[PDF]    http://dx.doi.org/10.3952/lithjphys.51101

Open access article / Atviros prieigos straipsnis

Lith. J. Phys. 51, 5–18 (2011)

AN EXPLICIT BASIS OF LOWERING OPERATORS FOR IRREDUCIBLE REPRESENTATIONS OF UNITARY GROUPS
D.S. Sage and L. Smolinsky
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
E-mail: sage@math.lsu.edu, smolinsk@math.lsu.edu

Received 2 July 2010; revised 13 December 2010; accepted 15 December 2010

The representation theory of the unitary groups is of fundamental significance in many areas of physics and chemistry. In order to label states in a physical system with unitary symmetry, it is necessary to have explicit bases for the irreducible representations. One systematic way of obtaining bases is to generalize the ladder operator approach to the representations of SU(2) by using the formalism of lowering operators. Here, one identifies a basis for the algebra of all lowering operators and, for each irreducible representation, gives a prescription for choosing a subcollection of lowering operators that yields a basis upon application to the highest weight vector. Bases obtained through lowering operators are particularly convenient for computing matrix coefficients of observables as the calculations reduce to the commutation relations for the standard matrix units. The best known examples of this approach are the extremal projector construction of the Gelfand–Zetlin basis and the crystal (or canonical) bases of Kashiwara and Lusztig. In this paper, we describe another simple method of obtaining bases for the irreducible representations via lowering operators. These bases do not have the algebraic canonicity of the Gelfand–Zetlin and crystal bases, but the combinatorics involved are much more straightforward, making the bases particularly suited for physical applications.
Keywords: unitary group, special unitary group, irreducible representations, lowering operators, spin-free quantum chemistry, many-body problem
PACS: 02.20.-a, 31.15.xh

ŽEMINANČIŲJŲ OPERATORIŲ, SKIRTŲ UNITARINIŲ GRUPIŲ NEREDUKUOTINIAMS ATVAIZDAMS, IŠREIKŠTINĖ BAZĖ
D.S. Sage, L. Smolinsky
Luizianos valstijos universitetas, Baton Ružas, JAV

Unitarinių grupių atvaizdų teorija fundamentaliai svarbi daugelyje fizikos ir chemijos sričių. Unitarinės simetrijos fizikinės sistemos būsenų žymėjimui reikia turėti išreikštines bazes neredukuotiniams atvaizdams. Vienas sisteminių būdų gauti bazes yra apibendrinti laiptinių operatorių metodą SU(2) atvaizdams, panaudojant žeminančiųjų operatorių formalizmą. Čia nustatoma bazė visų žeminančiųjų operatorių algebrai ir kiekvienam neredukuotiniam atvaizdui pateikiama instrukcija, kaip parinkti žeminančiųjų operatorių rinkinio dalį, kurią naudojant bazė gaunama iš didžiausio svorio vektoriaus. Bazės, gautos žeminančiaisiais operatoriais, ypač patogios skaičiuojant stebimų dydžių matricinius koeficientus, kadangi jie virsta komutacijos sąryšiais standartiniams matriciniams vienetams. Žinomiausi šito metodo pavyzdžiai yra Gelfando ir Cetlino bazės sukonstravimas naudojant kraštutinius projektorius bei kristalinės (arba kanoninės) Kašivaros ir Lustigo bazės. Straipsnyje aprašomas kitas paprastas būdas gauti neredukuotinių atvaizdų bazes naudojant žeminančiuosius operatorius. Šios bazės nepasižymi Gelfando ir Cetlino ar kristalinių bazių kanoniškumu, tačiau kombinatorika, su kuria susiduriama, yra daug paprastesnė ir dėl to šios bazės ypač tinka fizikiniams taikymams.

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