Lie Algebra Of SO Research Articles

Creation and annihilation operators for SU(3) in an SO(6,2) model

Creation and annihilation operators are defined which are Wigner operators (tensor shift operators) for SU(3). While the annihilation operators are simply boson operators, the creation operators are cubic polynomials in boson operators. Together they generate under commutation the Lie algebra of SO(6,2). The vector space generated from a vacuum vector by repeated application of the creation operators carries an irreducible representation of the SO(6,2) algebra, equivalent to an hermitian representation, and also carries in direct sum every different irreducible representation of SU(3)<SO(6,2) exactly once. A model for SU(3), in the sense of Bernstein, Gel'fand and Gel'fand, is therefore defined. The different SU(3) irreducible representations appear explicitly as manifestly covariant, irreducible tensors, whose orthogonality and normalisation properties are examined. Other Wigner operators for SU(3) can be constructed simply as products of the new creation and annihilation operators, or sums of such products.

Integrable Euler equations on SO(4) and their physical applications

For the Lie algebra SO(4) (and other six dimensional Lie algebras) we find some Euler's equations which have an additional fourth order integral and are algebraically integrable (in terms of elliptic functions) in a one parameter set of orbits. Integrable Euler's equations having an additional second order integral and generalizing Steklov's case are presented. Equations for rotation of a rigid body havingn ellipsoid cavities filled with the ideal incompressible fluid being in a state of homogeneous vortex motion are derived. It is shown that the obtained equations are Euler's equations for the Lie algebra of the groupGn+1=SO(3) × ... × SO(3). New physical applications of Euler's equations on SO(4) are discussed.

Connection between the hydrogen atom and the harmonic oscillator: The zero-energy case

The connection between the three-dimensional hydrogen atom and a four-dimensional harmonic oscillator obtained in previous works, from a hybridization of the infinitesimal Pauli approach to the hydrogen system with the Schwinger approach to spherical and hyperbolical angular momenta, is worked out in the case of the zero-energy point of the hydrogen atom. This leads to the equivalence of the three-dimensional hydrogen problem with a four-dimensional free-particle problem involving a constraint condition. For completeness, the latter result is also derived by using the Kustaanheimo-Stiefel transformation introduced in celestial mechanics. Finally, it is shown how the Lie algebra of SO(4,2) quite naturally arises for the whole spectrum (discrete plus continuum plus zero-energy point) of the three-dimensional hydrogen atom from the introduction of the constraint condition into the Lie algebra of Sp(8,R) associated with the four-dimensional harmonic oscillator.

Boson mappings for schematic nuclear models with the symmetry of SO(5)

In the past, several schematic nuclear models have been proposed with the symmetry of the Lie algebra SO(5). We derive a set of mappings from the shell model spaces in which these models are defined to suitable boson spaces. The derivations are carried out by two distinct algebraic methods, which are customarily associated with the names Holstein-Primakoff and Dyson, respectively. The relative utility of the two is seen to depend on the particular mapping. The mappings are useful for the study of the collective properties of the various models and of generalizations of each of them. This is illustrated with one of the models. Some algebraic details, some ideas concerning schematic model building, and an extension of the results of this paper are presented in appendices.

Lie algebra: the Delaunay similar elements in the produced eccentric anomaly

The concretisation of an abstract isomorphism between two Lie-algebras SO(4,2) leads to the transformation of the Keplerian problem from cartesian coordinates into Delaunay similar elements in the eccentric anomaly.

The algebraic structure of relativistic wave equations

The algebraic structure of relativistic wave equations of the formis considered. This leads to the problem of finding all Lie algebrasLwhich contain the Lorentz Lie algebraso(3, 1) and also contain a “four-vector” αμa such anLgives rise to a family of wave equations. The simplest possibility is the Bhabha equations whereL≅so(5). Some authors have claimed that this is theonlyone, but it is shown that there are many other possibilities still in accord with physical requirements. Known facts about representations, along with Dynkin's theory of the embeddings of Lie algebras, are used to obtain a partial classification of wave equations. The discrete transformationsC, P, Tare also discussed, along with reality properties. Finally, a simple example of a family of wave equations based onL=sp(12) is considered in detail. Theso(3, 1) content and mass spectra are given for the low order members of the family, and the problem of causality is briefly discussed.

Fermi-Bose Similarity

The usual theory of the Lie algebras and the Lie groups rs shown to be formally extended to the cases in which the group parameters commute and/or anticommute in the most general manner. It is then proved that the parafermi and parabose algebras for f degrees of freedom correspond to the Lie algebras of SO (2/+ 1) and of a graded version of Sp (2/, R), respectively. Further the trilinear and bilinear commutation relations for a general system comprising parafermi and parabose fields are shown to coincide with the Lie commutation relations of a certain group in the above-mentioned sense. Throughout our argumentation parafermi and parabose fields can formally be treated in an analogous man­ ner. It is thus concluded that the fermi-bose similarity that is known to hold in the ordi­ nary field theory persists also in parafielcl theory.

An interaction of three resonant modes in a nonlinear lattice

The purpose of this paper is to discuss the Hamiltonian H = J 1 + 2J 2 + 3J 3 + αJ 1(2J 2) 1 2 cos(2φ 1 − φ 2) + 2β(J 1J 2J 3 1 2 cos(φ 1 + φ 2 − φ 3 , where the J k 's are canonically conjugate to the φ k 's ( k = 1, 2, 3). In the case β = 0 or α = 0 the corresponding Hamilton equations are integrable. A computer study of the full Hamiltonian was made by Ford and Lunsford ( Phys. Rev. A. 1 (1970) , 59–70). The present paper obtains analytical results that are confronted with the computer study. The results are obtained by expanding the Hamiltonian into a power series about a certain equilibrium point and constructing the corresponding Gustavson normal form up to 4th-order terms. The Gustavson normal form appears as a member of the enveloping algebra of the Lie algebra SO(2, 1). It is shown that the normal form can be used to explain certain features of Figures 5–9 of the above-mentioned computer study. Moreover the Komogloroff-Arnold-Moser theory is invoked to prove that the quasiperiodic solutions of the β = 0 case can be analytically continued to nonzero but sufficiently small β α .

Vector Operators and a Polynomial Identity for SO(n)

It is shown that if α denotes an n × n antisymmetric matrix of operators αpq,p,q = 1, 2, …, n, which satisfy the commutation relations characteristic of the Lie algebra of SO(n), then α satisfies an nth degree polynomial identity. A method is presented for determining the form of this polynomial for any value of n. An indication is given of the simple significance of this identity with regard to the problem of resolving an arbitrary n-vector operator into n components, each of which is a vector shift operator for the invariants of the SO(n) Lie algebra.

Dynamical groups and spherical potentials in Classical Mechanics

The one particle problem in a spherical potential is examined in Classical Mechanics from a group theorical point of view. The constants of motion are classified according to their behaviour under the rotation groupSO(3), i.e. according to the irreducible representationsD j ofSO(3) (section 1). The Lie algebras ofSO(4) andSO(3) are explicitly built in terms of Poisson brackets for an arbitrary potential, from global considerations. The Kepler and the 3 dimensional oscillator problems are shown to play particular roles with respect to these groups (sections 2 and 3). In the last section, the Kepler problem is analyzed with the aid of theSO(4) group instead of the Lie algebra. It is proved that the transformations generated by the angular momentum and the Runge-Lenz vector form indeed a group of canonical transformations isomorphic toSO(4). Consequences with respect to the quantization problem are examined.