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Generating series and asymptotics of classical spin networks

Francesco Costantino, Julien Marché (2015)

Journal of the European Mathematical Society

We study classical spin networks with group SU 2 . In the first part, using Gaussian integrals, we compute their generating series in the case where the edges are equipped with holonomies; this generalizes Westbury’s formula. In the second part, we use an integral formula for the square of the spin network and perform stationary phase approximation under some non-degeneracy hypothesis. This gives a precise asymptotic behavior when the labels are rescaled by a constant going to infinity.

Internal Symmetries and Additional Quantum Numbers for Nanoparticles

V.G. Yarzhemsky (2013)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

Wavefunctions of symmetrical nanoparticles are considered making use of induced representation method. It is shown that when, at the same total symmetry, the order of local symmetry group decreases, additional quantum numbers are required for complete labelling of electron states. It is shown that the labels of irreducible representations of intermediate subgroups can be used for complete classification of states in the case of repeating IRs in symmetry adapted linear combinations. The intermediate...

Invariant symbolic calculus for semidirect products

Benjamin Cahen (2018)

Commentationes Mathematicae Universitatis Carolinae

Let G be the semidirect product V ⋊ K where K is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space V . Let π be a unitary irreducible representation of G which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of G whose little group is a maximal compact subgroup of K . We construct an invariant symbolic calculus for π , under some technical hypothesis. We give some examples including the Poincaré group.

On the contraction of the discrete series of S U ( 1 , 1 )

C. Cishahayo, S. De Bièvre (1993)

Annales de l'institut Fourier

It is shown, using techniques inspired by the method of orbits, that each non-zero mass, positive energy representation of the Poincaré group 𝒫 1 , 1 = S O ( 1 , 1 ) ⊗ s ℝ 2 can be obtained via contraction from the discrete series of representations of S U ( 1 , 1 ) .

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