Quantum Algebra Research Articles

Inductive Systems of $$\boldsymbol{C}^{\boldsymbol{*}}$$-Algebras over Posets: A Survey

We survey the research on the inductive systems of $$C^{*}$$ -algebras over arbitrary partially ordered sets. The motivation for our work comes from the theory of reduced semigroup $$C^{*}$$ -algebras and local quantum field theory. We study the inductive limits for the inductive systems of Toeplitz algebras over directed sets. The connecting $$\ast$$ -homomorphisms of such systems are defined by sets of natural numbers satisfying some coherent property. These inductive limits coincide up to isomorphisms with the reduced semigroup $$C^{*}$$ -algebras for the semigroups of non-negative rational numbers. By Zorn’s lemma, every partially ordered set $$K$$ is the union of the family of its maximal directed subsets $$K_{i}$$ indexed by elements of a set $$I$$ . For a given inductive system of $$C^{*}$$ -algebras over $$K$$ one can construct the inductive subsystems over $$K_{i}$$ and the inductive limits for these subsystems. We consider a topology on the set $$I$$ . It is shown that characteristics of this topology are closely related to properties of the limits for the inductive subsystems.

Algebraic classical and quantum field theory on causal sets

This paper provides the first construction of a wide class of algebraic quantum field theory (AQFT) models on causal sets. A deformation quantization of the underlying algebra is used to define the AQFT on a fixed causal set. The sensitivity of observables to changes in the underlying causal set is captured in a relative Cauchy evolution.

Operads for algebraic quantum field theory

We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed Lorentzian manifold, locally covariant theories and also chiral conformal and Euclidean theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain adjunctions between categories of different types of quantum field theories. These include novel and interesting constructions such as time-slicification and local-to-global extensions of quantum field theories. We compare the latter to Fredenhagen’s universal algebra.

Vibrational molecular and nuclear spectra in terms of quantum algebras

A generalized deformed oscillator giving the same spectrum as the Morse potential is con­ structed through the use of quantum algebraic techniques. The model of n coupled anharmonic oscillators of Iachello and Oss, suitable for the description of vibrational spectra of polyatomic molecules, is subsequently written in terms of such generalized deformed oscilla­tors. In addition to clarifying the relation of the model of Iachello and Oss to other models using coupled oscillators for the description of vibrational molecular spectra, the present formalism allows for the construction of a large class of exactly soluble models with no extra computational effort. As an example, the way of including a coupling of the Darling-Dennison type is shown. Nuclear models giving good descriptions of vibrational spectra are reviewed and the need of modifying the SUq(2) model in order to extend its region of applicability towards the vibrational limit is discussed.

QUANTUM ALGEBRAIC SYMMETRIES IN NUCLEAR AND MOLECULAR PHYSICS

The first realizations of quanttun algebraic symmetries in nuclear and molecular spectra are presented. Rotational spectra of even-even nuclei are described by the quantum algebra SUq(2). The two parameter formula given by the algebra is equivalent to an expan- sion in terms of powers of j(j + 1), similar to the expansion given by the Variable Moment of Inertia (VMI) model. The moment of inertia parameter in the two models, as well as the small parameter of the expansion, are found to have very similar numerical values. The same formalism is found to give very good results for superdeformed nuclear bands, which are closer to the classical SU(2) limit, as well as for rotational bands of diatomic molecules, in which a partial summation of the Dunham expansion for rotation-vibration spectra is achieved. Vibrational spectra of diatomic molecules can be described by the q-deformed anhannonic oscillator, having the symmetry Uq(2)>Oq(2). An alternative de- scription is obtained in terms of the quantum algebra SUq(1,1). In both cases the energy formula obtained is equivalent to an expansion in terms of powers of (v+½) , where ν is the vibrational quantum number, while in the classical ST(1,1) case only the first two powers appear. In all cases the improved description of the empirical data is obtained with q being a phase (and not a real number). Further applications of quantum algebraic symmetries in nuclei and molecules are discussed.

Emergent fractons and algebraic quantum liquid from plaquette melting transitions

Paramagnetic spin systems with spontaneously broken spatial symmetries, such as valence bond solid (VBS) phases, can host topological defects carrying non-trivial quantum numbers, which enables the paradigm of deconfined quantum criticality. In this work, we study the properties of topological defects in valence plaquette solid (VPS) phases on square and cubic lattices. We show that the defects of the VPS order parameter, in addition to possessing non-trivial quantum numbers, have fracton mobility constraints deep in the VPS phase, which has been overlooked previously. The spinon inside a single vortex cannot move freely in any direction, while a dipolar pair of vortices with spinon pairs can only move perpendicular to its dipole moment. These mobility constraints, while they persist, can potentially inhibit the condensation of vortices and preclude a continuous transition from the VPS to the N\'eel antiferromagnet. Instead, the VPS melting transition can be driven by proliferation of spinon dipoles. For example, we argue that a $2d$ VPS can melt into a stable gapless phase in the form of an algebraic bond liquid with algebraic correlations and long range entanglement. Such a bond liquid phase yields a concrete example of the elusive $2d$ Bose metal with symmetry fractionalization. We also study $3d$ valence plaquette and valence cube ordered phase, and demonstrate that the topological defects therein also have fractonic dynamics. Possible nearby phases after melting the valence plaquettes or cubes are also discussed.

The star product in interacting quantum field theory

We propose a new formula for the star product in deformation quantization of Poisson structures related in a specific way to a variational problem for a function S, interpreted as the action functional. Our approach is motivated by perturbative algebraic quantum field theory (pAQFT). We provide a direct combinatorial formula for the star product, and we show that it can be applied to a certain class of infinite-dimensional manifolds (e.g. regular observables in pAQFT). This is the first step towards understanding how pAQFT can be formulated such that the only formal parameter is hbar , while the coupling constant can be treated as a number. In the introductory part of the paper, apart from reviewing the framework, we make precise several statements present in the pAQFT literature and recast these in the language of (formal) deformation quantization. Finally, we use our formalism to streamline the proof of perturbative agreement provided by Drago, Hack, and Pinamonti and to generalize some of the results obtained in that work to the case of a nonlinear interaction.

Deformation quantization of constrained systems: a group averaging approach

Motivated by certain concepts introduced by the refined algebraic quantization formalism for constrained systems which has been successfully applied within the context of loop quantum gravity, in this paper we propose a phase space implementation of the Dirac quantization formalism to appropriately include systems with constraints. In particular, we propose a physically prescribed Wigner distribution which allows for the definition of a well-defined inner product by judiciously introducing a star version of the group averaging of the constraints. This star group averaging procedure is obtained by considering the star-exponential of the constraints and then integrating with respect to a suitable measure. In this manner, the proposed physical Wigner distribution explicitly solves the constraints by including generalized functions on phase space. Finally, in order to illustrate our approach, our proposal is tested in a couple of simple examples which recover standard results found in the literature.

The algebra of Wick polynomials of a scalar field on a Riemannian manifold

On a connected, oriented, smooth Riemannian manifold without boundary we consider a real scalar field whose dynamics is ruled by [Formula: see text], a second-order elliptic partial differential operator of Laplace type. Using the functional formalism and working within the framework of algebraic quantum field theory and of the principle of general local covariance, first we construct the algebra of locally covariant observables in terms of equivariant sections of a bundle of smooth, regular polynomial functionals over the affine space of the parametrices associated to [Formula: see text]. Subsequently, adapting to the case in hand a strategy first introduced by Hollands and Wald in a Lorentzian setting, we prove the existence of Wick powers of the underlying field, extending the procedure to smooth, local and polynomial functionals and discussing in the process the regularization ambiguities of such procedure. Subsequently we endow the space of Wick powers with an algebra structure, dubbed E-product, which plays in a Riemannian setting the same role of the time-ordered product for field theories on globally hyperbolic spacetimes. In particular, we prove the existence of the E-product and we discuss both its properties and the renormalization ambiguities in the underlying procedure. As the last step, we extend the whole analysis to observables admitting derivatives of the field configurations and we discuss the quantum Møller operator which is used to investigate interacting models at a perturbative level.

Solution of the κ-Deformed Dirac Equation with Vector and Scalar Interactions in the Context of Spin and Pseudospin Symmetries

The deformed Dirac equation invariant under the κ-Poincaré-Hopf quantum algebra in the context of minimal and scalar couplings under spin and pseudospin symmetry limits is considered. The κ-deformed Pauli-Dirac Hamiltonian allows us to study effects of quantum deformation in a class of physical systems, such as a Zeeman-like effect, Aharonov-Bohm effect, and an anomalous-like contribution to the electron magnetic moment, between others. In our analysis, we consider the motion of an electron in a uniform magnetic field and interacting with (i) a planar harmonic oscillator and (ii) a linear potential. We verify that the particular choice of a linear potential induces a Coulomb-type term in the equation of motion. Expressions for the energy eigenvalues and wave functions are determined taking into account both symmetry limits. We verify that the energies and wave functions of the particle are modified by the deformation parameter as well as by the element of spin.

The Regular Ground States of the Linear Boson Field in Terms of Soft Modes

We supplement the characterization of the regular ground states of the linear boson field, as detailed in Segal (Illinois J. Math. 6 (1962), 500–523), Weinless (J. Funct. Anal. 4 (1969), 350–379) and Baez, Segal & Zhou (Introduction to algebraic and constructive quantum field theory, Princeton University Press, 1992, Princeton), by using more developed operator algebraic methods for general Weyl systems and by continuing ideas of Honegger & Rieckers (Photons in Fock space and beyond I, World Scientific, 2015, Singapore). For a linear boson field in the sense of Segal (Illinois J. Math. 6 (1962), 500–523) the originally real symplectic test function dynamics is given in terms of a continuous unitary group. The quantized field dynamics is realized via a continuous unitary group in a representation space of the Weyl commutation relations and specifies a quasifree automorphism group, which does not act strongly continuous on the C*-Weyl algebra. If the generator of the test function dynamics is strictly positive, we show that the quasifree dynamical automorphism group is R -central. Under this condition, we determine the form of all “partially regular” ground states, which are regular only over a subspace of test functions, introduced in Segal (Illinois J. Math. 6 (1962), 500–523). All regular ground states are then functional integrals , in some weakened sense, over the bare vacua dressed by singular classical modes. If the ground states are nuclear continuous these functional integrals are proper ones. If there are also time-invariant test functions then the related invariant quantum modes intervene into these superpositions of the pure regular ground states and complicate particularly the fully regular case. Global features of the sets of (partially) regular ground states are detailed and compared with the ground state sets of C*-dynamical systems. By way of example, the vacuum dressing is interpreted as a soft-boson cloud, forming collective classical soft modes.

Unifying lattice models, links and quantum geometric Langlands via branes in string theory

We explain how, starting with a stack of D4-branes ending on an NS5-brane in type IIA string theory, one can, via T-duality and the topological-holomorphic nature of the relevant worldvolume theories, relate (i) the lattice models realized by Costello's 4d Chern-Simons theory, (ii) links in 3d analytically-continued Chern-Simons theory, (iii) the quantum geometric Langlands correspondence realized by Kapustin-Witten using 4d N = 4 gauge theory and its quantum group modification, and (iv) the Gaitsgory-Lurie conjecture relating quantum groups/affine Kac-Moody algebras to Whittaker D-modules/W-algebras. This furnishes, purely physically via branes in string theory, a novel bridge between the mathematics of integrable systems, geometric topology, geometric representation theory, and quantum algebras.

Entanglement wedge reconstruction using the Petz map

At the heart of recent progress in AdS/CFT is the question of subregion duality, or entanglement wedge reconstruction: which part(s) of the boundary CFT are dual to a given subregion of the bulk? This question can be answered by appealing to the quantum error correcting properties of holography, and it was recently shown that robust bulk (entanglement wedge) reconstruction can be achieved using a universal recovery channel known as the twirled Petz map. In short, one can use the twirled Petz map to recover bulk data from a subset of the boundary. However, this map involves an averaging procedure over bulk and boundary modular time, and hence it can be somewhat intractable to evaluate in practice. We show that a much simpler channel, the Petz map, is sufficient for entanglement wedge reconstruction for any code space of fixed finite dimension — no twirling is required. Moreover, the error in the reconstruction will always be non-perturbatively small. From a quantum information perspective, we prove a general theorem extending the use of the Petz map as a general-purpose recovery channel to subsystem and operator algebra quantum error correction.

From Hopf algebras to topological quantum groups. A short history, various aspects and some problems

Quantum groups have been studied within several areas of mathematics and mathematical physics. This has led to different approaches, each of them with their own techniques and conventions. Starting with Hopf algebras, where there is a general consensus, moving in the direction of topological quantum groups, where there is no such consensus, it is easy to get lost. Not only many difficulties have to be overcome, but also several choices must be made. The way this is done is often confusing. Some choices even turn out to be rather annoying. As an introductory lecture at the conference on Topological quantum groups and Hopf algebras in 2016, we have explained these choices, difficulties and annoyances, encountered on the road from Hopf algebras to topological quantum groups. In these notes, we discuss more aspects of the development of locally compact quantum groups. We not only explain some of these difficulties in greater detail, but we also give background information about the different steps, combined with some historical comments. We start with finite quantum groups and continue with discrete quantum groups, compact quantum groups and algebraic quantum groups. Multiplicative unitaries are an important side track before we finally arrive at locally compact quantum groups. Along the way, we also formulate some interesting remaining problems in the theory.

Relating Nets and Factorization Algebras of Observables: Free Field Theories

In this paper we relate two mathematical frameworks that make perturbative quantum field theory rigorous: perturbative algebraic quantum field theory (pAQFT) and the factorization algebras framework developed by Costello and Gwilliam. To make the comparison as explicit as possible, we use the free scalar field as our running example, while giving proofs that apply to any field theory whose equations of motion are Green-hyperbolic (which includes, for instance, free fermions). The main claim is that for such free theories, there is a natural transformation intertwining the two constructions. In fact, both approaches encode equivalent information if one assumes the time-slice axiom. The key technical ingredient is to use time-ordered products as an intermediate step between a net of associative algebras and a factorization algebra.

Generalized quantum cumulant dynamics.

A means of unifying some semiclassical models of computational chemistry is presented; these include quantized Hamiltonian dynamics, quantal cumulant dynamics, and semiclassical Moyal dynamics (SMD). A general method for creating the infinite hierarchy of operator dynamics in the Heisenberg picture is derived together with a general method for truncation (or closure) of that series, and in addition, we provide a simple link to the phase space methods of SMD. Operator equations of arbitrary order may be created readily, avoiding the tedious algebra identified previously. Truncation is based on a simple recurrence formula which is related to, but avoids the more complex contractions of, Wick's theorem. This generalized method is validated against a number of trial problems considered using the previous methods. We also touch on some of the limitations involved using such methods, noting, in particular, that any truncation will lead to a state which is in some sense unphysical. Finally, we briefly introduce our quantum algebra package QuantAL which provides an automated method for the generation of the required equation set, the initial conditions for all variables from any start, and all the higher order approximations necessary for truncation of the series, at essentially arbitrary order.

Quantum Groups in Nuclear Spectra and in Metal Clusters

Quantum algebras (quantum groups), which are nonlinear generalizations of the usual Lie algebras, provide a rich variety of symmetries finding applications in the description of several physical systems [1]. Using irreducible tensor operators under sug(2) a rotationally invariant Hamiltonian which provides a good description of nuclear rotational spectra is constructed and its relation to existing nuclear models is considered. Using the same techniques a 3-dimensional ç-deformed harmonic oscillator with ug(3)Dsoq(3) symmetry is constructed, compared to the modified oscillator of Nilsson, and used for the successful description of magic numbers [2] and supershells [3] in metal clusters.

Linear Yang\u2013Mills Theory as a Homotopy AQFT

It is observed that the shifted Poisson structure (antibracket) on the solution complex of Klein–Gordon and linear Yang–Mills theory on globally hyperbolic Lorentzian manifolds admits retarded/advanced trivializations (analogs of retarded/advanced Green’s operators). Quantization of the associated unshifted Poisson structure determines a unique (up to equivalence) homotopy algebraic quantum field theory (AQFT), i.e. a functor that assigns differential graded *-algebras of observables and fulfills homotopical analogs of the AQFT axioms. For Klein–Gordon theory the construction is equivalent to the standard one, while for linear Yang–Mills it is richer and reproduces the BRST/BV field content (gauge fields, ghosts and antifields).

Locality and causality in perturbative algebraic quantum field theory

In this paper, we discuss how seemingly different notions of locality and causality in quantum field theory can be unified using a non-Abelian generalization of the Hammerstein property (originally introduced as a weaker version of linearity). We also prove a generalization of the main theorem of renormalization, in which we do not require field independence.

Algebraic Gorkov solution in finite systems for the separable pairing interaction

An algebraic Quantum Field Theory formulation of separable pairing interaction for spherical finite systems is presented. The Lipkin formulation of the model Hamiltonian and model wave function is used. The Green function technique is applied to obtain the model energy through the spectral function. Closed equation for the many-body energy of the system is given and comparison with exact models are performed.