scholarly journals GENERALIZATION OF THE GELL–MANN DECONTRACTION FORMULA FOR sl(n,ℝ) AND su(n) ALGEBRAS

2011 ◽  
Vol 08 (02) ◽  
pp. 395-410 ◽  
Author(s):  
IGOR SALOM ◽  
DJORDJE ŠIJAČKI
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Algebraic Expression ◽  
Matrix Elements ◽  
Group Manifold ◽  
The Matrix

The so-called Gell–Mann or decontraction formula is proposed as an algebraic expression inverse to the Inönü–Wigner Lie algebra contraction. It is tailored to express the Lie algebra elements in terms of the corresponding contracted ones. In the case of sl (n,ℝ) and su (n) algebras, contracted w.r.t. so (n) subalgebras, this formula is generally not valid, and applies only in the cases of some algebra representations. A generalization of the Gell–Mann formula for sl (n,ℝ) and su (n) algebras, that is valid for all tensorial, spinorial, (non)unitary representations, is obtained in a group manifold framework of the SO(n) and/or Spin (n) group. The generalized formula is simple, concise and of ample application potentiality. The matrix elements of the [Formula: see text], i.e. SU(n)/SO(n), generators are determined, by making use of the generalized formula, in a closed form for all irreducible representations.

scholarly journals COHERENT STATES, DISPLACED NUMBER STATES AND LAGUERRE POLYNOMIAL STATES FOR su(1, 1) LIE ALGEBRA

2000 ◽  
Vol 14 (10) ◽  
pp. 1093-1103 ◽  
Author(s):  
XIAO-GUANG WANG
Keyword(s):  
Coherent State ◽  
Lie Algebra ◽  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Optical Systems ◽  
Matrix Elements ◽  
Operator Formalism ◽  
Ladder Operator ◽  
Deformed Oscillator ◽  
The Matrix

The ladder operator formalism of a general quantum state for su(1, 1) Lie algebra is obtained. The state bears the generally deformed oscillator algebraic structure. It is found that the Perelomov's coherent state is a su(1, 1) nonlinear coherent state. The expansion and the exponential form of the nonlinear coherent state are given. We obtain the matrix elements of the su(1, 1) displacement operator in terms of the hypergeometric functions and the expansions of the displaced number states and Laguerre polynomial states are followed. Finally some interesting su(1, 1) optical systems are discussed.

scholarly journals Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1120
Author(s):  
Carlo Heissenberg ◽  
Franco Strocchi
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Goldstone Bosons ◽  
The Matrix ◽  
Symmetry Breaking Term ◽  
Breaking Term

The matrix elements of operators transforming as irreducible representations of an unbroken symmetry group G are governed by the well-known Wigner–Eckart relations. In the case of infinite-dimensional systems, with G spontaneously broken, we prove that the corrections to such relations are provided by symmetry breaking Ward identities, and simply reduce to a tadpole term involving Goldstone bosons. The analysis extends to the case in which an explicit symmetry breaking term is present in the Hamiltonian, with the tadpole term now involving pseudo Goldstone bosons. An explicit example is discussed, illustrating the two cases.

scholarly journals BASIC PROPERTIES OF COHERENT AND GENERALIZED COHERENT OPERATORS REVISITED

2001 ◽  
Vol 16 (20) ◽  
pp. 1277-1286 ◽  
Author(s):  
KAZUYUKI FUJII
Keyword(s):  
Lie Algebra ◽  
Matrix Elements ◽  
Basic Properties ◽  
The Matrix

In this paper we make a brief review of some basic properties (the matrix elements, the trace, the Glauber formula) of coherent operators and study the corresponding ones for generalized coherent operators based on Lie algebra su(1,1). We also propose some problems.

scholarly journals One more method to construct the irreducible unitary representations of nipotent Lie groups

1986 ◽  
Vol 40 (1) ◽  
pp. 89-94
Author(s):  
A. A. Astaneh
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Nilpotent Lie Group ◽  
Irreducible Unitary Representations ◽  
Simply Connected

AbstractIn this paper one more canonical method to construct the irreducible unitary representations of a connected, simply connected nilpotent Lie group is introduced. Although we used Kirillov' analysis to deduce this procedure, the method obtained differs from that of Kirillov's, in that one does not need to consider the codjoint representation of the group in the dual of its Lie algebra (in fact, neither does one need to consider the Lie algebra of the group, provided one knows certain connected subgroups and their characters). The method also differs from that of Mackey's as one only needs to induce characters to obtain all irreducible representations of the group.

The matrix elements of tensor operators for configurations of three equivalent electrons

1959 ◽  
Vol 250 (1263) ◽  
pp. 562-574 ◽  
Keyword(s):  
Irreducible Representations ◽  
Matrix Elements ◽  
Irreducible Tensor ◽  
Quantum Numbers ◽  
Fractional Parentage ◽  
The Matrix ◽  
Tensor Operators

The formulae of Redmond are used to construct expressions for the fractional parentage coefficients relating the configurations l 3 and l 2 . The explicit occurrence of godparent states is avoided for the quartet states of f 3 and also for a sequence of doublet states. The latter are defined by the set of quantum numbers f 3 WUSLJJ 2 , where W and U are irreducible representations of the groups R 7 and G 2 . Matrix elements of the type ( f 3 WUSL || U k || f 3 W'U'SL' ), where U k is the sum of the three irreducible tensor operators u k corresponding to the three f electrons, are tabulated for k = 2, 4 and 6 and for all values of W, U, S and L .

scholarly journals SU(1,1) solution for the Dunkl–Coulomb problem in two dimensions and its coherent states

2018 ◽  
Vol 33 (20) ◽  
pp. 1850112 ◽  
Author(s):  
M. Salazar-Ramírez ◽  
D. Ojeda-Guillén ◽  
R. D. Mota ◽  
V. D. Granados
Keyword(s):  
Energy Spectrum ◽  
Closed Form ◽  
Lie Algebra ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Two Dimensions ◽  
Irreducible Representations ◽  
Radial Part ◽  
Coulomb Problem ◽  
Radial Schrödinger Equation

We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.

VALIDITY OF THE GELL-MANN FORMULA FOR sl(n, ℝ) AND su(n) ALGEBRAS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350017 ◽  
Author(s):  
IGOR SALOM ◽  
DJORDJE ŠIJAČKI
Keyword(s):  
Closed Form ◽  
Lie Algebra ◽  
Complete List ◽  
Matrix Elements ◽  
Operator Expression ◽  
General Validity ◽  
Potential Value ◽  
Explicit Expressions

The so-called Gell-Mann formula, a prescription designed to provide an inverse to the Inönü–Wigner Lie algebra contraction, has a great versatility and potential value. This formula has no general validity as an operator expression. The question of applicability of Gell-Mann's formula to various algebras and their representations was only partially treated. The validity constraints of the Gell-Mann formula for the case of sl(n, ℝ) and su(n) algebras are clarified, and the complete list of representations spaces for which this formula applies is given. Explicit expressions of the sl(n, ℝ) generators matrix elements are obtained for all these cases in a closed form by making use of the Gell-Mann formula.

On non-compact groups. II. Representations of the 2+1 Lorentz group

1965 ◽  
Vol 287 (1411) ◽  
pp. 532-548 ◽  
Keyword(s):  
Hilbert Space ◽  
Direct Product ◽  
Lorentz Group ◽  
Complex Number ◽  
Algebraic Method ◽  
Unitary Representations ◽  
Compact Groups ◽  
Matrix Elements ◽  
Linear Representations ◽  
The Matrix

A simple algebraic method based on multispinors with a complex number of indices is used to obtain the linear (and unitary) representations of non-com pact groups. The method is illustrated in the case of the 2+1 Lorentz group. All linear representations of this group, their various realizations in Hilbert space as well as the matrix elements of finite transformations have been found. The problem of reduction of the direct product is also briefly discussed.

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