Canonical Transformations and the Theory of Representations of Lie Groups

Leznov, A. N.; Malkin, I.A.; Man’ko, Vladimir I.

doi:10.1007/978-1-4684-0676-4_3

Cited by 4 publications

(5 citation statements)

References 14 publications

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“…As always, to pass from the classical expressions to the quantum ones it is necessary to order in some way the operators involved and replace the Poisson brackets by commutators. The equations (3) give us a hint about the very tempting possibility to rewrite them as…”

Section: Algebraic Casementioning

confidence: 99%

“…the constant term from the second equation ( 10) may be taken away. This corresponds to the transition from the algebra U(3) to SU (3).…”

Section: Classical Casementioning

confidence: 99%

“…Such general approach to the problem of the constructing the irreducible representations of the Lie algebras was proposed many years ago in [3] but up to now has not been used (to the best of our knowledge).…”

Section: Introductionmentioning

confidence: 99%

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To the Gel’fand–Tsetlin realization of irreducible representations of classical semisimple algebras

Leznov

2001

Journal of Mathematical Physics

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Section: Algebraic Casementioning

confidence: 99%

“…the constant term from the second equation ( 10) may be taken away. This corresponds to the transition from the algebra U(3) to SU (3).…”

Section: Classical Casementioning

confidence: 99%

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To the Gel’fand–Tsetlin realization of irreducible representations of classical semisimple algebras

Leznov

2001

Journal of Mathematical Physics

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“…Thus the components in each row characterize one of the possible irreducible representations of a specific subalgebra. The eigenvalues of the invariant (or Casimir) operators of so n+1 have been obtained by Perelomov, Popov [18] and Leznov, Malkin, and Man'ko [19] and are given by…”

Section: The Orthogonal Cayley-klein Algebrasmentioning

confidence: 99%

The Gel’fand–Tsetlin representations of the orthogonal Cayley–Klein algebras

Gromov¹

1992

Journal of Mathematical Physics

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The explicit expressions of the operators of the irreducible representations of the orthogonal Cayley-Klein algebras so n+1 (j) are obtained from the well-known Gel'fand-Tsetlin representations of so n+1 . The contractions and the analytic continuations of the representations of so 3 (j 1 , j 2 ), so 4 (j 1 , j 2 , j 3 ) regarded as examples. This approach gives the representations of the contracted and the analytically continued algebras in different bases, for example, in discrete and continuous ones. Possible contractions of the irreducible representations are discussed.

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“…In fact, this quantum system is a particular case of systems with Hamiltonians linear in Lie algebra generators. Such systems in the general case (integrals of motion, evolution operator and propagators) were proposed many years ago; for more details, see [21]. Since the Hamiltonian (1.10) is explicitly time-dependent, it is also not possible to find the exact expression for either the wave function or the solution for the equations of motion in the Heisenberg picture.…”

Section: Introductionmentioning

confidence: 99%

Nonclassical properties of the integrability condition of the time-dependent SU(2) quantum system

Abdalla¹,

Khalil²

2011

Phys. Scr.

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In this paper, we treat the problem of the multi-photon Tavis–Cummings classically, where the effect of time-dependence takes place. Under certain integrability conditions the time-dependent wave function in the Schrödinger picture is obtained. Some statistical properties of the system are discussed based on the solution for the equations of motion in the Heisenberg picture. Using the atomic coherent state as a basis, the phenomenon of squeezing is observed in the quadrature variances Fx and Fy; however, it was pronounced in the quadrature Fy. The phenomenon of squeezing is also reported for the usual atomic state case where squeezing is only observed in the first quadrature Fx. An examination of the correlation function using the atomic state showed us that the system is sensitive to variation in the integrability parameter β. The nonclassical effect is also found to be pronounced for small values of the parameter β.

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Canonical Transformations and the Theory of Representations of Lie Groups

Cited by 4 publications

References 14 publications

To the Gel’fand–Tsetlin realization of irreducible representations of classical semisimple algebras

To the Gel’fand–Tsetlin realization of irreducible representations of classical semisimple algebras

The Gel’fand–Tsetlin representations of the orthogonal Cayley–Klein algebras

Nonclassical properties of the integrability condition of the time-dependent SU(2) quantum system

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