### Abstract

We consider the "missing label" problem for basis vectors of SU(3) representations in a basis corresponding to the group reduction SU(3) ⊃ O(3) ⊃ O(2). We prove that only two independent O(3) scalars exist in the enveloping algebra of S U (3), in addition to the obvious ones, namely the angular momentum L^{2} and the two SU(3) Casimir operators C ^{(2)} and C^{(3)}. Any one of these two operators (of third and fourth order in the generators) can be added to C^{(2)}, C ^{(3)}, L^{2}, and L_{3} to form a complete set of commuting operators. The eigenvalues of the third and fourth order scalars X^{(3)} and X^{(4)} are calculated analytically or numerically for many cases of physical interest. The methods developed in this article can be used to resolve a missing label problem for any semisimple group G, when reduced to any semisimple subgroup H.

Original language | English (US) |
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Pages (from-to) | 1787-1799 |

Number of pages | 13 |

Journal | Journal of Mathematical Physics |

Volume | 15 |

Issue number | 10 |

DOIs | |

State | Published - Jan 1 1973 |

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### Cite this

*Journal of Mathematical Physics*,

*15*(10), 1787-1799. https://doi.org/10.1063/1.1666542

**Complete sets of commuting operators and O(3) scalars in the enveloping algebra of SU(3).** / Judd, B. R.; Miller, Willard; Patera, J.; Winternitz, P.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 15, no. 10, pp. 1787-1799. https://doi.org/10.1063/1.1666542

}

TY - JOUR

T1 - Complete sets of commuting operators and O(3) scalars in the enveloping algebra of SU(3)

AU - Judd, B. R.

AU - Miller, Willard

AU - Patera, J.

AU - Winternitz, P.

PY - 1973/1/1

Y1 - 1973/1/1

N2 - We consider the "missing label" problem for basis vectors of SU(3) representations in a basis corresponding to the group reduction SU(3) ⊃ O(3) ⊃ O(2). We prove that only two independent O(3) scalars exist in the enveloping algebra of S U (3), in addition to the obvious ones, namely the angular momentum L2 and the two SU(3) Casimir operators C (2) and C(3). Any one of these two operators (of third and fourth order in the generators) can be added to C(2), C (3), L2, and L3 to form a complete set of commuting operators. The eigenvalues of the third and fourth order scalars X(3) and X(4) are calculated analytically or numerically for many cases of physical interest. The methods developed in this article can be used to resolve a missing label problem for any semisimple group G, when reduced to any semisimple subgroup H.

AB - We consider the "missing label" problem for basis vectors of SU(3) representations in a basis corresponding to the group reduction SU(3) ⊃ O(3) ⊃ O(2). We prove that only two independent O(3) scalars exist in the enveloping algebra of S U (3), in addition to the obvious ones, namely the angular momentum L2 and the two SU(3) Casimir operators C (2) and C(3). Any one of these two operators (of third and fourth order in the generators) can be added to C(2), C (3), L2, and L3 to form a complete set of commuting operators. The eigenvalues of the third and fourth order scalars X(3) and X(4) are calculated analytically or numerically for many cases of physical interest. The methods developed in this article can be used to resolve a missing label problem for any semisimple group G, when reduced to any semisimple subgroup H.

UR - http://www.scopus.com/inward/record.url?scp=36849101564&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36849101564&partnerID=8YFLogxK

U2 - 10.1063/1.1666542

DO - 10.1063/1.1666542

M3 - Article

VL - 15

SP - 1787

EP - 1799

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 10

ER -