Quantum Loop Modules and Quantum Spin Chains

We construct level-0 modules of the quantum affine algebra \Uq, as the $q$-deformed version of the Lie algebra loop module construction. We give necessary and sufficient conditions for the modules to be irreducible. We construct the crystal base for some of these modules and find significant differences from the case of highest weight modules. We also consider the role of loop modules in the recent scheme for diagonalising certain quantum spin chains using their \Uq symmetry.Comment: 32 pages, 5 figures (appended), ENSLAPP-L-419/93, MRR2/9

Algebraic Quantum Mechanics and Pregeometry

We discuss the relation between the q-number approach to quantum mechanics suggested by Dirac and the notion of "pregeometry" introduced by Wheeler. By associating the q-numbers with the elements of an algebra and regarding the primitive idempotents as "generalized points" we suggest an approach that may make it possible to dispense with an a priori given space manifold. In this approach the algebra itself would carry the symmetries of translation, rotation, etc. Our suggestion is illustrated in a preliminary way by using a particular generalized Clifford algebra proposed originally by Weyl, which approaches the ordinary Heisenberg algebra a suitable limit. We thus obtain a certain insight into how quantum mechanics may be regarded as a purely algebraic theory, provided that we further introduce a new set of "neighbourhood operators", which remove an important kind of arbitrariness that has thus far been present in the attempt to treat quantum mechanics solely in terms of a Heisenberg algebra

Close Approach during Hard Binary--Binary Scattering

We report on an extensive series of numerical experiments of binary--binary scattering, analysing the cross--section for close approach during interactions for a range of hard binary parameters of interest in globular cluster cores. We consider the implied rate for tidal interactions for different globular clusters and compare our results with previous, complementary estimates of stellar collision rates in globular clusters. We find that the collision rate for binary--binary encounters dominates in low density clusters if the binary fraction in the cluster is larger than $0.2$ for wide main--sequence binaries. In dense clusters binary--single interactions dominate the collision rate and the core binary fraction must be \ltorder 0.1 per decade in semi--major axis or too many collisions take place compared to observations. The rates are consistent if binaries with semi--major axes $\sim 100 AU$ are overabundant in low density clusters or if breakup and ejection substantially lowers the binary fraction in denser clusters. Given reasonable assumptions about fractions of binaries in the cores of low density clusters such as NGC~5053, we cannot account for all the observed blue stragglers by stellar collisions during binary encounters, suggesting a substantial fraction may be due to coalescence of tight primordial binaries.Comment: 13 pages including 13 ps figures. MNRAS in pres

Knowledge development for organic systems: An example of weed management

Despite the large amount information on weed biology and specific weed control measures produced by researchers, organic farmers still prioritise weeds as an important area for further research. A recent project investigating weed management in organic farming systems has established that knowledge and learning are key requirements for this to be effective. Development of relevant, practically useful knowledge depends on access to information generated ‘scientifically’ by researchers and also to knowledge generated as a result of farmer experience with weeds. This requires that farmers, advisors and researchers take a participatory approach to collecting and processing information on weed management, using it to develop new and relevant knowledge. The appropriate framework for knowledge development is thus a collegiate one in which all stakeholders’ value and learn from the observations and experience of others. These findings have implications for the way in which research is conducted and funded

Determining the Nature of Late Gunn-Peterson Troughs with Galaxy Surveys

Recent observations have discovered long (up to ~110 Mpc/h), opaque Gunn-Peterson troughs in the z ~ 5.5 Lyman-alpha forest, which are challenging to explain with conventional models of the post-reionization intergalactic medium. Here we demonstrate that observations of the galaxy populations in the vicinity of the deepest troughs can distinguish two competing models for these features: deep voids where the ionizing background is weak due to fluctuations in the mean free path of ionizing photons would show a deficit of galaxies, while residual temperature variations from extended, inhomogeneous reionization would show an overdensity of galaxies. We use large (~550 Mpc/h) semi-numerical simulations of these competing explanations to predict the galaxy populations in the largest of the known troughs at z ~ 5.7. We quantify the strong correlation of Lyman-alpha effective optical depth and galaxy surface density in both models and estimate the degree to which realistic surveys can measure such a correlation. While a spectroscopic galaxy survey is ideal, we also show that a relatively inexpensive narrowband survey of Lyman-alpha-emitting galaxies is ~90% likely to distinguish between the competing models.Comment: 12 pages, 16 figures. Submitted to Ap

A numerical investigation of the solution of a class of fourth-order eigenvalue problems

This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corners on domain boundaries. Recent computational results of Bjorstad & Tjostheim, using a highly accurate spectral Legendre-Galerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite-element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained from this mixed finite-element approach are then presented for a variety of domains. These include a family of circular sector regions, for which the oscillatory behaviour is studied as a function of the internal angle, and another family of (symmetric and non-convex) domains, for which the parity of the least eigenfunction is investigated. The paper not only verifies existing asymptotic theory, but also allows us to make a new conjecture concerning the eigenfunctions of the biharmonic operator

Time-of-arrival probabilities and quantum measurements: II Application to tunneling times

We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance L from the tunneling region. For this purpose, we use a Positive-Operator-Valued-Measure (POVM) for the time-of-arrival determined in quant-ph/0509020 [JMP 47, 122106 (2006)]. This only depends on the initial state, the Hamiltonian and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement set-up: the phase space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime, in which these probabilities correspond to a tunneling-time operator.Comment: 26 pages--revised version, small changes, to appear in J. Math. Phy