10,209 research outputs found
Wigner-Eckart theorem for tensor operators of Hopf algebras
We prove Wigner-Eckart theorem for the irreducible tensor operators for
arbitrary Hopf algebras, provided that tensor product of their irreducible
representation is completely reducible. The proof is based on the properties of
the irreducible representations of Hopf algebras, in particular on Schur lemma.
Two classes of tensor operators for the Hopf algebra U(su(2)) are
considered. The reduced matrix elements for the class of irreducible tensor
operators are calculated. A construction of some elements of the center of
U(su(2)) is given.Comment: 14 pages, late
Describing neutrino oscillations in matter with Magnus expansion
We present new formalism for description of the neutrino oscillations in
matter with varying density. The formalism is based on the Magnus expansion and
has a virtue that the unitarity of the S-matrix is maintained in each order of
perturbation theory. We show that the Magnus expansion provides better
convergence of series: the restoration of unitarity leads to smaller deviations
from the exact results especially in the regions of large transition
probabilities. Various expansions are obtained depending on a basis of neutrino
states and a way one splits the Hamiltonian into the self-commuting and
non-commuting parts. In particular, we develop the Magnus expansion for the
adiabatic perturbation theory which gives the best approximation. We apply the
formalism to the neutrino oscillations in matter of the Earth and show that for
the solar oscillation parameters the second order Magnus adiabatic expansion
has better than 1% accuracy for all energies and trajectories. For the
atmospheric and small 1-3 mixing the approximation works well ( accuracy for ) outside the resonance region
(2.7 - 8) GeV.Comment: Discussions expanded, two figures and references added, the version
will appear in Nucl. Phys.
Attenuation effect and neutrino oscillation tomography
Attenuation effect is the effect of weakening of contributions to the
oscillation signal from remote structures of matter density profile. The effect
is a consequence of integration over the neutrino energy within the energy
resolution interval. Structures of a density profile situated at distances
larger than the attenuation length, , are not "seen". We show
that the origins of attenuation are (i) averaging of oscillations in certain
layer(s) of matter, (ii) smallness of matter effect: , where is the matter potential, and (iii) specific
initial and final states on neutrinos. We elaborate on the graphic description
of the attenuation which allows us to compute explicitly the effects in the
order for various density profiles and oscillation channels. The
attenuation in the case of partial averaging is described. The effect is
crucial for interpretation of oscillation data and for the oscillation
tomography of the Earth with low energy (solar, supernova, atmospheric, {\it
etc.}) neutrinos.Comment: 24 pages, 8 figures, typos corrected, more explanations adde
Solar neutrinos: global analysis and implications for SNO
We present a global analysis of all the available solar neutrino data
treating consistently the 8B and hep neutrino fluxes as free parameters. The
analysis reveals at 99.7% C.L. eight currently-allowed discrete regions in
two-neutrino oscillation space, five regions corresponding to active neutrinos
and three corresponding to sterile neutrinos. Most of the allowed solutions are
robust with respect to changes in the analysis procedure, but the traditional
vacuum solution is fragile. The globally-permitted range of the 8B neutrino
flux, 0.45 to 1.95 in units of the BP2000 flux, is comparable to the 3 sigma
range allowed by the standard solar model. We discuss the implications for SNO
of a low mass, Delta m^2 ~ 6 times 10^{-12} eV^2, vacuum oscillation solution,
previously found by Raghavan, and by Krastev and Petcov, but absent in recent
analyses that included Super-Kamiokande data. For the SNO experiment, we
present refined predictions for the charged-current rate and the ratio of the
neutral-current rate to charged-current rate. The predicted charged-current
rate can be clearly distinguished from the no-oscillation rate only for the LMA
solution. The predicted ratio of the neutral-current rate to charged-current
rate is distinguishable from the no-oscillation ratio for the LMA, SMA, LOW,
and VAC solutions for active neutrinos.Comment: viewgraphs and related material at http://www.sns.ias.ed
On a general analytical formula for U_q(su(3))-Clebsch-Gordan coefficients
We present the projection operator method in combination with the
Wigner-Racah calculus of the subalgebra U_q(su(2)) for calculation of
Clebsch-Gordan coefficients (CGCs) of the quantum algebra U_q(su(3)). The key
formulas of the method are couplings of the tensor and projection operators and
also a tensor form for the projection operator of U_q(su(3)). We obtain a very
compact general analytical formula for the U_q(su(3)) CGCs in terms of the
U_q(su(2)) Wigner 3nj-symbols.Comment: 9 pages, LaTeX; to be published in Yad. Fiz. (Phys. Atomic Nuclei),
(2001
q-Analog of Gelfand-Graev Basis for the Noncompact Quantum Algebra U_q(u(n,1))
For the quantum algebra U_q(gl(n+1)) in its reduction on the subalgebra
U_q(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction
Z-algebra Z_q(gl(n+1),gl(n)) is given in terms of the generators and their
defining relations. Using this Z-algebra we describe Hermitian irreducible
representations of a discrete series for the noncompact quantum algebra
U_q(u(n,1)) which is a real form of U_q(gl(n+1)), namely, an orthonormal
Gelfand-Graev basis is constructed in an explicit form.Comment: Invited talk given by V.N.T. at XVIII International Colloquium
"Integrable Systems and Quantum Symmetries", June 18--20, 2009, Prague, Czech
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