Dual pairing of symmetry groups and dynamical groups in physics

This article reviews many manifestations and applications of dual representations of pairs of groups, primarily in atomic and nuclear physics. Examples are given to show how such paired representations are powerful aids in understanding the dynamics associated with shell-model coupling schemes and in identifying the physical situations for which a given scheme is most appropriate. In particular, they suggest model Hamiltonians that are diagonal in the various coupling schemes. The dual pairing of group representations has been applied profitably in mathematics to the study of invariant theory. We show that parallel applications to the theory of symmetry and dynamical groups in physics are equally valuable. In particular, the pairing of the representations of a discrete group with those of a continuous Lie group or those of a compact Lie with those of a non-compact Lie group makes it possible to infer many properties of difficult groups from those of simpler groups. This review starts with the representations of the symmetric and unitary groups, which are used extensively in the many-particle quantum mechanics of bosonic and fermionic systems. It gives a summary of the many solutions and computational techniques for solving problems that arise in applications of symmetry methods in physics and which result from the famous Schur-Weyl duality theorem for the pairing of these representations. It continues to examine many chains of symmetry groups and dual chains of dynamical groups associated with the several coupling schemes in atomic and nuclear shell models and the valuable insights and applications that result from this examination.Comment: 51 pages, 5 figures and 5 table

Vector coherent state theory of the generic representations of so(5) in an so(3) basis

For applications of group theory in quantum mechanics, one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some Hamiltonian of interest. Here we use vector coherent state techniques to develop an algorithm for constructing the matrices for arbitrary finite-dimensional irreps of the SO(5) Lie algebra in an SO(3) basis. The SO(3) subgroup of SO(5) is defined by regarding SO(5) as linear transformations of the five-dimensional space of an SO(3) irrep of angular momentum two. A need for such irreps arises in the nuclear collective model of quadrupole vibrations and rotations. The algorithm has been implemented in MAPLE, and some tables of results are presented.Comment: 20 pages, uses multirow.sty, submitted to J. Math. Phy

Dissimilar responses of membrane potential (EM), permeability properties and respiration to cadmium and nickel in maize root cells

The short-term treatment with Cd2+ and Ni2+ triggered transient depolarization of transplasma membrane potential (EM) in the outer cortical root cells of two maize cultivars (cv. Premia and cv. Blitz), however, both metals changed the EM in a quantitatively different way. The magnitude and duration of EM depolarization were concentration dependent and were greater in the metal susceptible cv. Blitz. The highest EM depolarization was recorded with simultaneous application of Cd2+ + Ni2+ in both maize cultivars. The EM depolarization induced by Cd2+ or Cd2+ + Ni2+ but not Ni2+ alone was accompanied with a tremendous increase of membrane conductivity, but it was not accompanied with the effect of heavy metals (HM) on respiration. Simultaneous application of fusiccocin (FC) with Cd2+ or Cd2+ + Ni2+ during the EM depolarization, inability of FC to stop the depolarization by FC-enhanced proton extrusion and rapid restoration of EM, suggested a transient inhibition of the plasma membrane H+-ATPase by these toxic metals. Our data support the opinion that differences in the effects of the studied ions were not the result of their direct action on PM, but rather of their different influence on intracellular processes within root cells

Vector coherent state representations, induced representations, and geometric quantization: II. Vector coherent state representations

It is shown here and in the preceeding paper (quant-ph/0201129) that vector coherent state theory, the theory of induced representations, and geometric quantization provide alternative but equivalent quantizations of an algebraic model. The relationships are useful because some constructions are simpler and more natural from one perspective than another. More importantly, each approach suggests ways of generalizing its counterparts. In this paper, we focus on the construction of quantum models for algebraic systems with intrinsic degrees of freedom. Semi-classical partial quantizations, for which only the intrinsic degrees of freedom are quantized, arise naturally out of this construction. The quantization of the SU(3) and rigid rotor models are considered as examples.Comment: 31 pages, part 2 of two papers, published versio

Maximum rates of N2 fixation and primary production are out of phase in a developing cyanobacterial bloom in the Baltic Sea

Although N2-fixing cyanobacteria contribute significantly to oceanic sequestration of atmospheric CO2, little is known about how N2 fixation and carbon fixation (primary production) interact in natural populations of marine cyanobacteria. In a developing cyanobacterial bloom in the Baltic Sea, rates of N2 fixation (acetylene reduction) showed both diurnal and longer-term fluctuations. The latter reflected fluctuations in the nitrogen status of the cyanobacterial population and could be correlated with variations in the ratio of acetylene reduced to 15N2 assimilated. The value of this ratio may provide useful information about the release of newly fixed nitrogen by a cyanobacterial population. However, although the diurnal fluctuations in N2 fixation broadly paralleled diurnal fluctuations in carbon fixation, the longer-term fluctuations in these two processes were out of phase

Vector coherent state representations, induced representations, and geometric quantization: I. Scalar coherent state representations

Coherent state theory is shown to reproduce three categories of representations of the spectrum generating algebra for an algebraic model: (i) classical realizations which are the starting point for geometric quantization; (ii) induced unitary representations corresponding to prequantization; and (iii) irreducible unitary representations obtained in geometric quantization by choice of a polarization. These representations establish an intimate relation between coherent state theory and geometric quantization in the context of induced representations.Comment: 29 pages, part 1 of two papers, published versio

Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models

This is the third paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we analyze models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper: These are systems with an SL(2,\Rl) gauge symmetry and the complications arise because non -- compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the Master Constraint does not contain the point zero. However, the minimum of the spectrum is of order $\hbar^2$ which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to $\hbar$ normal ordering constants). The physical Hilbert space can then be be obtained after subtracting this normal ordering correction.Comment: 33 pages, no figure

Randomised controlled trial of video clips and interactive games to improve vision in children with amblyopia using the I-BiT system

Background Traditional treatment of amblyopia involves either wearing a patch or atropine penalisation of the better eye. A new treatment is being developed on the basis of virtual reality technology allowing either DVD footage or computer games which present a common background to both eyes and the foreground, containing the imagery of interest, only to the amblyopic eye. Methods A randomised control trial was performed on patients with amblyopia aged 4–8 years with three arms. All three arms had dichoptic stimulation using shutter glass technology. One arm had DVD footage shown to the amblyopic eye and common background to both, the second used a modified shooter game, Nux, with sprite and targets presented to the amblyopic eye (and background to both) while the third arm had both background and foreground presented to both eyes (non-interactive binocular treatment (non-I-BiT) games). Results Seventy-five patients were randomised; 67 were residual amblyopes and 70 had an associated strabismus. The visual acuity improved in all three arms by approximately 0.07 logMAR in the amblyopic eye at 6 weeks. There was no difference between I-BiT DVD and non-I-BiT games compared with I-BiT games (stated primary outcome) in terms of gain in vision. Conclusions There was a modest vision improvement in all three arms. Treatment was well tolerated and safe. There was no difference between the three treatments in terms of primary stated outcomes but treatment duration was short and the high proportion of previously treated amblyopia and strabismic amblyopia disadvantaged dichoptic stimulation treatment