Hidden Algebra Research Articles

Two-body Coulomb problem and hidden g (2) algebra: superintegrability and cubic polynomial algebra

It is shown that the two-body Coulomb problem in the Sturm representation leads to a new two-dimensional, exactly-solvable, superintegrable quantum system in curved space with a g (2) hidden algebra and a cubic polynomial algebra of integrals. The two integrals are of orders two and four, they are made from two components of the angular momentum and from the modified Laplace-Runge-Lenz vector, respectively. It is demonstrated that the cubic polynomial algebra is an infinite-dimensional subalgebra of the universal enveloping algebra Ug (2).

Conservation of all Lipkin’s zilches from symmetries of the standard electromagnetic action and a hidden algebra

In 1964, Lipkin discovered the zilches, a set of conserved quantities in free electromagnetism. Among the zilches, optical chirality was identified by Tang and Cohen in 2010, serving as a measure of the handedness of light and leading to investigations into light’s interactions with chiral matter. While the symmetries underlying the conservation of the zilches have been examined, the derivation of zilch conservation laws from symmetries of the standard free electromagnetic (EM) action using Noether’s theorem has only been addressed in the case of optical chirality. We provide the full answer by demonstrating that the zilch symmetry transformations of the four-potential, A_{mu }, preserve the standard free EM action. We also show that the zilch symmetries belong to the enveloping algebra of a “hidden” invariance algebra of free Maxwell’s equations. This “hidden” algebra is generated by familiar conformal transformations and certain “hidden” symmetry transformations of A_{mu }. Generalizations of the “hidden” symmetries are discussed in the presence of a material four-current, as well as in the theory of a complex Abelian gauge field. Additionally, we extend the zilch symmetries of the standard free EM action to the standard interacting action (with a non-dynamical four-current), allowing for a new derivation of the continuity equation for optical chirality in the presence of electric charges and currents. Furthermore, new continuity equations for the remaining zilches are derived.

Generalized three-body harmonic oscillator system: ground state

In this work we report on a 3-body system in a d−dimensional space ℝd with a quadratic harmonic potential in the relative distances rij = |ri −rj| between particles. Our study considers unequal masses, different spring constants and it is defined in the three-dimensional (sub)space of solutions characterized (globally) by zero total angular momentum. This system is exactly-solvable with hidden algebra sℓ4(ℝ). It is shown that in some particular cases the system becomes maximally (minimally) superintegrable. We pay special attention to a physically relevant generalization of the model where eventually the integrability is lost. In particular, the ground state and the first excited state are determined within a perturbative framework.

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model

A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied. We present a set of six operators $\{A^{\pm},B^{\pm},C^{\pm}\}$ that can be combined to build a ${\mathfrak{gl}}(3)$ hidden algebra. The latter can be embedded in an ${\mathfrak{sp}}(6)$ algebra, as well as in an ${\mathfrak{osp}}(1/6)$ superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.

Three-body closed chain of interactive (an)harmonic oscillators and the algebra

In this work we study 2- and 3-body oscillators with quadratic and sextic pairwise potentials which depend on relative distances, , between particles. The two-body harmonic oscillator is two-parametric and can be reduced to a one-dimensional radial Jacobi oscillator with hidden algebra , while in the 3-body case such a reduction is not possible in general. Our study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only (S-states). We pay special attention to the cases where all masses of the particles and spring constants are unequal as well as to the atomic, where one mass is infinite, and molecular, where two masses are infinite, limits; it is a non-integrable system. In general, a three-body harmonic oscillator is 7-parametric depending on three masses and three spring constants, and frequency; it is an exactly-solvable problem with spectra linear in three quantum numbers and with hidden algebra . In particular, the first and second order integrals of the 3-body oscillator for unequal masses are searched: it is shown that for certain relations involving masses and spring constants the system becomes maximally (minimally) superintegrable in the case of two (one) relations.

Three-body problem in d-dimensional space: Ground state, (quasi)-exact-solvability

As a straightforward generalization and extension of our previous paper [A. V. Turbiner et al., “Three-body problem in 3D space: Ground state, (quasi)-exact-solvability,” J. Phys. A: Math. Theor. 50, 215201 (2017)], we study the aspects of the quantum and classical dynamics of a 3-body system with equal masses, each body with d degrees of freedom, with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories (which are in the interaction plane) in the classical case are of this type. The quantum (and classical) Hamiltonian for which these states are eigenfunctions is derived. It corresponds to a three-dimensional quantum particle moving in a curved space with special d-dimension-independent metric in a certain d-dependent singular potential, while at d = 1, it elegantly degenerates to a two-dimensional particle moving in flat space. It admits a description in terms of pure geometrical characteristics of the interaction triangle which is defined by the three relative distances. The kinetic energy of the system is d-independent; it has a hidden sl(4, R) Lie (Poisson) algebra structure, alternatively, the hidden algebra h(3) typical for the H3 Calogero model as in the d = 3 case. We find an exactly solvable three-body S3-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly solvable three-body sextic polynomial type potential with singular terms. For both models, an extra first order integral exists. For d = 1, the whole family of 3-body (two-dimensional) Calogero-Moser-Sutherland systems as well as the Tremblay-Turbiner-Winternitz model is reproduced. It is shown that a straightforward generalization of the 3-body (rational) Calogero model to d > 1 leads to two primitive quasi-exactly solvable problems. The extension to the case of non-equal masses is straightforward and is briefly discussed.

Three-body problem in 3D space: ground state, (quasi)-exact-solvability

We study aspects of the quantum and classical dynamics of a 3-body system in 3D space with interaction depending only on mutual distances. The study is restricted to solutions in the space of relative motion which are functions of mutual distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories in the classical case are of this type. The quantum (and classical) system for which these states are eigenstates is found and its Hamiltonian is constructed. It corresponds to a three-dimensional quantum particle moving in a curved space with special metric. The kinetic energy of the system has a hidden sl(4, R) Lie (Poisson) algebra structure, alternatively, the hidden algebra h(3) typical for the H3 Calogero model. We find an exactly solvable three-body generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential; both models have an extra integral.

Hidden gauge structure of supersymmetric free differential algebras

The aim of this paper is to clarify the role of the nilpotent fermionic generator Q' introduced in Ref. [3] and appearing in the hidden supergroup underlying the free differential algebra (FDA) of D=11 supergravity. We give a physical explanation of its role by looking at the gauge properties of the theory. We find that its presence is necessary, in order that the extra 1-forms of the hidden supergroup give rise to the correct gauge transformations of the p-forms of the FDA. This interpretation is actually valid for any supergravity containing antisymmetric tensor fields, and any supersymmetric FDA can always be traded for a hidden Lie superalgebra containing extra fermionic nilpotent generators. As an interesting example we construct the hidden superalgebra associated with the FDA of N=2, D=7 supergravity. In this case we are able to parametrize the mutually non local 2- and 3-form B^(2) and B^(3) in terms of hidden 1-forms and find that supersymmetry and gauge invariance require in general the presence of two nilpotent fermionic generators in the hidden algebra. We propose that our approach, where all the invariances of the FDA are expressed as Lie derivatives of the $p$-forms in the hidden supergroup manifold, could be an appropriate framework to discuss theories defined in enlarged versions of superspace recently considered in the literature, such us double field theory and its generalizations.

(Quasi)-exact-solvability on the sphere Sn

An Exactly Solvable (ES) potential on the sphere Sn is reviewed and a related Quasi-Exactly Solvable (QES) potential is found and studied. After mapping the sphere to a simplex, it is found that the metric (of constant curvature) is in polynomial form, and both the ES and the QES potentials are rational functions. Their hidden algebra is gln, realized in a finite-dimensional representation by first order differential operators acting on RPn. It is shown that variables in the Schrödinger eigenvalue equation can be separated in polyspherical coordinates and there is always complete integrability. The QES system is completely integrable for n = 2 and non-maximally superintegrable for n ≥ 3. There is no separable coordinate system in which it is exactly solvable. We point out that by taking contractions of superintegrable systems, such as induced by Inönü-Wigner Lie algebra contractions, we can find other QES superintegrable systems, and we illustrate this by contracting our Sn system to a QES non-maximal superintegrable system on Euclidean space En, an extension of the Smorodinsky-Winternitz potential.

G 2 Algebra and two-dimensional quasiexactly solvable Hamiltonian related to Poschl–Teller potential

In this article, we write the general form of the quasiexactly solvable Hamiltonian of g 2 algebra via one special representation in the x–y two-dimensional space. Then, by choosing an appropriate set of coefficients and making a gauge rotation, we show that this Hamiltonian leads to the solvable Poschl–Teller model on an open infinite strip. Finally, we assign g 2 hidden algebra to the Poschl–Teller potential and obtain its spectrum by using the representation space of g 2 algebra.

Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions

The quantum mechanics of two Coulomb charges on a plane (e1, m1) and (e2, m2) subject to a constant magnetic field B perpendicular to the plane is considered. Four integrals of motion are explicitly indicated. It is shown that for two physically important particular cases, namely that of two particles of equal Larmor frequencies, (e.g. two electrons) and one of a neutral system (e.g. the electron–positron pair, hydrogen atom) at rest (the center-of-mass momentum is zero) some outstanding properties occur. They are the most visible in double polar coordinates in CMS (R, ϕ) and relative (ρ, φ) coordinate systems: (i) eigenfunctions are factorizable, all factors except one with the explicit ρ-dependence are found analytically, they have definite relative angular momentum, (ii) dynamics in the ρ-direction is the same for both systems, it corresponds to a funnel-type potential and it has hidden sl(2) algebra, at some discrete values of dimensionless magnetic fields b ⩽ 1, (iii) particular integral(s) occur, (iv) the hidden sl(2) algebra emerges in finite-dimensional representation, thus, the system becomes quasi-exactly-solvable and (v) a finite number of polynomial eigenfunctions in ρ appear. Nine families of eigenfunctions are presented explicitly.

Coinduction for preordered algebra

We develop a combination, called hidden preordered algebra, between preordered algebra, which is an algebraic framework supporting specification and reasoning about transitions, and hidden algebra, which is the algebraic framework for behavioural specification. This combination arises naturally within the heterogeneous framework of the modern formal specification language CafeOBJ. The novel specification concept arising from this combination, and which constitutes its single unique feature, is that of behavioural transition. We extend the coinduction proof method for behavioural equivalence to coinduction for proving behavioural transitions.

An Algebraic Framework for Modeling of Mobile Systems

We present MobileOBJ, a formal framework for specifying and verifying mobile systems. Based on hidden algebra, the components of a mobile system are specified as behavioral objects or Observational Transition Systems, a kind of transition system, enriched with special action and observation operators related to the distinct characteristics of mobile computing systems. The whole system comes up as the concurrent composition of these components. The implementation of the abstract model is achieved using CafeOBJ, an executable, industrial strength algebraic specification language. The visualization of the specification can be done using CafeOBJ graphical notation. In addition, invariant and behavioral properties of mobile systems can be proved through theorem proving techniques, such as structural induction and coinduction that are fully supported by the CafeOBJ system. The application of the proposed framework is presented through the modeling of a mobile computing environment and the services that need to be supported by the former.

Object oriented institutions to specify symbolic computation systems

The specification of the data structures used in EAT, a software system for symbolic computation in algebraic topology, is based on an operation that defines a link among different specification frameworks like hidden algebras and coalgebras. In this paper, this operation is extended using the notion of institution, giving rise to three institution encodings. These morphisms define a commutative diagram which shows three possible views of the same construction, placing it in an equational algebraic institution, in a hidden institution or in a coalgebraic institution. Moreover, these morphisms can be used to obtain a new description of the final objects of the categories of algebras in these frameworks, which are suitable abstract models for the EAT data structures. Thus, our main contribution is a formalization allowing us to encode a family of data structures by means of a single algebra (which can be described as a coproduct on the image of the institution morphisms). With this aim, new particular definitions of hidden and coalgebraic institutions are presented.

Quasi exactly solvable operators and Lie superalgebras

Linear operators preserving the direct sum of polynomial rings P(m)⊕ P(n) are constructed. In the case | m− n|=1 they correspond to atypical representations of the superalgebra osp(2,2). For | m− n|=2 the generic, finite-dimensional representations of the superalgebra q(2) are recovered. An example of a Hamiltonian possessing such a hidden algebra is analyzed.

A hidden Herbrand theorem: combining the object and logic paradigms

The benefits of the object, logic (or relational), functional, and constraint paradigms can be obtained from our previous combination of the object and functional paradigms in hidden algebra, by combining it with existential queries over the states and attributes of objects, and then lifting to hidden Horn clause logic with equality, using an extension of a result due to Diaconescu. We call this novel programming paradigm active constraint object programming, suggest some applications for it, and show that it is computationally feasible by reducing it to familiar problems over term algebras (i.e., Herbrand universes). Our main result is a version of Herbrand's Theorem, lifted from hidden algebra by the extended result of Diaconescu. This paper also contains new results on the existence of initial and final models, and on the consistency of hidden theories.

What is the coalgebraic analogue of Birkhoff's variety theorem?

Logical definability is investigated for certain classes of coalgebras related to state-transition systems, hidden algebras and Kripke models. The filter enlargement of a coalgebra A is introduced as a new coalgebra A + whose states are special “observationally rich” filters on the state set of A. The ultrafilter enlargement is the subcoalgebra A ∗ of A + whose states are ultrafilters. Boolean combinations of equations between terms of observable (or output) type are identified as a natural class of formulas for specifying properties of coalgebras. These observable formulas are permitted to have a single-state variable, and form a language in which modalities describing the effects of state transitions are implicitly present. A ∗ and A + validate the same observable formulas. It is shown that a class of coalgebras is definable by observable formulas iff the class is closed under disjoint unions, images of bisimulations, and (ultra)filter enlargements. (Closure under images of bisimulations is equivalent to closure under images and domains of coalgebraic morphisms.) Moreover, every set of observable formulas has the same models as some set of conditional equations. Examples are constructed to show that the use of enlargements is essential in these characterisations, and that there are classes of coalgebras definable by conditional observable equations, but not by equations alone. The main conclusion of the paper is that to structurally characterise classes of coalgebras that are logically definable by modal languages requires a new construction, of “Stone space” type, in addition to the coalgebraic duals of the three constructions (homomorphisms, subalgebras, direct products) that occur in Birkhoff's original variety theorem for algebras.

Semantic constructions for the specification of objects

Hidden algebra is a behavioural algebraic specification formalism for objects. It captures their constructional aspect (concerned with the initialisation and evolution of their states), their observational aspect (concerned with the observable behaviour of such states), and the relationship between these two aspects. When attention is restricted to the observational aspect, final/cofree algebras provide suitable denotations for the specification techniques employed by hidden algebra. However, when the constructional aspect is integrated with the observational one, the possibility of underspecification prevents the existence of such algebras. It is shown here that final/cofree familiesof algebras exist in this case, with each algebra in such a family resolving the nondeterminism arising from underspecification in a particular way. The existence of final/cofree families also yields a canonical way of constructing algebras of structured specifications from algebras of their component specifications.

Equational axiomatizability for coalgebra

A characterization result for equationally definable classes of certain coalgebras (including basic hidden algebra) shows that a class of coalgebras is definable by equations if and only if it is closed under coproducts, quotients, sources of morphisms and representative inclusions. The notions of equation and satisfaction are axiomatized in order to include the different approaches in the literature.

Web-Based Support for Cooperative Software Engineering

The Tatami project is building a system to support software engineering over the internet, exploiting recent advances in Web technology, interface design, and specification. Our effort to improve the usability of such systems has led us into algebraic semiotics, while our effort to develop better formal methods for distributed concurrent systems has led us into hidden algebra and fuzzy logic. This paper discusses the Tatami system design, especially its software architecture, and its user interface principles. New work in the latter area includes an extension of algebraic semiotics to dynamic multimedia interfaces, and integrating Gibsonian affordances with algebraic semiotics.