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Lowering and Raising Operators for the Orthogonal Group in the Chain O(n) ⊃ O(n − 1) ⊃ … , and their Graphs

dc.contributor.authorPang, Sing-Chinen_US
dc.contributor.authorHecht, Karl T.en_US
dc.date.accessioned2010-05-06T22:20:33Z
dc.date.available2010-05-06T22:20:33Z
dc.date.issued1967-06en_US
dc.identifier.citationPang, Sing Chin; Hecht, K. T. (1967). "Lowering and Raising Operators for the Orthogonal Group in the Chain O(n) ⊃ O(n − 1) ⊃ … , and their Graphs." Journal of Mathematical Physics 8(6): 1233-1251. <http://hdl.handle.net/2027.42/70553>en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/70553
dc.description.abstractNormalized lowering and raising operators are constructed for the orthogonal group in the canonical group chain O(n) ⊃ O(n − 1) ⊃ … ⊃ O(2) with the aid of graphs which simplify their construction. By successive application of such lowering operators for O(n), O(n − 1), … on the highest weight states for each step of the chain, an explicit construction is given for the normalized basis vectors. To illustrate the usefulness of the construction, a derivation is given of the Gel'fand‐Zetlin matrix elements of the infinitesimal generators of O(n).en_US
dc.format.extent3102 bytes
dc.format.extent1071974 bytes
dc.format.mimetypetext/plain
dc.format.mimetypeapplication/pdf
dc.publisherThe American Institute of Physicsen_US
dc.rights© The American Institute of Physicsen_US
dc.titleLowering and Raising Operators for the Orthogonal Group in the Chain O(n) ⊃ O(n − 1) ⊃ … , and their Graphsen_US
dc.typeArticleen_US
dc.subject.hlbsecondlevelPhysicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.peerreviewedPeer Revieweden_US
dc.contributor.affiliationumPhysics Department, University of Michigan, Ann Arbor, Michiganen_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/70553/2/JMAPAQ-8-6-1233-1.pdf
dc.identifier.doi10.1063/1.1705340en_US
dc.identifier.sourceJournal of Mathematical Physicsen_US
dc.identifier.citedreferenceJ. G. Nagel and M. Moshinsky, J. Math. Phys. 6, 682 (1965).en_US
dc.identifier.citedreferenceM. Moshinsky, J. Math. Phys. 4, 1128 (1963); G. E. Baird and L. C. Biedenharn, 4, 1449 (1963). For earlier references consult these references.en_US
dc.identifier.citedreferenceB. H. Flowers and S. Szpikowski, Proc. Phys. Soc. (London) 84, 193 (1964); J. C. Parikh, Nucl. Phys. 63, 214 (1965). J. N. Ginocchio, 74, 321 (1965); M. Ichimura, Progr. Theoret. Phys. (Kyoto) 32, 757 (1964); 33, 215 (1965). K. T. Hecht, Phys. Rev. 139, B794 (1965).en_US
dc.identifier.citedreferenceB. H. Flowers and S. Szpikowski, Proc. Phys. Soc. (London) 84, 673 (1964).en_US
dc.identifier.citedreferenceP. Kramer and M. Moshinsky, Nucl. Phys. 82, 241 (1966).en_US
dc.identifier.citedreferenceI. M. Gel’fand and M. L. Zetlin, Dokl. Akad. Nauk. USSR 71, 1017 (1950). I. M. Gel’fand, R. A. Minlos, and Z. Ya. Shapiro Representations of the Rotation and Lorentz Groups and Their Application (The Macmillan Company, New York, 1963), p. 353.en_US
dc.identifier.citedreferenceThe Gel’fand‐Zetlin result has also been derived by algebraic techniques by J. D. Louck, Los Alamos Scientific Laboratory Reports LA 2451 (1960).en_US
dc.identifier.citedreferenceG. Racah, CERN reprint 61‐8 (1961).en_US
dc.identifier.citedreferenceThe raising and lowering generators are not to be confused with the raising and lowering operators which are the subject of this paper. Except for O(3) the lowering and raising operators are complicated polynomial functions of the lowering and raising generators.en_US
dc.identifier.citedreferenceA slight change has been made in the Gel’fand‐Zetlin notation. The first index has been shifted up by one unit so that mν1,mν1, mν2,mν2, … characterize the irreducible representation of O(v). The chain of numbers thus ends with m21m21 [irreducible representation of O(2)], rather than with m11m11.en_US
dc.identifier.citedreferenceFor the specific cases n  =  5n=5 and 6 explicit expressions for raising and lowering operators have been given previously. J. Flores, E. Chacon, P. A. Mello, and M. de Llano, Nucl. Phys. 72, 352 (1965), and (n  =  5)(n=5)K. T. Hecht, 63, 177 (1965).en_US
dc.identifier.citedreferenceS. C. Pang, University of Michigan dissertation (to be published).en_US
dc.identifier.citedreference† Note that O63O63 is an example of a neutral or zero‐step operator of type O2k,k.O2k,k.en_US
dc.identifier.citedreferenceThe superscript (2k+1)(2k+1) will be omitted whenever it is obvious.en_US
dc.owningcollnamePhysics, Department of


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