct their Drinfeld double algebra.

Quantum B‐modules

AbstractQuantum B‐algebras are partially ordered algebras characterizing the residuated structure of a quantale. Examples arise in algebraic logic, non‐commutative arithmetic, and quantum theory. A quantum B‐algebra with trivial partial order is equivalent to a group. The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B‐module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi‐poset version of the completion of Dedekind and MacNeille.

On the second realization for the positive part of $$U_q(\widehat{sl_2})$$ of equitable type

The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is considered. This presentation was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works the (nonstandard) positive part of $U_q(\widehat{sl_2})$ of equitable type $U_q^{IT,+}$ and its second realization (current algebra) $U_q^{T,+}$ are introduced and studied. A presentation for $U_q^{T,+}$ is given in terms of a K-operator satisfying a Freidel-Maillet type equation and a condition on its quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel L-operators are considered, from which an explicit injective homomorphism from $U_q^{T,+}$ to a subalgebra of Drinfeld's second realization (current algebra) of $U_q(\widehat{sl_2})$ is derived, and the comodule algebra structure of $U_q^{T,+}$ is characterized. The central extension of $U_q^{T,+}$ and its relation with Drinfeld's second realization of $U_q(\widehat{gl_2})$ is also described using the framework of Freidel-Maillet algebras.

Floquet engineering of Lie algebraic quantum systems

We propose a `Floquet engineering' formalism to systematically design a periodic driving protocol in order to stroboscopically realize the desired system starting from a given static Hamiltonian. The formalism is applicable to quantum systems which have an underlying closed Lie-algebraic structure, for example, solid-state systems with noninteracting particles moving on a lattice or its variant described by the ultra-cold atoms moving on an optical lattice. Unlike previous attempts at Floquet engineering, our method produces the desired Floquet Hamiltonian at any driving frequency and is not restricted to the fast or slow driving regimes. The approach is based on Wei-Norman ansatz, which was originally proposed to construct a time-evolution operator for any arbitrary driving. Here, we apply this ansatz to the micro-motion dynamics, defined within one period of the driving, and obtain the driving protocol by fixing the gauge of the micro-motion. To illustrate our idea, we use a two-band system or the systems consisting of two sub-lattices as a testbed. Particularly, we focus on engineering the cross-stitched lattice model that has been a paradigmatic flat-band model.

On quantum toroidal algebra of type A1

In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.

Entanglement Entropy and Non-Local Duality: Quantum Channels and Quantum Algebras

Entanglement Entropy and Non-Local Duality: Quantum Channels and Quantum Algebras

A geometric construction of representations of the Berezin–Toeplitz quantization

For a K\"ahler manifold $X$ equipped with a prequantum line bundle $L$, we give a geometric construction of a family of representations of the Berezin-Toeplitz deformation quantization algebra $(C^\infty(X)[[\hbar]],\star_{BT})$ parametrized by points $z_0 \in X$. The key idea is to use peak sections to suitably localize the Hilbert spaces $H^{0}\left(X,L^{\otimes m}\right)$ around $z_{0}$ in the large volume limit.

A state-space approach to quantum permutations

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.

Smooth 1-Dimensional Algebraic Quantum Field Theories

This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a smooth family of observable algebras. Using stacks of categories, this proposal is realized concretely for the simplest case of 1-dimensional spacetimes, leading to a stack of smooth 1-dimensional AQFTs. Concrete examples of smooth AQFTs, of smooth families of smooth AQFTs and of equivariant smooth AQFTs are constructed. The main open problems that arise in upgrading this approach to higher dimensions and gauge theories are identified and discussed.

On the hydrogen atom in the holographic universe

We investigate the holographic bound utilizing a homogeneous, isotropic, and non-relativistic neutral hydrogen gas present in the de Sitter space. Concretely, we propose to employ de Sitter holography intertwined with quantum deformation of the hydrogen atom using the framework of quantum groups. Particularly, the quantum algebra is used to construct a finite-dimensional Hilbert space of the hydrogen atom. As a consequence of the quantum deformation of the hydrogen atom, we demonstrate that the Rydberg constant is dependent on the de Sitter radius, L Λ. This feature is then extended to obtain a finite-dimensional Hilbert space for the full set of all hydrogen atoms in the de Sitter universe. We then show that the dimension of the latter Hilbert space satisfies the holographic bound. We further show that the mass of a hydrogen atom , the total number of hydrogen atoms at the Universe, N, and the retrieved dimension of the Hilbert space of neutral hydrogen gas, , are related to the de Sitter entropy, , the Planck mass, , the electron mass, , and the proton mass , by , and , respectively.

The CBH characterisation theorem beyond algebraic quantum theory

The CBH theorem characterises quantum theory within a C*-algebraic framework. Namely, mathematical properties of C*-algebras modelling quantum systems are equivalent to constraints that are information-theoretic in nature: (1) noncommutativity of subalgebras is equivalent to impossibility of signalling; (2) noncommutativity of the whole algebra is equivalent to impossibility of broadcasting; (3) the existence of entangled states is implied by the impossibility of secure bit commitment (with the converse conjectured). However, the C*-algebraic framework has drawn criticism as it already contains much of the mathematical structure of quantum theory such as complex linearity. We address this issue by a generalising C*-algebras categorically. In this framework, equivalence (1) holds, equivalence (2) becomes a strict implication, and implication (3) fails in general. Thus we identify exactly what work is being done by the complex-linear structure of C*-algebras. In doing so, we uncover a richer hierarchy of notions of ‘classicality’ and ‘quantumness’ of information than visible in the concrete case.

Bargmann–Fock Sheaves on Kähler Manifolds

Fedosov used flat sections of the Weyl bundle on a symplectic manifold to construct a star product $\star$ which gives rise to a deformation quantization. By extending Fedosov's method, we give an explicit, analytic construction of a sheaf of Bargmann-Fock modules over the Weyl bundle of a K\"ahler manifold $X$ equipped with a compatible Fedosov abelian connection, and show that the sheaf of flat sections forms a module sheaf over the sheaf of deformation quantization algebras defined $(C^\infty_X[[\hbar]], \star)$. This sheaf can be viewed as the $\hbar$-expansion of $L^{\otimes k}$ as $k \to \infty$, where $L$ is a prequantum line bundle on $X$ and $\hbar = 1/k$.

Spinor vacuum and C, P, T inversions

We have developed the theory of Clifford reflections and extended spacetime inversions. This extended spacetime has two additional dimensions associated with the presence of internal degrees of freedom of spinors. Inversions C, P, and T contain not only reflections of the basis Clifford vectors and transformations of basis spinors, but also transformations of the components of vector and spinor quantities. The research is carried out on the basis of algebraic quantum field theory using the superalgebraic representation of spinors as well as the 8-component matrix representation of spinors. We have proved that due to the presence of internal degrees of freedom of spinors, there are two vacua, the vacuum of our Universe and an alternative vacuum. The inversion operators C and T transform the vacuum into an alternative one, and therefore cannot be operators of the exact symmetry of our Universe.

The quantum gravity disk: Discrete current algebra

We study the quantization of the corner symmetry algebra of 3D gravity, that is, the algebra of observables associated with 1D spatial boundaries. In the continuum field theory, at the classical level, this symmetry algebra is given by the central extension of the Poincaré loop algebra. At the quantum level, we construct a discrete current algebra based on a quantum symmetry group given by the Drinfeld double DSU(2). Those discrete currents depend on an integer N, a discreteness parameter, understood as the number of quanta of geometry on the 1D boundary: low N is the deep quantum regime, while large N should lead back to a continuum picture. We show that this algebra satisfies two fundamental properties. First, it is compatible with the quantum space-time picture given by the Ponzano–Regge state-sum model, which provides discrete path integral amplitudes for 3D quantum gravity. The integer N then counts the flux lines attached to the boundary. Second, we analyze the refinement, coarse-graining, and fusion processes as N changes, and we show that the N → ∞ limit is a classical limit where we recover the Poincaré current algebra. Identifying such a discrete current algebra on quantum boundaries is an important step toward understanding how conformal field theories arise on spatial boundaries in quantized space-times such as in loop quantum gravity.

Conditional Distributions for Quantum Systems

Conditional distributions, as defined by the Markov category framework, are studied in the setting of matrix algebras (quantum systems). Their construction as linear unital maps are obtained via a categorical Bayesian inversion procedure. Simple criteria establishing when such linear maps are positive are obtained. Several examples are provided, including the standard EPR scenario, where the EPR correlations are reproduced in a purely compositional (categorical) manner. A comparison between the Bayes map, the Petz recovery map, and the Leifer-Spekkens acausal belief propagation is provided, illustrating some similarities and key differences.

Gravitational edge modes, coadjoint orbits, and hydrodynamics

The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite-dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner’s famous classification of representations of the Poincaré group since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincaré group. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid.

Quadratic algebras based on $$SL(NM)$$ elliptic quantum $$R$$-matrices

We construct quadratic quantum algebra based on the dynamical RLL-relation for the quantum $R$-matrix related to $SL(NM)$-bundles with nontrivial characteristic class over elliptic curve. This $R$-matrix generalizes simultaneously the elliptic nondynamical Baxter--Belavin and the dynamical Felder $R$-matrices,and the obtained quadratic relations generalize both -- the Sklyanin algebra and the relations in the Felder-Tarasov-Varchenko elliptic quantum group, which are reproduced in the particular cases $M=1$ and $N=1$ respectively.