Lie Algebra Su Research Articles

So(3) ⊂ su(3) revisited

This paper reproduces the result of Elliot, namely that the irreducible finite dimensional representation of the Lie algebra su(3) of highest weight (m, n) is decomposed according to the embedding so(3) ⊂ su(3). First, a realisation (a representation in terms of vector fields) of the Lie algebra su(3) is constructed on a space of polynomials of three variables. The special polynomial basis of the representation space is given. In this basis, we find the highest weight vectors of the representation of the Lie subalgebra so(3) and in this way the representation space is decomposed to the direct sum of invariant subspaces. The process is illustrated by the example of the decomposition of the representation of highest weight (2, 2). As an additional result, the generating function of the decomposition is given.

Eulerian and Lagrangian Stability in Zeitlin’s Model of Hydrodynamics

The two-dimensional (2-D) Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant Riemannian metric. This structure enables insights to Eulerian and Lagrangian stability via sectional curvature and Jacobi equations. The Zeitlin model is a finite-dimensional analogue of the 2-D Euler equations; the only known discretization that preserves the rich geometric structure. Theoretical and numerical studies indicate that Zeitlin’s model provides consistent long-time behaviour on large scales, but to which extent it truly reflects the Euler equations is mainly open. Towards progress, we give here two results. First, convergence of the sectional curvature in the Euler–Zeitlin equations on the Lie algebra su(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {su}(N)$$\\end{document} to that of the Euler equations on the sphere. Second, L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-convergence of the corresponding Jacobi equations for Lagrangian and Eulerian stability. The results allow geometric conclusions about Zeitlin’s model to be transferred to Euler’s equations and vice versa, which could expedite the ultimate aim: to characterize the generic long-time behaviour of perfect 2-D fluids.

On different approaches to integrable lattice models

Interaction-round the face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e. the models the Boltzmann weights of which satisfy the quantum Yang–Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras su(2)k and su(3)k are computed for fundamental and adjoint representations for some fixed levels k. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on su(3)k (for general k) and discuss how it can be used to find Boltzmann weights in terms of the quantum R^ matrix when the adjoint representation defines the admissibility conditions.

Normal ordering of the angular momentum cylindrical ladder operators and their products with Wigner D0,ε 1 functions.

The operator canonical perturbation theory (CPT) is an efficient tool for solving the molecular vibration-rotation Schrödinger equation. The corresponding Watson Hamiltonian can be written using angular momentum cylindrical ladder operators (Jz, J± = Jx ∓ iJy) possessing the Lie algebra su(2) commutation relations [J+, J-] = 2Jz, [Jz, J±] = ±J±. The reduced effective Hamiltonians suitable for fitting to observed spectra are traditionally based on Hermitian basis sets, e.g., (J2)lJz n,(J+ m+J- m)+. It is beneficial to re-express such Hamiltonians using sums of normal ordered products of ladder operators Jz aJ+ bJ- c. For instance, in the CPT, the unitary transformations reduce to the normal ordering problem. Similarly, the dipole moment transition probabilities can be evaluated using Wigner functions, D0,ε 1(ε=0,±1), possessing complex commutation relations with Jα-operators. The related line strengths are proportional to the squared matrix elements of the unitary transformed dipole moment operator given by a polynomial in normal ordered products D0,ε 1Jz aJ+ bJ- c. We have applied the technique of combinatorial calculations associated with the classical representation theory and obtained compact formulas for normal ordering of the typical raw products Jz aJ+ bJ- cJz dJ+ eJ- f and Jz aJ+ bJ- cD0,ε 1. The theory of universal enveloping Lie algebras and representation theory show that the resulting formulas cannot be further improved from the mathematical point of view. The results have a wide scope of applications in other fields of physics, including other operators with the same commutation properties. The resulting expressions for normal orderings and routines for evaluation of matrix elements, as well as the systematic tests of working formulas, were coded in Fortran and the software is available through the GitHub hosting service.

Algebraic representation of Three Qubit Quantum Circuit Problems

The evolution of quantum states serves as good fundamental studies in understanding the quantum information systems which finally lead to the research on quantum computation. To carry out such a study, mathematical tools such as the Lie group and their associated Lie algebra is of great importance. In this study, the Lie algebra of su(8) is represented in a tensor product operation between three Pauli matrices. This can be realized by constructing the generalized Gell-Mann matrices and comparing them to the Pauli bases. It is shown that there is a one-to-one correlation of the Gell-Mann matrices with the Pauli basis which resembled the change of coordinates. Together with the commutator relations and the frequency analysis of the structure constant via the algebra, the Lie bracket operation will be highlighted providing insight into relating quantum circuit model with Lie Algebra. These are particularly useful when dealing with three-qubit quantum circuit problems which involve quantum gates that is derived from the SU(8) Lie group.

Octions: An E8 description of the Standard Model

We interpret the elements of the exceptional Lie algebra e8(−24) as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra su(3)⊕su(2)⊕u(1), and the Lorentz Lie algebra so(3,1). Our construction relies on identifying a complex structure on spinors and then working in the enveloping algebra. The resulting model naturally contains Grand Unified Theories based on SO(10) (Georgi), SU(5) (Georgi–Glashow), and SU(4) ×SU(2) ×SU(2) (Pati–Salam). We then briefly speculate on the role of the remaining elements of e8 and propose a mechanism leading to exactly three generations of particles.

Topology and Elastic Theory of DNA Molecule

We have determined at this point that the DNA molecule may be considered as a long chain of an antiferromagnetic spin adjustment when the spin is accurate to be situated on the axis of the supercoil. Furthermore, we examine that as two polynucleotide chains are coiled regarding the same axis with an exact helical sense in a DNA molecule, this can be considered as if a spin with a specific point of reference is confined on the axis of the coil such that two contiguous coils have reverse direction of the spin. Actually, with each turn two strands move in the opposite side of the axis and so the spin direction accredited for the two adjacent coils should be opposite to each other. As soon as spins are defined in the Lie algebra of SU (2) the linking number can be established from the Chern-Simon’s topology linked with the spin system. We shall study the topological property of a DNA supercoil from the angle of such a spin chain. Also, we shall put together the elastic properties of a DNA molecule from on analysis of the spin degrees of freedom. As spins are considered at this time as gauge currents formulated from SU (2) gauge fields we map the elastic properties onto the space of gauge potentials. In quantum field theory the divergence of this current is non-zero which gives increase to chiral anomaly caused by chiral balance contravention.

Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of SU(N) Gauge Symmetry

In this paper, we study possible mathematical connections of the Clifford algebra with the su(N)-Lie algebra, or in more physical terms the links between space-time symmetry (Lorentz invariance) and internal SU(N) gauge-symmetry for a massive spin one-half fermion described by the Dirac equation. The related matrix algebra is worked out in particular for the SU(2) symmetry and outlined as well for the color gauge group SU(3). Possible perspectives of this approach to unification of symmetries are briefly discussed. The calculations make extensive use of tensor multiplication of the matrices involved, whereby our focus is on revisiting the Coleman–Mandula theorem. This permits us to construct unified symmetries between Lorentz invariance and gauge symmetry in a direct product sense.

Эволюция кутрита и контракция алгебры Ли su(3)

The evolution of a three-level quantum system interacting with the environment with the symmetry of the Lie algebra su(3) (qutrit) is considered. The evolution of qutrit is obtained when interacting with the environment. The transverse relaxation of the qutrit (due to the diagonal Lindblad operators) has been studied, leading to zero (with the same speedfor the real and imaginary parts) off-diagonal elements of the density matrix. The dynamics of qutrit is obtained in the presence of both transverse and longitudinal relaxation, i.e. when the Lindblad operators are off-diagonal and there are transitions in the system between all energy levels. It is shown that the addition of the longitudinal relaxation ofthe qutrit in addition to the conversion of the offdiagonallements to zero, also affects the diagonal elements of its density matrix. As a result of dissipation, the quantum system becomes incoherent over time. In both cases, initially non-commuting quantum observables in the process of evolution acquire the properties of commuting classical observables,which algebraically manifests itself in the contraction of the algebra su(3).

Fock Spaces for the Complex Dunkl Operator and Deformed (1, 1) Algebra

In this paper, we study a class of generalized Fock spaces for the complex Dunkl operator associated with the cyclic group, generalizing the Bargmann representations for Lie algebra su(1,1)-generators. We present an explicit realization of Cr-extended oscillator in terms of this operator. Furthermore, we construct the wave functions and we determine the coherent states. We also determine an explicit realization of a polynomial deformation sud(1,1) of the classical Lie algebra su(1,1). Finally, we give a deforming functional which maps sud(1,1) to su(1,1).

A Space-Frame Periodic Table Representation System Testing Relativity in Nucleosynthesis of the Elements

The geometric Lie algebra SU(3) isotropic vector matrix realization of the periodic table reported at PIRT 2017 has now been broken up to a disjoint-set modular R3×SO(3) building kit, exactly sufficing to stage the Big Bang and ensuing nucleosynthesis events: First the ultrashort radiation/plasma inflation of the Big Bang phase transition moment with release of photons, neutrinos and module precursors, which in next seconds recombine to the Protium proton/electron compound and its neutron conversion to continue separately or in fusion of the two get on to Deuterium and from there to Tritium, Helium in isotope and α form, and traces of Lithium and possibly Beryllium. That is, literally the whole primordial start-gas delivered within a few minutes to billion-year wait for sufficiently energetic perturbations with itself for astrophysical/cosmogenic/experimental nucleosynthesis of the full periodic table and likewise replicable by systematic space-filling assembly/disassembly of the here disclosed neutrino and photon lattice vector, β particle, α wave-packet and neutron building bricks, providing clues also on isotope/neutron excess, shell/subshell, spectroscopy, and chemical bond structural make-up and disposition. Furthermore, the absolute trigonometric sharpness of the nucleosynthesis phase transition burst and expansion is reciprocal to the absolute speed of light and hence a specific test and verification of the relativity theory.

Normal ordering of the su(1, 1) ladder operators for the quasi-number states of the Morse oscillator

The Morse oscillator is an adequate zero-order model for describing the highly excited vibrational states and large-amplitude vibrational motion. The corresponding Schrödinger equations can be conveniently solved by algebraic methods using the so-called quasi-number states (QNS) resembling the true wave functions of the Morse oscillator. The associated QNS ladder operators are occupation number, creation and annihilation operators K0,K+,K− that satisfy the commutation relations [K0,K±]=±K± and [K+,K−]=−2K0 of the Lie algebra su(1, 1). The efficient applications of the corresponding algebraic techniques within the variational or perturbative approximations require reduction of arbitrary multiplicative combinations of the su(1,1) ladder operators to the linear combinations of the normalized products K0aK+bK−c. The closed form solution of this problem for a typical complex form of the operator product is derived and proven mathematically. The validity and the numerical stability of the derived expressions are demonstrated by the systematic numerical tests for the product of six-term multiplicative operators.

Algebraic structure of classical integrability for complex sine-Gordon

The algebraic structure underlying the classical rr-matrix formulation of the complex sine-Gordon model is fully elucidated. It is characterized by two matrices aa and ss, components of the rr matrix as r=a-sr=a−s. They obey a modified classical reflection/Yang–Baxter set of equations, further deformed by non-abelian dynamical shift terms along the dual Lie algebra su(2)^*su(2)*. The sign shift pattern of this deformation has the signature of the twisted boundary dynamical algebra. Issues related to the quantization of this algebraic structure and the formulation of quantum complex sine-Gordon on those lines are introduced and discussed.

The algebraic structure of the SU(7) Lie group

In recent years, the Lie group SU(7) has been featured prominently in a number of grand unification proposals involving the Standard Model as a low energy effective theory. This note investigates the framework of the SU(7) group. The antisymmetric and symmetric structure constants for the Lie algebra su (7) have been explicitly calculated from the generators for the fundamental representation of SU(7), which are also cataloged.

Potential algebra approach to quantum mechanics with generalized uncertainty principle

In this note, we study the potential algebra for several models arising out of quantum mechanics with generalized uncertainty principle. We first show that the eigenvalue equation corresponding to the momentum-space HamiltonianH=−(1+βp2)ddp(1+βp2)ddp+g(g−1)β2p2−gβ, which is associated with some one-dimensional models with minimal length uncertainty, can be solved by the unitary representations of the Lie algebra su(2) if g∈{12,1,32,2,⋯}. We then apply this result to spectral problems for the non-relativistic harmonic oscillator as well as the relativistic Dirac oscillator in the presence of a minimal length and show that these problems can be solved solely in terms of su(2).

Zernike functions, rigged Hilbert spaces, and potential applications

We revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1, 1) ⊕ su(1, 1). We show how they induce discrete as well as continuous bases that coexist in the framework of rigged Hilbert spaces. We also discuss some other interesting properties of Zernike polynomials and Zernike functions. One of the areas of interest of Zernike functions has been their applications in optics. Here, we suggest that operators on the spaces of Zernike functions may play a role in optical image processing.

Fermi surface enlargement on the Kondo lattice

The Kondo lattice model is a paradigmatic model for the description of local moment systems, a class of materials exhibiting a range of strongly correlated phenomena including heavy fermion formation, magnetism, quantum criticality and unconventional superconductivity. Conventional theoretical approaches invoke fractionalization of the local moment spin through large-N and slave particle methods. In this work we develop a new formalism, based instead on non-canonical degrees of freedom. We demonstrate that the graded Lie algebra su(2|2) provides a powerful means of organizing correlations on the Kondo lattice through a splitting of the electronic degree of freedom, in a manner which entwines the conduction electrons with the local moment spins. This offers a novel perspective on heavy fermion formation. Unlike slave-particle methods, non-canonical degrees of freedom generically allow for a violation of the Luttinger sum rule, and we interpret recent angle resolved photoemission experiments on Ce-115 systems in view of this.

On a Microscopic Representation of Space-Time III

Using the Dirac (Clifford) algebra $$\gamma ^{\mu }$$ as initial stage of our discussion, we summarize previous work with respect to the isomorphic 15 dimensional Lie algebra su*(4) as complex embedding of sl(2, $$\mathbb {H}$$ ), the relation to the compact group SU(4) as well as subgroups and group chains. The main subject, however, is to relate these technical procedures to the geometrical (and physical) background which we see in projective and especially in line geometry of $$\mathbb {R}^{3}$$ . This line geometrical description, however, leads to applications and identifications of line Complexe and the discussion of technicalities versus identifications of classical line geometrical concepts, Dirac’s ‘square root of $$p^{2}$$ ’, the discussion of dynamics and the association of physical concepts like electromagnetism and relativity. We outline a generalizable framework and concept, and we close with a short summary and outlook.

Fractional supersymmetric su(2) algebras

In this paper, the third root of the Lie algebra su(2) based on the permutation group S3 is formulated in the Hopf algebra formalism.

Splitting of electrons and violation of the Luttinger sum rule

We obtain a controlled description of a strongly correlated regime of electronic behaviour. We begin by arguing that there are two ways to characterise the electronic degree of freedom, either by the canonical fermion algebra or the graded Lie algebra su(2|2). The first underlies the Fermi liquid description of correlated matter, and we identify a novel regime governed by the latter. We exploit an exceptional central extension of su(2|2) to employ a perturbative scheme recently developed by Shastry, and obtain a series of successive approximations for the electronic Green's function. We then focus on the leading approximation, which reveals a splitting in two of the electronic dispersion. The Luttinger sum rule is violated, and a Mott metal-insulator transition is exhibited. We offer a perspective.