Kirillov’s character formula,the holomorphic Peter–Weyl theorem,and the Blattner–Kostant–Sternberg pairing

By means of the orbit method we show that, for a compact Lie group, the Blattner–Kostant–Sternberg pairing map, with the constants being appropriately fixed, is unitary. Along the way we establish a holomorphic Peter–Weyl theorem for the complexification of a compact Lie group. Among our crucial tools is Kirillov’s character formula. The basic observation is that the Weyl vector is lurking behind the Kirillov character formula, as well as behind the requisite half-form correction on which the Blatter–Kostant–Sternberg-pairing for the compact Lie group relies, and thus produces the appropriate shift which, in turn, controls the unitarity of the BKS-pairing map. Our methods are independent of heat kernel harmonic analysis, which is used by B. C. Hall to obtain a number of these results [B.C. Hall, The Segal–Bargmann Coherent State Transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103–151; B.C. Hall, Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type, Comm. Math. Phys. 226 (2002) 233–268, quant.ph/0012015].     

Analytic continuation,the Chern–Gauss–Bonnet theorem,and the Euler–Lagrange equations in Lovelock theory for indefinite signature metrics

We use analytic continuation to derive the Euler–Lagrange equations associated to the Pfaffian in indefinite signature $(p,q)$ directly from the corresponding result in the Riemannian setting. We also use analytic continuation to derive the Chern–Gauss–Bonnet theorem for pseudo-Riemannian manifolds with boundary directly from the corresponding result in the Riemannian setting. Complex metrics on the tangent bundle play a crucial role in our analysis and we obtain a version of the Chern–Gauss–Bonnet theorem in this setting for certain complex metrics.     

A note on the Holst action, the time gauge, and the Barbero–Immirzi parameter

In this note, we review the canonical analysis of the Holst action in the time gauge, with a special emphasis on the Hamiltonian equations of motion and the fixation of the Lagrange multipliers. This enables us to identify at the Hamiltonian level the various components of the covariant torsion tensor, which have to be vanishing in order for the classical theory not to depend upon the Barbero–Immirzi parameter. We also introduce a formulation of three-dimensional gravity with an explicit phase space dependency on the Barbero–Immirzi parameter as a potential way to investigate its fate and relevance in the quantum theory.     

Atomic structure of the Zr–He,Zr–vac,and Zr–vac–He systems: First-principles calculation

The ab initio investigations have been performed for the atomic structure of the Zr–He, Zr–vac, and Zr–vac–He systems with concentrations of helium atoms and vacancies (vac) of ~6 at %. A heliuminduced instability of the zirconia lattice has been revealed in the Zr–He system, which disappears with the formation of vacancies. The most preferred positions of impurities in the metal lattice have been determined. The energy of helium dissolution and the excess volume introduced by helium have been calculated. It has been established that the presence of helium in the Zr lattice leads to a significant decrease in the energy of vacancy formation.     

Franck–Condon Factors and r‐Centroids for the B–, C–, F–, and G–X Band Systems of YF Molecule

The Franck–Condon factors and r‐centroids, which are very closely related to relative transition probabilities, have been evaluated by a more reliable numerical integration procedure for the B¹π–X¹Σ⁺, C¹Σ⁺–X¹Σ⁺, F¹Σ⁺–X¹Σ⁺, and G¹π–X¹Σ⁺ band systems of the YF molecule, using suitable potentials.     

Analysis of the common characteristics of the Hartmann,Ronchi, and Shack–Hartmann tests

An overview of the settings of the planes for the filters and observed patterns in the Hartmann and Ronchi tests is presented. Also a new set of filters for both test were developed. In a similar way, it is easy to extend this analysis to the Shack–Hartmann test, and to propose a new Null Shack–Hartmann filter.     

Internal-friction study of the interstitial–substitutional effect on the deformation behaviour of Nb–O,Nb–Ta–O and Nb–Mo–O single crystals

To investigate the interstitial-substitutional interaction in dislocations, the effect of 0 on the temperature, frequency and amplitude dependence of the internal friction Q ^?1 in Nb–O, Nb-20 mol% Ta–O and Nb-20 mol% Mo–O single crystals has been studied (f = 1.55–8.2 Hz) in the temperature range from 298 to 1473 K. In our previous study, Nb–Mo and Nb–Ta single crystals were found to be strengthened by solute O. It was also suggested that the interstitial–substitutional interaction in dislocations contributes to the increase in their critical resolved shear stress (CRSS). In this study, Snoek-type relaxation peaks due to O are observed between 500 and 700 K in all the single crystals. The Snoek peak of Nb-20mol% Mo–O consisting of several peaks is analysed. The activation energy of the Snoek peak in Nb-20mol% Mo–O is higher than that of Nb–O. These results are attributable to the existence of the interstitial–substitutional complexes. The amplitude dependence of Q ^?1 at intermediate and high temperatures decreases as the O content increases. Moreover, the breakaway stress of Nb-20 mol% Mo–O still has a high value at 1200K and does not decrease much even at 1473 K. This suggests that the formation of Mo–O complexes reduces the dislocation mobility at high temperatures. From the results, the effect of the interstitial-substitutional interaction on the CRSS was discussed.     

Nanotoxicology: characterizing the scientific literature, 2000–2007

Understanding the toxicity of nanomaterials and nano-enabled products is important for human and environmental health and safety as well as public acceptance. Assessing the state of knowledge about nanotoxicology is an important step in promoting comprehensive understanding of the health and environmental implications of these new materials. To this end, we employed bibliometric techniques to characterize the prevalence and distribution of the current scientific literature. We found that the nano-toxicological literature is dispersed across a range of disciplines and sub-fields; focused on in vitro testing; often does not specify an exposure pathway; and tends to emphasize acute toxicity and mortality rather than chronic exposure and morbidity. Finally, there is very little research on consumer products, particularly on their environmental fate, and most research is on the toxicity of basic nanomaterials. The implications for toxicologists, regulators and social scientists studying nanotechnology and society are discussed.

Mutual transformations between the P–Q,Q–P,and generalized Weyl ordering of operators

Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ok （p, q） with a real k parameter and can unify the P-Q, Q-P, and Weyl ordering of operators in k = 1, - 1,0, respectively, we find the mutual transformations between 6 （p - P） （q - Q）, （q - Q） 3 （p - P）, and （p, q）, which are, respectively, the integration kernels of the P-Q, Q-P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The - and - ordered forms of （p, q） are also derived, which helps us to put the operators into their - and - ordering, respectively.     

Loop Groups, Anyons and the Calogero–Sutherland Model

The positive energy representations of the loop group of U(1) are used to construct a boson-anyon correspondence. We compute all the correlation functions of our anyon fields and study an anyonic W-algebra of unbounded operators with a common dense domain. This algebra contains an operator with peculiar exchange relations with the anyon fields. This operator can be interpreted as a second quantized Calogero–Sutherland (CS) Hamiltonian and may be used to solve the CS model. In particular, we inductively construct all eigenfunctions of the CS model from anyon correlation functions, for all particle numbers and positive couplings. Received: 12 May 1998 / Accepted: 4 August 1998     

MAP,MAC, and vortex-rings configurations in the Weinberg–Salam model

We report on the presence of new axially symmetric monopoles, antimonopoles and vortex-rings solutions of the SU(2)×U(1) Weinberg–Salam model of electromagnetic and weak interactions. When the ?$?$-winding number n=1$n=1$, and 2, the configurations are monopole–antimonopole pair (MAP) and monopole–antimonopole chain (MAC) with poles of alternating sign magnetic charge arranged along the z$z$-axis. Vortex-rings start to appear from the MAP and MAC configurations when the winding number n=3$n=3$. The MAP configurations possess zero net magnetic charge whereas the MAC configurations possess net magnetic charge of 4πn/e$4\pi n/e$.     

Einstein–Cartan gravity,Asymptotic Safety,and the running Immirzi parameter

In this paper we analyze the functional renormalization group flow of quantum gravity on the Einstein–Cartan theory space. The latter consists of all action functionals depending on the spin connection and the vielbein field (co-frame) which are invariant under both spacetime diffeomorphisms and local frame rotations. In the first part of the paper we develop a general methodology and corresponding calculational tools which can be used to analyze the flow equation for the pertinent effective average action for any truncation of this theory space. In the second part we apply it to a specific three-dimensional truncated theory space which is parametrized by Newton’s constant, the cosmological constant, and the Immirzi parameter. A comprehensive analysis of their scale dependences is performed, and the possibility of defining an asymptotically safe theory on this hitherto unexplored theory space is investigated. In principle Asymptotic Safety of metric gravity (at least at the level of the effective average action) is neither necessary nor sufficient for Asymptotic Safety on the Einstein–Cartan theory space which might accommodate different “universality classes” of microscopic quantum gravity theories. Nevertheless, we do find evidence for the existence of at least one non-Gaussian renormalization group fixed point which seems suitable for the Asymptotic Safety construction in a setting where the spin connection and the vielbein are the fundamental field variables.     

Low temperature properties of the Fermi–Dirac,Boltzmann and Bose–Einstein equations

We investigate low temperature (T  ) properties of three classical quantum statistics models: (I) the Fermi–Dirac equation, (II) the Boltzmann equation, and (III) the Bose–Einstein equation. It is widely assumed that each of these equations is valid for all T>0$T>0$. For each equation we prove that this assumption leads to erroneous predictions as T→0⁺$T\to 0^{+}$. Our approach to correct these errors gives new low temperature predictions which contradict previous theory. We examine a two-state paramagnetism system and show how our new low temperature prediction compares favorably with experimental data.     

Bond properties of the chalcopyrite and stannite phases in the Cu–(In,Ga)–Se system

Bond properties of the chalcopyrite and (defect) stannite phases in the Cu–(In,Ga)–Se system are compared in view of the bond overlap population calculated by the molecular orbital calculation of the DV-Xα method. Bond stretching force constant α is estimated for the stannite phases through the bond Ovlp. The Cu–Se and In(Ga)–Se bonds in defect stannite structure are considered to be mechanically weakened by the 2b-site vacancies. We estimate the weakened force constants to be 60–70% of those in the chalcopyrite structure. On the other hand, in In(Ga)-rich stannite, In(Ga)_4d–Se_8i and In(Ga)_2b–Se_8i bonds are estimated to be tighter by 23–25 and 8–9%, respectively, than In(Ga)_4b–Se_8d bond in the chalcopyrite structure. The Γ₁ frequencies of the stannite phases are also calculated using the estimated force constants. Characteristic Raman signals peaked at 160–175 cm⁻¹ observed for the Cu(In_1−xGa_x)₃Se₅ system are explained by the Cu-rich phase for the Cu–In–Se system, and the phase combination of Cu-rich and Ga_2aV_2b types for the Cu–Ga–Se system from these calculations.     

Electromagnetism,Local Covariance,the Aharonov–Bohm Effect and Gauss’ Law

We quantise the massless vector potential A of electromagnetism in the presence of a classical electromagnetic (background) current, j, in a generally covariant way on arbitrary globally hyperbolic spacetimes M. By carefully following general principles and procedures we clarify a number of topological issues. First we combine the interpretation of A as a connection on a principal U(1)-bundle with the perspective of general covariance to deduce a physical gauge equivalence relation, which is intimately related to the Aharonov–Bohm effect. By Peierls’ method we subsequently find a Poisson bracket on the space of local, affine observables of the theory. This Poisson bracket is in general degenerate, leading to a quantum theory with non-local behaviour. We show that this non-local behaviour can be fully explained in terms of Gauss’ law. Thus our analysis establishes a relationship, via the Poisson bracket, between the Aharonov–Bohm effect and Gauss’ law – a relationship which seems to have gone unnoticed so far. Furthermore, we find a formula for the space of electric monopole charges in terms of the topology of the underlying spacetime. Because it costs little extra effort, we emphasise the cohomological perspective and derive our results for general p-form fields A (p <  dim(M)), modulo exact fields, for the Lagrangian density ${\mathcal{L} = \frac{1}{2} dA\wedge*dA+ A\wedge*j}$ . In conclusion we note that the theory is not locally covariant, in the sense of Brunetti–Fredenhagen–Verch. It is not possible to obtain such a theory by dividing out the centre of the algebras, nor is it physically desirable to do so. Instead we argue that electromagnetism forces us to weaken the axioms of the framework of local covariance, because the failure of locality is physically well-understood and should be accommodated.     

Symmetry Reduction and Exact Solutions of the Euler–Lagrange–Born–Infeld,Multidimensional Monge–Ampere and Eikonal Equations

Abstract

Using the subgroup structure of the generalized Poincaré group P (1, 4), ansatzes which reduce the Euler–Lagrange–Born–Infeld, multidimensional Monge–Ampere and eikonal equations to differential equations with fewer independent variables have been constructed. Among these ansatzes there are ones which reduce the considered equations to linear ordinary differential equations. The corresponding symmetry reduction has been done. Using the solutions of the reduced equations, some classes of exact solutions of the investigated equation have been presented.