PROGRAM SUMMARY
Title of program:
exactU3clebsch
Catalogue identifier:
ADOL
Ref. in CPC:
139(2001)246
Distribution format: tar gzip file
Operating system: MS Windows 95/98/NT
Number of lines in distributed program, including test data, etc:
2558
Keywords:
U(3), SU(3), SO(3), Clebsch-Gordan coefficients, Wigner coefficients,
Isoscalar factors, Matrix elements reduced, Nuclear physics,
Fractional parentage.
Programming language used: Macsyma 2.4 (pro)
Computer:
PC 300 MHz .
Nature of problem:
Elliott's nuclear SU(3) model [1,2] describes collective nuclear motion
by considering the leading SU(3) irrep within the totally antisymmetric
tensor product space of the valence nucleons. A rotational spectrum is
expected, hence the basis for the irrep space should be symmetry adapted
to the rotation group SO(3). The reduced matrix elements of the
spherical tensor operator Q are needed to compute the electromagnetic
transition rates. When coupling two SU(3) contained in SO(3) irreps to
good SU(3) symmetry, the isoscalar factors together with the known SO(3)
Clebsch-Gordan coefficients provide a unitary basis transformation from
the uncoupled to the coupled basis.
In the interacting boson model [3,4], one of the subgroup chains
containing the rotation group is U(6) contained in SU(3) contained in
SO(3). When writing the U(6) generators as SU(3) tensor operators,
their matrix elements can be expressed in terms of the triple-barred
matrix elements and the isoscalar factors [5].
Method of solution:
It follows from the Borel-Weil theorem [6], that every irreducible
representation of the compact group U(n) can be realized on the space of
polynomials in the coordinates of GL(n,C). This space can be endowed
with a 'differentiation' inner product [7], which is equivalent to the
usual 'integration' inner product but more time-efficient for computer
algebra programs like Macsyma. Applying the lowering operators to the
highest weight and all subsequently generated polynomials while doing
Gram-Schmidt orthogonalization, an orthonormal polynomial basis is
generated. Then the representation when restricted to the generators of
the desired subgroup (in this case SO(3) contains U(3)) is decomposed by
diagonalizing the Cartan subalgebra of that subgroup, identifying the
weight spaces and 'peeling off' the irreps of the subgroup. This is the
standard mathematical procedure according to the Cartan-Weyl theory.
Macsyma's sparse matrix routines allow for large matrix manipulation by
way of keeping track only of the nonzero elements.
Restrictions:
When working with a noncanonical basis, the reduction of an irrep to a
subgroup is usually not multiplicity free, that is, there are missing
labels. Racah has shown that there is no operator with integer
eigenvalues that could be used as the missing label. However operators
in the enveloping algebra with noninteger eigenvalues can be used.
Since it is in general mathematically impossible to find the roots of a
polynomial of degree greater than four using analytical methods (i.e.
exactly), the computer algebra routines can fail if the degeneracy of
the missing label is greater than four. Another limitation is the
amount of memory of the PC. The larger the dimension of the U(3) irreps
the more space is allocated by Macsyma for the different kinds of memory
(cons, binary, ...). See Section 1.5 for details.
Typical running time:
A 300 MHz processor with 64 MB memory needs for the computation of all
reduced matrix elements of Q about 30 minutes for the 125-dimensional
(8,4,0) and about 2 days for the 315-dimensional (12,4,0). To compute
all isoscalars, it needs about 20 minutes for the coupling of (3,1,0)
with (2,1,0) and about 2 days for the coupling of (4,2,0) with (4,2,0).
Unusual features:
When Macsyma computes in batch mode, it dynamically allocates space to
the different types of memory while computing, but it cannot decrease
the spaces already allocated unless one starts a new math engine. When
the allocation levels get to high, more and more time is spent on
garbage collecting up the the point where no more progress is made.
That is why the program periodically writes intermediate results to
temporary files, so that the computation can be interrupted at any time
(except when saving). Then the computation can be resumed at the point
of the most recent saving in a new notebook.
References:
[1] J.P. Elliott, Collective motion in the nuclear shell model, I, Proc.
Roy. Soc. (London) A 245 (1958) 128-145.
[2] M. Harvey, The nuclear SU(3) model, Adv. in Nuclear Phys.
1 (1968) 67-182.
[3] F.Iachello, A. Arima, The Interacting Boson Model, Cambridge
University Press, 1987.
[4] F. Alejandro, P. Van Isacker, Algebraic Methods in Molecular and
Nuclear Structure Physics, Wiley-Interscience, New York, 1994.
[5] G. Rosensteel, Analytic formulae for interacting boson model matrix
elements in the SU(3) basis, Phys. Rev. C 41 (2) (1990) 730-735.
[6] A. Knapp, Representation Theory of Semisimple Groups, and Overview
Based on Examples, Princeton University Press, 1986.
[7] W.H. Klink, T. Ton-That, Holomorphic induction and the tensor
product decomposition of irreducible representations of compact
groups. I. SU(n) groups, Ann. Inst. H. Poincare, Section A XXXI (2)
(1979) 77-97.