ON THE DEGENERACY PROBLEM IN SU(3)

1966 ◽  
Vol 44 (11) ◽  
pp. 2789-2795 ◽  
Author(s):  
C. K. Chew ◽  
R. T. Sharp
Keyword(s):  
Simple Form ◽  
Casimir Operator ◽  
Recent Suggestion ◽  
The Matrix ◽  
Casimir Operators ◽  
Antisymmetric Functions ◽  
Degeneracy Problem

The recent suggestion of Macfarlane, O'Raifeartaigh, and Rao that mixed Casimir operators be used to resolve the external degeneracy problem is applied to the group SU(3). The significant operator is the part of the cubic mixed Casimir operator which is antisymmetric with respect to the two factor spaces and also with respect to starred and unstarred variables. The matrix of this operator is derived with respect to a convenient set of functions and is found to have a simple form. The eigenfunctions are simply related to the usual symmetric and antisymmetric functions in the case of degeneracy 2 with equal factor representations.

Generalized Casimir Operators

1969 ◽  
Vol 21 ◽  
pp. 1496-1505
Author(s):  
A. J. Douglas
Keyword(s):  
Finite Group ◽  
Casimir Operator ◽  
Identity Element ◽  
Ring Homomorphism ◽  
Fixed Ring ◽  
Injective Modules ◽  
Casimir Operators ◽  
Maschke’S Theorem ◽  
A Finite Group ◽  
Special Case

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

scholarly journals Algebraic DVR Approaches Applied to Describe the Stark Effect

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1719
Author(s):  
Marisol Bermúdez-Montaña ◽  
Marisol Rodríguez-Arcos ◽  
Renato Lemus ◽  
José M. Arias ◽  
Joaquín Gómez-Camacho ◽  
...  
Keyword(s):  
Hydrogen Atom ◽  
Harmonic Oscillator ◽  
Simple Form ◽  
Stark Effect ◽  
Matrix Representation ◽  
Discrete Variable ◽  
Discrete Variable Representation ◽  
Branching Rules ◽  
The Matrix ◽  
Variable Representation

Two algebraic approaches based on a discrete variable representation are introduced and applied to describe the Stark effect in the non-relativistic Hydrogen atom. One approach consists of considering an algebraic representation of a cutoff 3D harmonic oscillator where the matrix representation of the operators r2 and p2 are diagonalized to define useful bases to obtain the matrix representation of the Hamiltonian in a simple form in terms of diagonal matrices. The second approach is based on the U(4) dynamical algebra which consists of the addition of a scalar boson to the 3D harmonic oscillator space keeping constant the total number of bosons. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations, whose involved branching rules define the discrete variable representation. Both methods, although originating from the harmonic oscillator basis, provide different convergence tests due to the fact that the associated discrete bases turn out to be different. These approaches provide powerful tools to obtain the matrix representation of 3D general Hamiltonians in a simple form. In particular, the Hydrogen atom interacting with a static electric field is described. To accomplish this task, the diagonalization of the exact matrix representation of the Hamiltonian is carried out. Particular attention is paid to the subspaces associated with the quantum numbers n=2,3 with m=0.

Generalized Casimir operators

2019 ◽  
Vol 18 (11) ◽  
pp. 1950219 ◽  
Author(s):  
S. Eswara Rao
Keyword(s):  
Lie Algebra ◽  
Casimir Operator ◽  
New Class ◽  
Highest Weight ◽  
Casimir Operators

Let [Formula: see text] be symmetrizable Kac–Moody Lie algebra. In this paper, we describe a new class of central operators generalizing the Casimir operator. We also prove some properties of these operators and show that these operators move highest weight vectors to new highest weight vectors.

scholarly journals Split Casimir Operator and Universal Formulation of the Simple Lie Algebras

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1046
Author(s):  
Alexey Isaev ◽  
Sergey Krivonos
Keyword(s):  
Lie Algebras ◽  
Casimir Operator ◽  
Adjoint Representation ◽  
Simple Lie Algebras ◽  
Casimir Operators ◽  
Explicit Formulae

We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in adjoint representation. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T⊗2 in the case when T is the adjoint representation. These projectors and characteristic identities are considered from the viewpoint of the universal description of the simple Lie algebras in terms of the Vogel parameters.

Tensor operators under semi-simple groups

1961 ◽  
Vol 57 (3) ◽  
pp. 460-468 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Simple Groups ◽  
Matrix Elements ◽  
Coupling Coefficients ◽  
The Matrix ◽  
Casimir Operators ◽  
Tensor Operators

ABSTRACTTensor operators under any group are defined and the theory is developed for semi-simple continuous groups. Coupled tensor operators are introduced and the matrix elements of tensor operators are expressed in terms of the coupling coefficients. The structure of generalized Casimir operators is investigated.

On the kinematics of particle interactions

1969 ◽  
Vol 312 (1511) ◽  
pp. 467-472
Keyword(s):  
Simple Form ◽  
Mass Shell ◽  
Particle Interaction ◽  
Frame Of Reference ◽  
Particle Interactions ◽  
Natural Coordinates ◽  
Canonical Frame ◽  
The Matrix ◽  
Conservation Of Momentum ◽  
Scalar Products

For a 2-particle interaction, a new canonical frame of reference is defined, namely, one making diagonal the matrix of quadratic moments of the four 4-momenta of the incoming and outgoing particles. When we use constraint coordinates based on the above canonical frame, on the mass-shell, and on the conservation of momentum and energy, it is found that the Jacobian of the transformation from the natural coordinates to the constraint coordinates has an interesting simple form. In fact, to within a trivial numerical factor, it is a simple symmetric function of the eigenvalues of the 4 x 4 matrix formed by the scalar products of the 4-momenta.

Crystalloid Inclusions in Hepatocyte Mitochondria and Their Similarity to Cytoplasmic Crystalloids

1974 ◽  
Vol 32 ◽  
pp. 198-199
Author(s):  
Odell T. Minick ◽  
Hidejiro Yokoo
Keyword(s):  
Liver Disease ◽  
Alcoholic Liver Disease ◽  
Special Interest ◽  
Structural Variations ◽  
Mitochondrial Alterations ◽  
The Matrix ◽  
Crystalloid Inclusions ◽  
Liver Biopsies

Mitochondrial alterations were studied in 25 liver biopsies from patients with alcoholic liver disease. Of special interest were the morphologic resemblance of certain fine structural variations in mitochondria and crystalloid inclusions. Four types of alterations within mitochondria were found that seemed to relate to cytoplasmic crystalloids.Type 1 alteration consisted of localized groups of cristae, usually oriented in the long direction of the organelle (Fig. 1A). In this plane they appeared serrated at the periphery with blind endings in the matrix. Other sections revealed a system of equally-spaced diagonal lines lengthwise in the mitochondrion with cristae protruding from both ends (Fig. 1B). Profiles of this inclusion were not unlike tangential cuts of a crystalloid structure frequently seen in enlarged mitochondria described below.

Coherency loss in γ' precipitates in nickel-base superalloy

1983 ◽  
Vol 41 ◽  
pp. 244-245
Author(s):  
R. A. Ricks ◽  
Angus J. Porter
Keyword(s):  
Lattice Mismatch ◽  
Lattice Parameter ◽  
Nickel Base Superalloy ◽  
Large Size ◽  
The Matrix ◽  
Nimonic 80A ◽  
Interfacial Dislocations ◽  
Nickel Base ◽  
Coherency Loss ◽  
Nimonic 115

During a recent investigation concerning the growth of γ' precipitates in nickel-base superalloys it was observed that the sign of the lattice mismatch between the coherent particles and the matrix (γ) was important in determining the ease with which matrix dislocations could be incorporated into the interface to relieve coherency strains. Thus alloys with a negative misfit (ie. the γ' lattice parameter was smaller than the matrix) could lose coherency easily and γ/γ' interfaces would exhibit regularly spaced networks of dislocations, as shown in figure 1 for the case of Nimonic 115 (misfit = -0.15%). In contrast, γ' particles in alloys with a positive misfit could grow to a large size and not show any such dislocation arrangements in the interface, thus indicating that coherency had not been lost. Figure 2 depicts a large γ' precipitate in Nimonic 80A (misfit = +0.32%) showing few interfacial dislocations.

Influence of Heat Treatments on Microstructures in an Fe-Co-V Alloy

1974 ◽  
Vol 32 ◽  
pp. 512-513
Author(s):  
S. Mahajan ◽  
M. R. Pinnel ◽  
J. E. Bennett
Keyword(s):  
Single Phase ◽  
Alloy Composition ◽  
Supersaturated Solid Solution ◽  
Cell Formation ◽  
Heat Treatments ◽  
Air Cooling ◽  
Iced Brine ◽  
Transmission Electron ◽  
The Matrix ◽  
Bcc Structure

The microstructural changes in an Fe-Co-V alloy (composition by wt.%: 2.97 V, 48.70 Co, 47.34 Fe and balance impurities, such as C, P and Ni) resulting from different heat treatments have been evaluated by optical metallography and transmission electron microscopy. Results indicate that, on air cooling or quenching into iced-brine from the high temperature single phase ϒ (fcc) field, vanadium can be retained in a supersaturated solid solution (α2) which has bcc structure. For the range of cooling rates employed, a portion of the material appears to undergo the γ-α2 transformation massively and the remainder martensitically. Figure 1 shows dislocation topology in a region that may have transformed martensitically. Dislocations are homogeneously distributed throughout the matrix, and there is no evidence for cell formation. The majority of the dislocations project along the projections of <111> vectors onto the (111) plane, implying that they are predominantly of screw character.

Heterogeneity of Size and Distribution of Intramembrane Particles in the Plasma Membrane of Yeast

1977 ◽  
Vol 35 ◽  
pp. 596-597
Author(s):  
E. Keyhani
Keyword(s):  
Plasma Membrane ◽  
Candida Utilis ◽  
Synthetic Medium ◽  
Freeze Fracture ◽  
Translational Diffusion ◽  
Yeast Cells ◽  
Distilled Water ◽  
Intramembrane Particles ◽  
The Matrix ◽  
Water Cell

The matrix of biological membranes consists of a lipid bilayer into which proteins or protein aggregates are intercalated. Freeze-fracture techni- ques permit these proteins, perhaps in association with lipids, to be visualized in the hydrophobic regions of the membrane. Thus, numerous intramembrane particles (IMP) have been found on the fracture faces of membranes from a wide variety of cells (1-3). A recognized property of IMP is their tendency to form aggregates in response to changes in experi- mental conditions (4,5), perhaps as a result of translational diffusion through the viscous plane of the membrane. The purpose of this communica- tion is to describe the distribution and size of IMP in the plasma membrane of yeast (Candida utilis).Yeast cells (ATCC 8205) were grown in synthetic medium (6), and then harvested after 16 hours of culture, and washed twice in distilled water. Cell pellets were suspended in growth medium supplemented with 30% glycerol and incubated for 30 minutes at 0°C, centrifuged, and prepared for freeze-fracture, as described earlier (2,3).

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