Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms

We study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons and the so-called exchange term are neglected. A state of the system is described by its one-body density matrix, an infinite rank self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator (the Dirac sea). We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant α. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an L ¹ function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to α by the formula α_phys ≃ α(1 + 2α/(3π) log Λ)⁻¹, where Λ is the ultraviolet cut-off. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge Z. In the nonrelativistic limit, we obtain that this number is  ≤  2Z, recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Séré and Solovej on the mean-field approximation of no-photon QED.     

$$(O( V \oplus F), O( V))$$ is a Gelfand pair for any quadratic space V over a local field F

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let with the form Q extending q with Q(e) = 1. Consider the standard embedding and the two-sided action of on . In this note we show that any -invariant distribution on is invariant with respect to transposition. This result was earlier proven in a bit different form in van Dijk (Math Z 193:581–593, 1986) for , in Aparicio and van Dijk (Complex generalized Gelfand pairs. Tambov University, 2006) for and in Bosman and van Dijk (Geometriae Dedicata 50:261–282, 1994) for p-adic fields. Here we give a different proof. Using results from Aizenbud et al. (arXiv:0709.1273 (math.RT), submitted), we show that this result on invariant distributions implies that the pair (O(V), O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Fréchet representation (π, E) of we have A stronger result for p-adic fields is obtained in Aizenbud et al. (arXiv:0709.4215 (math.RT), submitted).     

Multiplicity one theorem for ({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))

Let F be either or . Consider the standard embedding and the action of GL_n(F) on GL_n+1(F) by conjugation. We show that any GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Fréchet representations π of GL_n+1(F) and of GL_n(F),

Cyclic voltammetry of Th(IV) in the room-temperature ionic liquid [Me3NnBu][N(SO2CF3)2

A Th(IV) compound, [Th(TFSI)4(HTFSI)].2H2O [where TFSI = N(SO2CF3)2], has been synthesized and characterized using elemental analysis, thermogravimetric analysis, and vibrational spectroscopy. The analysis suggests that the TFSI anion coordinates to the metal center via the sulfonyl oxygens as well as provides evidence for the coordination of HTFSI. The voltammetric behavior of this compound has been studied in the room-temperature ionic liquid [Me3NnBu][TFSI], and results show that Th(IV) is reduced to Th(0) in this ionic liquid in a single reduction step. Analysis of cyclic voltammograms shows that an insoluble product is being formed at the electrode surface, which is attributed to the formation of ThO2 by reaction with water. The E0 value for the reduction of Th(IV) to Th(0) has been determined to be -2.20 V (vs Fc+/Fc; -1.80 V vs SHE). A comparison of this E0 value with those obtained for Th(IV) reduction in a LiCl-KCl eutectic (400 degrees C), water, and nonaqueous solvents shows that the reduction in [Me3NnBu][TFSI] is easier to accomplish than that in these other solvents.     

Intrinsic fluorescence polarization of amniotic fluid: II. Toward a noninvasive method for the determination of fetal lung maturity

The present study compares two methods for the determination of fetal lung maturity: the novel intrinsic fluorescence polarization ratio (IFPR) and the commercial TDx-FLMII. Amniotic fluid (AF) samples were collected from 69 women during the second and third trimesters of singleton pregnancies. Thirty-three samples were tested for IFPR only after centrifugation, and the rest were examined both before and after centrifugation. Of the latter 33 samples, 29 were assessed for lung maturity with the TDx-FLMII method as well. The results showed that IFPR values decreased with the advance in gestational age (r = 0.77, p < 0.05, n = 69). A significant correlation was found between IFPR of centrifuged and noncentrifuged samples (r = 0.94, p < 0.05, n = 36). A significant correlation was demonstrated between IFPR and TDx-FLMII values of centrifuged (r = 0.75, p < 0.05, n = 29) and noncentrifuged (r = 0.63, p < 0.05, n = 29) samples and moreover, samples considered mature by TDx-FLMII had low values of IFPR (n = 10). It can be concluded that the IFPR method can utilize noncentrifuged AF, thus suggested as a potential noninvasive method.     

Stereoselectivity in the synthesis of polyprenylphosphoryl β-d-ribofuranoses

Decaprenylphosphoryl β-d-arabinofuranose (DPA) is a key arabinose donor in mycobacteria. The ribo analog of DPA (DPR) has also been found in mycobacteria. It has recently been confirmed that DPA is formed via a two-step epimerization of DPR. The stereoselective synthesis of DPR as well as two shorter analogs of DPR is described.     

Existence of Hartree–Fock excited states for atoms and molecules

For neutral and positively charged atoms and molecules, we prove the existence of infinitely many Hartree–Fock critical points below the first energy threshold (that is, the lowest energy of the same system with one electron removed). This is the equivalent, in Hartree–Fock theory, of the famous Zhislin–Sigalov theorem which states the existence of infinitely many eigenvalues below the bottom of the essential spectrum of the N-particle linear Schrödinger operator. Our result improves a theorem of Lions in 1987 who already constructed infinitely many Hartree–Fock critical points, but with much higher energy. Our main contribution is the proof that the Hartree–Fock functional satisfies the Palais–Smale property below the first energy threshold. We then use minimax methods in the N-particle space, instead of working in the one-particle space.     

A Family of Monotone Quantum Relative Entropies

We study here the elementary properties of the relative entropy ${\mathcal{H}_\varphi(A, B) = {\rm Tr}[\varphi(A) - \varphi(B) - \varphi'(B)(A - B)]}$ for φ a convex function and A, B bounded self-adjoint operators. In particular, we prove that this relative entropy is monotone if and only if φ′ is operator monotone. We use this to appropriately define ${\mathcal{H}_\varphi(A, B)}$ in infinite dimension.     

Scale-invariant correlations in dynamic bacterial clusters

In Bacillus subtilis colonies, motile bacteria move collectively, spontaneously forming dynamic clusters. These bacterial clusters share similarities with other systems exhibiting polarized collective motion, such as bird flocks or fish schools. Here we study experimentally how velocity and orientation fluctuations within clusters are spatially correlated. For a range of cell density and cluster size, the correlation length is shown to be 30% of the spatial size of clusters, and the correlation functions collapse onto a master curve after rescaling the separation with correlation length. Our results demonstrate that correlations of velocity and orientation fluctuations are scale invariant in dynamic bacterial clusters.