Abstract
In applying symmetry concepts to the description of a physical system, the following situation arises. One is given a Hilbert space ℍ which carries a reducible representation of some dynamical symmetry group G. The observables of the system are expressible as tensors under the group G and the basic problem is to carry out the following steps:
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(i)
Construct a basis for ℍ which reduces G and its relevant subgroups.
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(ii)
Calculate the matrix representations of the observables in this basis.
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(iii)
Diagonalize the Hamiltonian for the system, calculate the observable properties of the eigenstates and compare them with experiment.
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References
L.C. Biedenharn and J.D. Louck, A pattern calculus for tensor operators in the unitary groups, Commun. Math. Phys. 8:89 (1968).
R. Le Blanc, K.T. Hecht and L.C. Biedenharn, Screening and vertex operators for u(n), J. Phys. A: Math. Gen. 24:1393 (1991).
D.J. Rowe, Coherent state theory of the noncompact symplectic group, J. Math. Phys. 25:2662 (1984)
D.J. Rowe, G. Rosensteel and R. Carr, Analytical expressions for the matrix elements of the noncompact symplectic algebra, J. Phys. A: Math. Gen. 17:L399 (1984)
D.J. Rowe, G. Rosensteel and R. Gilmore, Vector coherent state representation theory, J. Math. Phys. 26: 2787 (1985)
K.T. Hecht, The vector coherent state method and its application to problems of higher symmetries, Lecture Notes in Physics, Springer, Berlin (1987)
D.J. Rowe, R. Le Blanc and K.T. Hecht, Vector coherent state theory and its application to the orthogonal groups, J. Math. Phys. 29:287 (1988).
P. Bouwknegt, J. McCarthy and K. Pilch, Free field realizations of WZNW models. The BRST complex and its quantum group structure, Phys. Lett. B, 234:297 (1990); Quantum group structure in the Fock space resolutions of sl(n) representations, Commun. Math. Phys. 131:125 (1990).
K.T. Hecht, A new generalization of vector coherent state theory and the quantized Bogoliubov transformation of Abraham Klein, in “The Proceedings of the Symposium on Contemporary Physics,” M Vallieres, ed., in press.
D.J. Rowe and J. Repka, Vector-coherent-state theory as a theory of induced representations, J. Math. Phys. 32:2614 (1991).
L.C. Biedenharn, Symetry properties of nuclei, in “Proceedings of the Fifteenth Solvay Conference on Physics,” Gordan and Breach, London, 1970.
J.-P. Serre, Representations linéaires et espaces homogènes Kählerians des groupes de Lie compacts in “Séminaires Bourbaki, 6{sue} année, 1953/54,” Inst. Henri Poincaré, Paris, 1954.
D.-N. Verma, Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74: 160, 628 (1968).
Harish Chandra, Representations of semisimple Lie groups IV, Amer. J. Math. 77:743 (1955)
Representations of semisimple Lie groups V, Amer. J. Math. 78:1 (1956)
Representations of semisimple Lie groups VI, Amer. J. Math. 78:564 (1956).
R. Godement, Seminaire Cartan, École Normale Supérieure, Paris, 1958
G. Rosensteel and D.J. Rowe, The discrete series of Sp(n,ℝ), Int. J. Theor. Phys. 16:63 (1977).
J. Deenen and C. Quesne, Partially coherent states of the real symplectic group, J. Math. Phys. 25:2354 (1984)
C. Quesne, Coherent states of the real symplectic group in a complex analytic parametrization. I. Unitary-operator coherent states, J. Math. Phys. 27:428 (1986).
D.J. Rowe, R. Le Blanc and J. Repka, A rotor expansion of the su(3) Lie algebra, J. Phys. A: Math. Gen. 22:L3O9 (1989)
D.J. Rowe and R. Le Blanc; Superfield and matrix realizations of highest weight representations for osp(m/2n), J. Math. Phys. 31:14 (1990).
B. Kostant, Verma modules and the existence of quasi-invariant differential operators, in “Non-commutative Harmonic Analysis,” Lect. Notes in Math. 466:101 (1974).
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Rowe, D.J., Repka, J. (1993). The Biedenharn-Louck Tensor Calculus From a Vector Coherent State Perspective. In: Gruber, B. (eds) Symmetries in Science VI. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1219-0_53
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DOI: https://doi.org/10.1007/978-1-4899-1219-0_53
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