scholarly journals Construction and Characterization of Representations of SU(7) for GUT Model Builders

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1044
Author(s):  
Daniel Jones ◽  
Jeffery A. Secrest
Keyword(s):  
Gauge Symmetry ◽  
Symmetry Group ◽  
Lie Group ◽  
Cartan Subalgebra ◽  
Natural Extension ◽  
Grand Unification ◽  
Raising And Lowering Operators ◽  
Higher Dimensional ◽  
Lowering Operators

The natural extension to the SU(5) Georgi-Glashow grand unification model is to enlarge the gauge symmetry group. In this work, the SU(7) symmetry group is examined. The Cartan subalgebra is determined along with their commutation relations. The associated roots and weights of the SU(7) algebra are derived and discussed. The raising and lowering operators are explicitly constructed and presented. Higher dimensional representations are developed by graphical as well as tensorial methods. Applications of the SU(7) Lie group to supersymmetric grand unification as well as applications are discussed.

scholarly journals On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

HAPPER'S CURIOUS DEGENERACIES AND YANGIAN

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1867-1873 ◽  
Author(s):  
CHENG-MING BAI ◽  
MO-LIN GE ◽  
KANG XUE
Keyword(s):  
Representation Theory ◽  
Algebraic Structure ◽  
Zeeman Effect ◽  
Tensor Space ◽  
Commutation Relations ◽  
Raising And Lowering Operators ◽  
Degenerate States ◽  
Lowering Operators

We find raising and lowering operators distinguishing the degenerate states for the Hamiltonian [Formula: see text] at x = ± 1 for spin 1 that was given by Happer et al.1,2 to interpret the curious degeneracies of the Zeeman effect for condensed vapor of 87 Rb . The operators obey Yangian commutation relations. We show that the curious degeneracies seem to verify the Yangian algebraic structure for quantum tensor space and are consistent with the representation theory of Y(sl(2)).

scholarly journals A demonstration that electroweak theory can violate parity automatically (leptonic case)

2018 ◽  
Vol 33 (04) ◽  
pp. 1830005 ◽  
Author(s):  
C. Furey
Keyword(s):  
Clifford Algebra ◽  
Ad Hoc ◽  
Higgs Bosons ◽  
Electroweak Theory ◽  
Ladder Operators ◽  
Raising And Lowering Operators ◽  
Ladder Operator ◽  
Left And Right ◽  
Electroweak Model ◽  
Lowering Operators

We bring to light an electroweak model which has been reappearing in the literature under various guises.[Formula: see text] In this model, weak isospin is shown to act automatically on states of only a single chirality (left). This is achieved by building the model exclusively from the raising and lowering operators of the Clifford algebra [Formula: see text]. That is, states constructed from these ladder operators mimic the behaviour of left- and right-handed electrons and neutrinos under unitary ladder operator symmetry. This ladder operator symmetry is found to be generated uniquely by [Formula: see text] and [Formula: see text]. Crucially, the model demonstrates how parity can be maximally violated, without the usual step of introducing extra gauge and extra Higgs bosons, or ad hoc projectors.

scholarly journals Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 717 ◽  
Author(s):  
Maël Dugast ◽  
Guillaume Bouleux ◽  
Eric Marcon
Keyword(s):  
Lie Group ◽  
Correlation Coefficients ◽  
Shape Space ◽  
Rotation Matrices ◽  
Dilation Matrix ◽  
Naimark Dilation ◽  
Periodically Correlated Processes ◽  
New Vision

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes; this enables a comparison between curves’ shapes and allows then the classification of random processes’ measures.

Factorization method for exact solution of the non-central modified Killingbeck potential plus a ring-shaped-like potential

2019 ◽  
Vol 34 (10) ◽  
pp. 1950072 ◽  
Author(s):  
B. Tchana Mbadjoun ◽  
J. M. Ema’a Ema’a ◽  
Jean Yomi ◽  
P. Ele Abiama ◽  
G. H. Ben-Bolie ◽  
...  
Keyword(s):  
Exact Solution ◽  
Potential Problem ◽  
Energy Levels ◽  
Factorization Method ◽  
Spherically Symmetric ◽  
Laguerre Function ◽  
Radial Part ◽  
Raising And Lowering Operators ◽  
Structure Of Molecules ◽  
Lowering Operators

In this paper, we study the Schrödinger equation with non-central modified Killingbeck potential plus a ring-shaped-like potential problem, which is not spherically symmetric. The factorization method is used to solve the hypergeometric equation types which lead to solutions with the associate Laguerre function for the radial part and Jacobi polynomial for the polar part. We introduce the raising and lowering operators to calculate the energies eigenvalues, which show that the lack of spherical symmetry removes the degeneracy of second quantum number m which is completely expected. These obtained energies are better to explain the superposition of the energy levels of the atoms in the crystalline structure of molecules.

scholarly journals The commutator of raising and lowering operators for angular momentum to the free particle’s hamiltonian

2020 ◽  
Vol 1538 ◽  
pp. 012037
Author(s):  
A F Sugihartin ◽  
B Supriadi ◽  
Subiki ◽  
V Rizqiyah ◽  
N Rizky ◽  
...  
Keyword(s):  
Angular Momentum ◽  
Raising And Lowering Operators ◽  
Lowering Operators
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