Combinatorial and Geometrical Origins of Regge Symmetries: Their Manifestations from Spin-Networks to Classical Mechanisms, and Beyond

  • Vincenzo Aquilanti
  • Manuela S. Arruda
  • Cecilia ColettiEmail author
  • Robert Littlejohn
  • Robenilson F. Santos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10408)


Tullio Regge discovered new symmetries in 1958, hidden in formulas for calculations of the coupling and recoupling coefficients of quantum angular momentum theory, as developed principally by Wigner and Racah: the only known (limited) application appeared computational. Ten years later, in a paper with Ponzano, Regge provided a semiclassical interpretation showing relevance to the basic geometry of quadrilaterals and tetrahedra, and opening also a promising road to quantum gravity, still currently being explored. New facets are here indicated, continuing a sequence of papers in this Lecture Notes series and elsewhere. We emphasize how an integrated combinatorial and geometrical interpretation is emerging, and also examples from the quantum mechanics of atoms and molecules are briefly documented. Attention is dedicated to the recently pointed out connection between the quantum mechanics of spin recouplings and the Grashof analysis of four-bar linkages, with perspective implications at the molecular level.


Quantum angular momentum Semiclassical asymptotics Wigner-Racah coupling and recoupling 3j and 6j symbols Kepler-Coulomb Sturmian orbitals 



Robenilson Ferreira is grateful to Brazilian CAPES for a sandwich doctoral (PDSE88881.134388/2016-01) fellowship to the Perugia University. Manuela Arruda is grateful to Brazilian CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for a post doctoral fellowship to the Perugia University. Vincenzo Aquilanti and Frederico Vasconcellos Prudente thank Brazilian CAPES for a Special Visiting Professorship at the Bahia Federal University (PVE 027/2013) and the Italian MIUR for Grant SIR 2014 (RBSI14U3VF).


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© Springer International Publishing AG 2017

Authors and Affiliations

  • Vincenzo Aquilanti
    • 1
    • 2
  • Manuela S. Arruda
    • 3
  • Cecilia Coletti
    • 4
    Email author
  • Robert Littlejohn
    • 5
  • Robenilson F. Santos
    • 6
    • 7
  1. 1.Dipartimento di Chimica, Biologia e BiotecnologieUniversità di PerugiaPerugiaItaly
  2. 2.Consiglio Nazionale Delle RicercheRomeItaly
  3. 3.Centro de Ciências Exatas e TecnológicasUniversidade Federal do Recôncavo da BahiaCruz Das AlmasBrazil
  4. 4.Dipartimento di FarmaciaUniversità G. d’AnnunzioChietiItaly
  5. 5.Department of PhysicsUniversity of CaliforniaBerkeleyUSA
  6. 6.Instituto de FísicaUniversidade Federal da BahiaSalvadorBrazil
  7. 7.Instituto Federal de Alagoas, Campus PiranhasPiranhasBrazil

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