Spectrum-generating Algebras Research Articles

Transmuted spectrum-generating algebras and detectable parastatistics of the Superconformal Quantum Mechanics

In a recent paper (Balbino-de Freitas-Rana-FT, arXiv:2309.00965) we proved that the supercharges of the supersymmetric quantum mechanics can be statistically transmuted and accommodated into a -graded parastatistics. In this talk I derive the 6 = 1 + 2 + 3 transmuted spectrum-generating algebras (whose respective gradings are n = 0, 1, 2) of the 𝒩 = 2 Superconformal Quantum Mechanics. These spectrum-generating algebras allow to compute, in the corresponding multiparticle sectors of the de Alfaro-Fubini-Furlan deformed oscillator, the degeneracies of each energy level. The levels induced by the Z 2 × Z 2-graded paraparticles cannot be reproduced by the ordinary bosons/fermions statistics. This implies the theoretical detectability of the Z 2 × Z 2-graded parastatistics.

Algebraic models of many-body systems and their dynamic symmetries and supersymmetries

An overview of the method of spectrum generating algebras (SGA) and dynamical symmetries (DS) is given and applied to a class of models (the so-called s-b boson models) with SGA≡u(n). Quantum phase transitions (QPT) in these systems are discussed by introducing the coset spaces U(n)/U(n-1)⊗U(1). Finally, spectrum generating superalgebras (SGSA) and dynamical supersymmetries (SUSY) of a class of Bose-Fermi systems with SGA≡u(n/m) are introduced and applied to s-b Bose-Fermi models.

Towards classical spectrum generating algebras for f-deformations

In this paper we revise the classical analog of f-oscillators, a generalization of q-oscillators given in Man'ko et al. (1997) [8], in the framework of classical spectrum generating algebras (SGA) introduced in Kuru and Negro (2008) [9]. We write down the deformed Poisson algebra characterizing the entire family of non-linear oscillators and construct its general solution algebraically. The latter, covering the full range of f-deformations, shows an energy dependence both in the amplitude and the frequency of the motion.

Approach of spherical harmonics to the representation of the deformed su(1,1) algebra

The m-shifting generators of su(2) algebra together with a pair of l-shifting ladder symmetry operators have been used in the space of all spherical harmonics Ylm(θ,ϕ) in order to introduce a new set of operators, expressing the transitions between them. It is shown that the space of spherical harmonics whose l+2m or l−2m is given presents negative and positive irreducible representations of a deformed su(1,1) algebra, respectively. These internal symmetries also suggest new algebraic methods to construct the spherical harmonics in the framework of the spectrum-generating algebras.

Exactly solvable models in supersymmetric quantum mechanics and connection with spectrum-generating algebras

For nonrelativistic Hamiltonians which are shape invariant, analytic expressions for the eigenvalues and eigenvectors can be derived using the well known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess spectrum generating algebras and are hence solvable by an independent group theoretic method. In this paper, we demonstrate the equivalence of the two methods of solution by developing an algebraic framework for shape invariant Hamiltonians with a general change of parameters, which involves nonlinear extensions of Lie algebras.

TWO NEW CLASSES OF CONDITIONALLY EXACTLY SOLVABLE POTENTIAL AND THEIR NONLINEAR SPECTRUM-GENERATING ALGEBRAS

Using ideas of supersymmetric quantum mechanics, two new classes of conditionally exactly solvable potentials are constructed from the displaced harmonic oscillator potential. The supersymmetry as well as the space symmetry is broken in one class of them. Furthermore we construct the nonlinear spectrum-generating algebras for them.

On the Spectrum-Generating Algebras of Oscillator and Coulomb Problems

The three-dimensional oscillator and Coulomb problems are compared from the point of view of their spectrum-generating algebras. The representation independence (duality) of their shift operators and their classical limits are discussed. A "Bertrand type" theorem for bifactorizability of vector operators is conjectured.

The role of dynamic symmetries in hadronic structure

The role of spectrum generating algebras (SGA) and dynamic symmetries (DS) in hadronic structure is briefly reviewed.

Universal enveloping algebras of Toda field theories and the light-cone asymmetry parameter

The generators of the universal enveloping algebras in Toda field theories associated with Lie algebras are constructed. These form spectrum-generating algebras of the system which survive the constraints acting on the larger current algebra structure. It is found that the same generators fail to be a symmetry in the case of affine Toda field theory despite their close relationship with Mandelstam's soliton operators. We introduce the light-cone asymmetry parameter; its significance and utility are demostrated.

BRS cohomology in string theory: Geometry of Abelization and the quartet mechanism

The Abelian representation of the string Becchi–Rouet–Stora (BRS) cohomology in terms of basic operators naturally forming two dual Kugo–Ojima quartets is found. Furthermore, using a gauge-fixing scheme, a contracting homotopy operator associated with the string BRS operator is constructed. This formalism gives the explicit realizations of the physical and unphysical subspaces. The passage between Abelian and non-Abelian quantities is realized geometrically through the use of the moments of the vertex operator, which act as vielbeins between infinite-dimensional algebras. The connection between the Virasoro and spectrum-generating algebras is clarified and the algebraic duality relation between them is uncovered.

Dynamical superalgebra and supersymmetry for a many-fermion system.

A linearization scheme is proposed for the general Hamiltonian of an interacting fermion system, consisting of a mean-field approximation in which pairing and umklapp play dominant roles. A variety of models emerge, characterized by a hierarchy of spectrum-generating algebras and superalgebras, of which some are supersymmetric. The superconducting phase transition is shown to be related to supersymmetry breaking.

Finite-Dimensional Spectrum-Generating Algebras

It is suggested that the generators of a spectrum-generating algebra are all constants of the motion, some of them having an explicit time dependence. Also suggested is a specific form of the Hamiltonian action on the spectrum-generating algebra for systems with a finite number of degrees of freedom. Well-known examples of spectrum-generating algebras are shown to fit into this framework. The stability of the suggested structure against small perturbation is discussed. The question of the generalization of the suggested structure to systems with an infinite number of degrees of freedom is briefly commented upon.

Algebraic Aspects of Regge Recurrences

The algebraic structure of Regge recurrences is examined, and its relation to symmetry schemes underlying various models is established. The relationship between the Spectrum Generating Algebras (SGA) and the noncompact subalgebras of the rotational states are discussed for the Coulomb potential, the three-dimensional oscillator, the signature which appears in exchange potentials, as well as the U(6, 6) and the other suggested SGA of hadron physics.