Ramanujan’s ‘‘Lost Notebook’’ and the Virasoro Algebra

By using the theory of vertex operator algebras, we gave a new proof of the famous Ramanujans modulus 5 modular equation from his Lost Notebook (p. 139 in [R]). Furthermore, we obtained an infinite list of q-identities for all odd moduli; thus, we generalized the result of Ramanujan.Acknowledgements It was indeed hard to trace all the known proofs of (1.1), (1.2) and (1.3). We apologize if some important references are omitted. We would like to thank Jim Lepowsky for conversations on many related subjects. A few years ago Lepowsky and the author were trying to relate classical Rogers-Ramanujan identities and Zhus work [Z]. We also thank Bruce Berndt for pointing us to [BrO] and Steve Milne for bringing [Mi] to our attention.     

Undulators and ‘free-electron lasers’

This article outlines the early development of the undulator in which an electron beam travelling through a system of magnetic fields with directions alternating in space emits a species of linearly polarized synchrotron radiation peaked at a wavelength which is short compared to the magnet period when the electron energy is high compared with the rest energy. The conditions of synchronism and the departure from synchronism that leads to gain during a single passage of the beam are examined. In later developments, stemming from this, single pass gain of circularly polarized radiation was experimentally demonstrated by means of a ‘twister’ structure and an optical cavity, obtained by adding mirrors to the magnet assembly, was used to achieve build-up of the radiation in an oscillator device known as the ‘free-electron laser’. Applications of this device and further theoretical developments are discussed.     

Simulation of quantum wells with ‘spikes’ and ‘dips’

The use of thin high-bandgap ‘spikes’ or thin low-bandgap ‘dips’ inside conventional rectangular quantum wells (QWs) gives supplementary flexibility in engineering intra- and inter-band energy level separation. The paper presents simulation and experimental studies on the effects of ‘spikes’ and ‘dips’ on the fundamental quantum well properties.     

‘‘Extrinsic’’ and ‘‘Intrinsic’’ Data in Quantum Measurements: Asymptotic Convex Decomposition of Positive Operator Valued Measures

We study the problem of separating the data produced by a given quantum measurement (on states from a memoryless source which is unknown except for its average state), described by a positive operator valued measure (POVM), into a meaningful (intrinsic) and a not meaningful (extrinsic) part. We are able to give an asymptotically tight separation of this form, with the intrinsic data quantified by the Holevo mutual information of a certain state ensemble associated to the POVM and the source, in a model that can be viewed as the asymptotic version of the convex decomposition of POVMs into extremal ones. This result is applied to a similar separation therorem for quantum instruments and quantum operations, in their Kraus form. Finally we comment on links to related subjects: we stress the difference between data and information (in particular by pointing out that information typically is strictly less than data), derive the Holevo bound from our main result, and look at its classical case: we show that this includes the solution to the problem of extrinsic/intrinsic data separation with a known source, then compare with the well–known notion of sufficient statistics. The result on decomposition of quantum operations is used to exhibit a new aspect of the concept of entropy exchange of an open dynamics. An appendix collects several estimates for mixed state fidelity and trace norm distance, that seem to be new, in particular a construction of canonical purification of mixed states that turns out to be valuable to analyze their fidelity.     

Interference and ‘which way’ information

An overview is given of typical examples of interference experiments that confirm the general quantum mechanical rule that distinguishability of the paths destroys interference. It will be shown that path information can be gained in different ways: from the screen's or the atom's recoil in Young-type optical experiments; from the spontaneously emitted photon in atom optics; and from the idler photon, in a striking interference experiment with a photon pair being involved. In all cases, the physical mechanism that actually makes the interference pattern disappear is elucidated.     

Mixing and its rate in ‘soft’ and ‘hard’ billiards motivated by the Lorentz process

Billiards corresponding to planar periodic Lorentz processes are considered in the usual (hard) sense and in the case when the hard core potential of the scatterers is replaced by some other circularly symmetric potential. A review on certain important aspects of the history of the subject is given and some new results on exponential decay of correlations are formulated. Both the results from the literature and those of our own mentioned are mathematically rigorous, nevertheless, proofs are only briefly sketched. For further details, see the preprint [Correlation decay in certain soft billiards, Commun. Math. Phys., in press].     

BRS ‘Symmetry’, prehistory and history

Prehistory ?C Starting from ??t Hooft??s (1971) we have a short look at Taylor??s and Slavnov??s works (1971?C72) and at the lectures given by Rouet and Stora in Lausanne (1973) which determine the transition from pre-history to history. History ?C We give a brief account of the main analyses and results of the BRS collaboration concerning the renormalized gauge theories, in particular the method of the regularization-independent, algebraic renormalization, the algebraic proof of S-matrix unitarity and that of gauge choice independence of the renormalized physics. We conclude this report with a suggestion to the crucial question: what could remain of BRS invariance beyond perturbation theory.     

‘In’ and ‘out’ vertex operators in a class of UV-finite nonlinear σ-models

In and out scalar vertex operators are constructed perturbatively in a class of recently discovered UV finite nonlinear -models describing the string evolution in gravitational plane wave backgrounds. They exhibit peculiar singularities in the target space related to the focusing phenomena in such backgrounds well known from the classical and quantum gravity theories. The computation is performed up to three loops of the usual perturbation expansion and to all loops of the weak field limit. An argument is given that the vertex operator singularities should persist, even when summing up the all perturbation expansions.     

Contextuality and Nonlocality in ‘No Signaling’ Theories

We define a family of ‘no signaling’ bipartite boxes with arbitrary inputs and binary outputs, and with a range of marginal probabilities. The defining correlations are motivated by the Klyachko version of the Kochen-Specker theorem, so we call these boxes Kochen-Specker-Klyachko boxes or, briefly, KS-boxes. The marginals cover a variety of cases, from those that can be simulated classically to the superquantum correlations that saturate the Clauser-Horne-Shimony-Holt inequality, when the KS-box is a generalized PR-box (hence a vertex of the ‘no signaling’ polytope). We show that for certain marginal probabilities a KS-box is classical with respect to nonlocality as measured by the Clauser-Horne-Shimony-Holt correlation, i.e., no better than shared randomness as a resource in simulating a PR-box, even though such KS-boxes cannot be perfectly simulated by classical or quantum resources for all inputs. We comment on the significance of these results for contextuality and nonlocality in ‘no signaling’ theories.     

Aggregation and mesomorphic properties of ‘double-headed’ carbohydrate amphiphiles

Sugar-based amphiphiles, consisting of two sugar head groups and an alkylene chain within the molecules, are synthesized and their aggregation and mesomorphic properties are evaluated. The hydrophilic sugar head groups, constituted with β-D-glucopyranoside units, and the lyophilic alkylene units, are coupled to a glycerol backbone to afford the ‘double-headed’ sugar amphiphiles. Aggregation studies in aqueous solutions provided their critical micellar concentrations and the aggregation numbers. Mesophase characterizations by polarizing optical microscopy and differential scanning calorimetry (DSC) revealed the phase-transition behaviour of these new ‘double-headed’ glycolipids.     

‘Twisted’ electrons

Electrons have played a significant role in the development of many fields of physics during the last century. The interest surrounding them mostly involved their wave-like features prescribed by the quantum theory. In particular, these features correctly predict the behaviour of electrons in various physical systems including atoms, molecules, solid-state materials, and even in free space. Ten years ago, new breakthroughs were made, arising from the new ability to bestow orbital angular momentum (OAM) to the wave function of electrons. This quantity, in conjunction with the electron’s charge, results in an additional magnetic property. Owing to these features, OAM-carrying, or twisted, electrons can effectively interact with magnetic fields in unprecedented ways and have motivated materials scientists to find new methods for generating twisted electrons and measuring their OAM content. Here, we provide an overview of such techniques along with an introduction to the exciting dynamics of twisted electrons.     

Out-of-roundness and thermal energy effects on ‘ideally soft ferromagnets’

Infinite initial susceptibility is predicted in certain micro-magnetic computations of sufficiently small spheres and circular discs made of isotropic magnetic material. In this letter, it is argued that this result pertains only to perfectly round samples in the absence of thermal energy. It is expected, therefore, that much lower values of the initial susceptibility will be found on experimentally realizable samples at room temperature.     

Proton spin relaxation and self-diffusion in some ‘isotropic’ nematogens

Angular distributions of reactive scattering arising from a long-lived collision complex dissociating via a transition state that approximates to a linear rotator are predicted for H atom displacement occurring over a distribution of bending angles β with respect to the rotator axis. A conical angular distribution may be generated when only a narrow range of bending angles contributes to the reactive scattering. However, the angular distribution may take on quite different character as the distribution of bending angles becomes broader. Thus, an isotropic angular distribution is generated by a bending angle distribution which follows the spherical polar weighting P(β) ∝ sin β. Forward and backward scattering is generated by bending angle distributions favouring less strongly bent displacement, while sideways scattering is generated by a bending angle distribution favouring more strongly bent displacement. The experimental angular distribution for OH + CO reactive scattering which shows mild forward and backward peaking, corresponds to a range of bending angles for H atom displacement from the H-OCO transition state which is broadened about the nominal preferred direction β = 45° but avoids strongly bent directions with β ≈ 90°.     

Quasi-integrable and superintegrable systems from a ‘deformed-mass’ two-photon algebra

A quantum deformation of the two-photon (or Schrödinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n?2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n?1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.     

Soft mode dispersion and ‘waterfall’ phenomenon in relaxors revisited

Results of recent inelastic neutron scattering studies of lead-based relaxor ferroelectrics by Gvasaliya et al. [J. Phys.: Condens. Matter 17, 4343 (2005); J. Phys.: Condens. Matter 19, 016219 (2007)] have put in question the existence of the “waterfall” anomaly–an apparent vertical dispersion segment joining the TA and TO branches–observed earlier in low-energy [ξ00] phonon dispersion curves of these materials. In the present article, we review the results of earlier experiments and model calculations together with the outcome of our recent measurements on PMN using the same instrumental set-up as Gvasaliya et al. to conclude that the “waterfall” feature is not an experimental artefact. We also give some hints on a possible explanation of the results of Gvasaliya et al., by exploring the fact that the reported dispersion of the underdamped transverse optic branch follows the longitudinal acoustic (LA) branch dispersion surprisingly closely.     

High-energy strong interactions: from ‘hard’ to ‘soft’

We discuss the qualitative features of the recent data on multiparticle production observed at the LHC. The tolerable agreement with Monte Carlos based on LO DGLAP evolution indicates that there is no qualitative difference between ‘hard’ and ‘soft’ interactions; and that a perturbative QCD approach may be extended into the soft domain. However, in order to describe the data, these Monte Carlos need an additional infrared cutoff k _min with a value k _min∼2–3 GeV which is not small, and which increases with collider energy. Here we explain the physical origin of the large k _min. Using an alternative model which matches the ‘soft’ high-energy hadron interactions smoothly on to perturbative QCD at small x, we demonstrate that this effective cutoff k _min is actually due to the strong absorption of low k _t partons. The model embodies the main features of the BFKL approach, including the diffusion in transverse momenta, ln k _t , and an intercept consistent with resummed next-to-leading log corrections. Moreover, the model uses a two-channel eikonal framework, and includes the contributions from the multi-Pomeron exchange diagrams, both non-enhanced and enhanced. The values of a small number of physically-motivated parameters are chosen to reproduce the available total, elastic and proton dissociation cross section (pre-LHC) data. Predictions are made for the LHC, and the relevance to ultra-high-energy cosmic rays is briefly discussed. The low x inclusive integrated gluon PDF, and the diffractive gluon PDF, are calculated in this framework, using the parameters which describe the high-energy pp and p[`(p)]p\bar{p} ‘soft’ data. Comparison with the PDFs obtained from the global parton analyses of deep inelastic and related hard scattering data and from diffractive deep inelastic data looks encouraging.     

Distinguishing ‘Higgs’ spin hypotheses using γγ and WW ∗ decays

A multi-step setup for heavy-flavor studies in high-energy nucleus-nucleus (AA) collisions—addressing within a comprehensive framework the initial $Q\overline{Q}$ production, the propagation in the hot medium until decoupling and the final hadronization and decays—is presented. The initial hard production of $Q\overline{Q}$ pairs is simulated using the POWHEG pQCD event generator, interfaced with the PYTHIA parton shower. Outcomes of the calculations are compared to experimental data in pp collisions and are used as a validated benchmark for the study of medium effects. In the AA case, the propagation of the heavy quarks in the medium is described in a framework provided by the relativistic Langevin equation. For the latter, different choices of transport coefficients are explored (either provided by a perturbative calculation or extracted from lattice-QCD simulations) and the corresponding numerical results are compared to experimental data from RHIC and the LHC. In particular, outcomes for the nuclear modification factor R _AA and for the elliptic flow v ₂ of D/B mesons, heavy-flavor electrons and non-prompt J/ψ’s are displayed.     

Per cent ‘covalency’ and the nephelauxetic effect in lanthanide complexes

The use of the nephelauxetic ratio for the calculation of percent covalency is examined. Values of F_k and E^k obtained from four parameter fits (e.g. F ₂, F ₄, F ₆ and ζ_4f) neglecting configuration interaction for Pr³⁺ in various host lattices are compared to those of the free ion. By including the effects of 4f-ligand overlap one can account for the empirically obtained order of sensitivity to the environment for F_k : F ₂ > F ₄ > F ₆ and for E^k : E ² > E ³ > E ¹. The results further indicate that by using the nephelauxetic effect and the relationship b = (1 - β)/2 one is likely to overestimate the degree of covalency of a complex.