Two-dimensional models as testing ground for principles and concepts of local quantum physics

In the past two-dimensional models of QFT have served as theoretical laboratories for testing new concepts under mathematically controllable condition. In more recent times low-dimensional models (e.g., chiral models, factorizing models) often have been treated by special recipes in a way which sometimes led to a loss of unity of QFT. In the present work, I try to counteract this apartheid tendency by reviewing past results within the setting of the general principles of QFT. To this I add two new ideas: (1) a modular interpretation of the chiral model Diff(S)-covariance with a close connection to the recently formulated local covariance principle for QFT in curved spacetime and (2) a derivation of the chiral model temperature duality from a suitable operator formulation of the angular Wick rotation (in analogy to the Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational chiral theories. The SL (2, Z) modular Verlinde relation is a special case of this thermal duality and (within the family of rational models) the matrix S appearing in the thermal duality relation becomes identified with the statistics character matrix S. The relevant angular “Euclideanization” is done in the setting of the Tomita-Takesaki modular formalism of operator algebras. I find it appropriate to dedicate this work to the memory of J.A. Swieca with whom I shared the interest in two-dimensional models as a testing ground for QFT for more than one decade. This is a significantly extended version of an “Encyclopedia of Mathematical Physics” contribution hep-th/0502125.     

Instability of two-phase flows: A lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system

We consider a model for the flow of a mixture of two viscous and incompressible fluids in a two or three dimensional channel-like domain. The model consists of the Navier-Stokes equations governing the fluid velocity coupled with a convective Cahn-Hilliard equation for the relative density of atoms of one of the fluids. We prove the instability of certain stationary solutions for such a system endowed with periodic boundary conditions on elongated domains (0,2π/α₀)×(0,2π) or (0,2π/α₀)×(0,2π)×(0,2π/β₀) for a special class of periodic body forces, provided that α₀ and β₀ are small enough. As a consequence, we deduce a lower bound for the Hausdorff dimension of the global attractor.     

Deformations and shape changes in81Y

High spin states in⁸¹Y up to the probable 33/2⁺ and 29/2^? yrast states have been measured via the reaction⁵⁸Ni(²⁸Si,αp, using the Cologne tandem accelerator. Directional correlations of oriented nuclei were determined and the level scheme was extended with lifetimes measured by means of DSA and recoil distance techniques. Reduced transition strengths inferred from lifetimes were consistent with the results of Hartree-Fock-Bogolyubov calculations, which predict a nearly axially symmetric deformation ofβ ₂≈0.37 for theπ=+ band. Predictions reveal an alignment driving the system to a smaller and triaxial deformation withβ ₂=0.23 andγ=?30°, although the data show this crossing to be somewhat delayed. Particle-rotor calculations for the one-quasiparticle bands corroborate the predictions of deformation parameters based on the cranking model.     

Algebraic realization of a coupled rotor picture

The relation between the shape variables (β, γ) of the collective model and the (λ, μ) labels which define the irreducible representations of the SU(3) shell model is extended to a coupled rotor picture where one rotor represents protons (π) and the other one neutrons (ν). The joint distribution, (β, γ), emerges as the overlap of the initial distributions, (β _π ,γ _π ) and (β _ν ,γ _ν ), where three Euler angles define the relative orientation of proton and neutron subsystems. It is shown analytically that the rotor construction for triaxial and axially symmetric shapes corresponds to a (λ _π ,μ _π = 0) ? (λ _ν ,μ _ν → (λ, μ) _ρ=1 coupling in the SU(3) model.     

The β-decay of98Y and the level scheme of98Zr

Study of theβ-decay of⁹⁸Y and of the level scheme of⁹⁸Zr was undertaken by the use of the two recoil fission product separators JOSEF and LOHENGRIN. Twoβ-decay modes have been detected for⁹⁸Y, the half-lives of which are 0.65±0.05 s and 2.0±0.2 s. The spins and parities of the two states which undergoβ-decay are proposed to be 1⁺ and 4^?, respectively, with the configurations (π1g _9/2)(ν1g _7/2) and (π2p _1/2 (ν1g _7/2). The measuredQ _β -value is 7300±400 keV for the 0.65 s decay. The level scheme of⁹⁸Zr was deduced fromγ,γ-coincidence data. Absoluteγ-intensities were obtained from two independent measurements. The half-life of the first excited state of⁹⁸Zr at 853.0 keV with spin and parity 0⁺ was determined through delayedγ, e^?-measurements to be 63 ±7ns. No indication of the coexistence of two different nuclear shapes of⁹⁸Zr was found.     

Application of nonperturbative quantum field theoretic methods to hadron-nucleus collisions: Background theory

An exposition of the background theory necessary for understanding the application of nonperturbative QFT methods (LSZ reduction formalism) to hadron-nucleus collisions, for example the derivation of pi-nucleus (πn) Low and Chew Low equations, is given. The many channels and complex targets on the one hand, and the quantized field interactions on the other, introduce subtleties not well covered in particle theory or potential scattering literature. By specializing the πn Low Equation we derive a “pi-nucleon Low equation in the nuclear medium.” The second main goal of the paper is to compare this equation with Dover and Lemmer's analogous equation arrived at by graphical arguments. This requires especially making explicit the analyticity properties left tacit in their work. In is concluded that the two are essentially different.     

AQFT from <Emphasis Type="Italic">n</Emphasis>-Functorial QFT

There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended” functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra “of observables”, the latter uses n-functors which assign to each patch a “propagator of states”.In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2-dimensional extended Minkowskian QFT 2-functor (“parallel surface transport”) naturally yields a local net, whose locality derives from the 2-categorical exchange law, and which is covariant if the 2-functor is equivariant. This is obtained by postcomposing the propagation 2-functor with an operation that mimics the passage from the Schrödinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general Lorentzian structure, bare lightcone structure and higher dimensions. It does not, however, by itself imply anything about the existence of a vacuum state or about positive energy representations.     

The level structure of 184Os from 184Ir decay and 185Re(p, 2nγ) in-beam spectroscopy

Levels of ¹⁸⁴Os populated in the decay of 3.1 h ¹⁸⁴Ir and in the ¹⁸⁵Re(p, 2nγ) reaction have been investigated. The measurements included γ-ray singles, β⁺ ray endpoint, conversion coefficient, β⁺-γ coincidence and detailed γ-γ coincidence determinations. The results have established an extensive ¹⁸⁴Os level scheme, which includes well developed ground state, γ-vibrational and K = 3 octupole bands and which accommodates all the intense transitions observed in both the radioactivity and in-beam γ-ray measurements. Deviations of the level energies in the K^π = 0⁺and K^π = 2⁺ bands and of the interband reduced transition probabilities from the predictions of the strong-coupling model are discussed in terms of the rotationvibration interaction, and the systematics of the octupole vibrational excitations in even-mass W and Os nuclei are reviewed. It is concluded that the ¹⁸⁴Ir ground state configuration has a spin of 5, and that it contains large admixtures of K = 0 or K = 1 character.     

Renormalization of two-dimensional R2-gravity

Two-dimensional gravity with an R²-term is quantized around the R2-Liouville solution in the semiclassical way. Renormalization, regularization (infra-red, ultra-violet) and a topological term ∂ (ϕ∂ϕ)) ) are carefully treated. All (1-loop) divergences are renormalized by the cosmological constant (μ) and the R2-coupling-constant (β) for the case β > 0. The quantum meaning of the topological term is clarified. The renormalization group beta-functions of the couplings β and μ are obtained. It is found that the theory is conformal (i.e. the if beta-functions = 0) for w = (β/A) · (16π · 48π/(26 − c_m)) ⩾ 1 (where A is a fixed area) exactly when the coupling constant ξ of the topological term takes the value of 1. As for 0 < w < 1, β is asymptotically free for c_m < 26 and μ is asymptotically non-free.     

Pionic deuterium

Data taking of the PIONIC HYDROGEN project has been completed with a high statistics study of the strong-interaction effects in πD by measuring the Kβ X-radiation for three different target densities with a high resolution Bragg crystal spectrometer. The πD hadronic shift will provide a constraint for the πN isospin scattering lengths extracted from the πH measurement. The hadronic width is directly related to pion production at threshold.     

Thermodynamic,elastic, elastic anisotropy and minimum thermal conductivity of β-GaN under high temperature

The thermodynamic, elastic, elastic anisotropy and minimum thermal conductivity of β-GaN are investigated at ambient pressure and high temperature by using first-principles calculations method with the ultrasoft psedopotential scheme. The elastic constants calculations reveal β-GaN is mechanically stability at ambient pressure and high temperature. The elastic modulus (Poisson's ratio, shear modulus and Young's modulus) decreases with increasing temperature. The calculations of anisotropy show that β-GaN has a larger elastic anisotropy in Poisson's ratio, shear modulus, Young's modulus and Zener anisotropy index. In addition, when the temperature increases from 0 to 1500 K, the elastic anisotropy decreases for β-GaN. The quasi-harmonic Debye model is successfully applied to determine the thermodynamic properties at different pressures and temperatures. Using the quasi-harmonic Debye model, the thermodynamic properties including the Debye temperature, Grüneisen parameter, the heat capacity, adiabatic bulk modulus, and the thermal expansion coefficients of β-GaN are predicted under high temperature and high pressure.     

The reaction γp → pπ+π−π+π− for photon energies from 25 to 70 GeV

Measurements of the reaction γp → pπ⁺π^?π⁺π^? are presented, in which π⁺π^?π⁺π^? systems with masses up to 3 GeV are produced from fragmentation of the incident photon. The reaction is dominated by production of the large peak of the ?′(1600) meson and, at higher masses ?2 GeV, y production of jet-like 4π systems. The ?′(1600) meson is produced by a predominantly s-channel helicity conserving mechanism. At higher masses there are also indications of ?π peaks, of masses 1.3 GeV (the A₂ meson) and 1.75 GeV, produced with a recoiling π meson by a mechanism consistent with the Deck effect.     

Collective states in 230Ra fed by β− decay of 230Fr

The γ-rays following the β⁻ decay of ²³⁰Fr have been investigated by means of γ-ray singles including multispectrum analysis and γγ coincidence measurements using Ge(Li) detectors. The half-life of ²³⁰Fr was measured to be 19.1 ± 0.5 s. Most of the observed transitions could be placed in a level scheme comprising 23 new excited states of ²³⁰Ra, ten of them grouped into the K^π = 0⁺ ground-state band, the K^π = 0⁻ band with its 1⁻ state at 710.9 keV and a K^π = 2⁺ γ-vibrational band with its head at 734.8 keV. It is concluded that ²³⁰Ra is a better rotator than the lighter radium isotopes, and has no ground-state octupole deformation.     

Beta delayed proton emission from the 14s 21/2+ spin gap isomer in95Pd

The residual activities of the projectile-target system⁴⁰Ca→⁵⁸Ni have been investigated with gamma and proton multi spectrum analyses,γγ andpγ coincidences. Aβ decaying high-spin isomer in⁹⁵Pd withJ ^π=(21/2⁺), a half-life of 14±1 s and an excitation energy of about 2 MeV has been identified. It has been found that⁹⁵Pd ^m is aβ delayed proton precursor. The totalβ branching to proton emitting states has been determined to be (0.74±0.19)·10^?2. It has been observed that the final state in⁹⁴Ru, populated after proton emission, is dominantly the 8⁺ state. A decay scheme of⁹⁵Pd ^m has been derived.     

Resonance production in the γγ→ππ process and low energy parameters

The amplitudes of the γγ→ππ reaction are obtained describing the process up to the energy of 1.5 GeV. It is shown that in thef-meson region there exists large background with the isospinI=2. This background and the backgrounds of the known mechanisms deal with the threshold expansion of the amplitude. This allows one to give polarizability estimate of the pion (α +β) ^π+ =0.45·10^?+fm, (α+β) ^π0. We predict a very small cross-section for the production of a broad ε (700) by using the low energy normalization.     

The decay of 4 min 158Tm

The decay of 4 min ¹⁵⁸Tm has been investigated with on-line mass-separated samples obtained from the Orsay ISOCELE separator. Measurements of γ-rays, conversion electron lines and γ-γ bi-dimensional coincidences were performed. About 180 transitions were ascribed to the decay and two thirds of them were placed in a decay scheme. The β-band and the γ-band were identified with bandheads situated at 806.40 and 820.13 keV respectively. In addition, a number of other vibrational bands (β-γ, β-β, K^π = 0^? and 1^?) are proposed. The decay properties of those bands are discussed in the framework of current nuclear models. The log ft values suggest a 2^? assignment for ¹⁵⁸Tm with the possible configuration (p404J↓-n521↑).     

Observation of diffraction excitation in pp → (pπ+π−)+X at the CERN-ISR and tests of limiting fragmentation

In a single-arm magnetic spectrometer experiment at the CERN Intersecting Storage Rings (ISR), a prominent diffractive enhancement is observed for inclusive production of (pπ⁺π^?) systems near x ～ 1 at √s = 53 and 35 GeV. In testing limiting fragmentation and scaling for this system, inclusive pπ⁺π^? diffractive excitation into a fixed invariant mass range is found to possess a cross-section independent of s to within (3 ± 5)%. The main component in the (pπ⁺π^?) system is Δ(1236)π, with a charge structure compatible with the break-up of an isotopic spin $\text{1}\text{2}$ system.     

The decay scheme of 228Fr and K = 0 BANDS in 228Ra

The γ-rays following the β^?-decay of ²²⁸Fr have been studied by means of γ-ray singles including multi-spectrum analysis and γ-γ coincidence measurements using Ge(Li) spectrometers. Most of the observed γ-transitions could be placed in the level scheme of ²²⁸Ra. The accuracy the energy of the first-excited state in ²²⁸Ra has been improved and 35 new excited levels have been established, 11 of them grouped into the ground-state band, the low-lying K^π = 0^? band with the head at 474.14 keV and two excited K^π = 0⁺ bands with heads at 721.17 and 1041.9 keV. Candidates for two close-lying K^π = 2⁺ bands have also been found. It is concluded that the ground state octupole deformation, if any, is less pronounced in ²²⁸Ra than in lighter radium isotopes.     

Spectroscopy of neutron deficient 65Ge

Excited states of ⁶⁵Ge were populated via the ¹²C + ⁵⁸Ni (261 MeV) reaction using the NORDBALL detector array equipped with charged-particle and neutron detector systems for reaction channel separation. On the basis of γγ-coincidence relations and angular distribution ratios a significantly extended level scheme was constructed up to E _x = 9 MeV and J ^π = (33/2^?). The low-energy states of the nucleus are discussed in the framework of the interacting boson-fermion model.     

Growing networks with preferential deletion and addition of edges

A preferential attachment model for a growing network incorporating the deletion of edges is studied and the expected asymptotic degree distribution is analyzed. At each time step t=1,2,…, with probability π₁>0 a new vertex with one edge attached to it is added to the network and the edge is connected to an existing vertex chosen proportionally to its degree, with probability π₂ a vertex is chosen proportionally to its degree and an edge is added between this vertex and a randomly chosen other vertex, and with probability π₃=1−π₁−π₂<1/2 a vertex is chosen proportionally to its degree and a random edge of this vertex is deleted. The model is intended to capture a situation where high-degree vertices are more dynamic than low-degree vertices in the sense that their connections tend to be changing. A recursion formula is derived for the expected asymptotic fraction p_k of vertices with degree k, and solving this recursion reveals that, for π₃<1/3, we have p_k∼k^{−(3−7π₃)/(1−3π₃)}, while, for π₃>1/3, the fraction p_k decays exponentially at rate (π₁+π₂)/2π₃. There is hence a non-trivial upper bound for how much deletion the network can incorporate without losing the power-law behavior of the degree distribution. The analytical results are supported by simulations.