Bubble Divergences from Twisted Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory and 3d Riemannian quantum gravity, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined because of a phenomenon known as ‘bubble divergences’. In this paper, we extend recent results of the authors to the cases where these divergences cannot be understood in terms of cellular cohomology. We introduce in its place the relevant twisted cohomology, and use it to compute the divergence degree of the partition function. We also relate its dominant part to the Reidemeister torsion of the complex, thereby generalizing previous results of Barrett and Naish-Guzman. The main limitation to our approach is the presence of singularities in the representation variety of the fundamental group of the complex; we illustrate this issue in the well-known case of two-dimensional manifolds.     

An invitation to higher gauge theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity.     

Time-dependent transport through a quantum dot: influence of the Coulomb interaction on the quantum dot charge states

We consider the electron transport through one-level quantum dot, out of the Kondo regime, under the influence of the external microwave fields. The influence of the intra-dot Coulomb electron-electron interaction is studied using the equation of motion method for appropriate correlation functions. The formula for the current and the closed set of the integro-differential equations for the expectation values of the quantum dot charge states are given. The most characteristic feature of these time-averaged expectation values is an appearance of the additional structure (sidebands) on the curves of the derivatives of the expectation values with respect to the gate voltage. The sidebands structure formed on both sides of the ‘ionization’ and ‘affinity’ quantum dot levels are also found on the current and differential conductance curves.     

Conformal invariance and scalar-tensor theories

A new generalisation of Einstein’s theory is proposed which is invariant under conformal mappings. Two scalar fields are introduced in addition to the metric tensor field, so that two special choices of gauge are available for physical interpretation, the ‘Einstein gauge’ and the ‘atomic gauge’. The theory is not unique but contains two adjustable parameters ζ anda. Witha=1 the theory viewed from the atomic gauge is Brans-Dicke theory (ω=−3/2+ζ/4). Any other choice ofa leads to a creation-field theory. In particular the theory given by the choicea=−3 possesses a cosmological solution satisfying Dirac’s ‘large numbers’ hypothesis.     

SPDM: light microscopy with single-molecule resolution at?the?nanoscale

Far-field fluorescence techniques based on the precise determination of object positions have the potential to circumvent the optical resolution limit of direct imaging given by diffraction theory. In order to use localization to obtain structural information far below the diffraction limit, the ‘point-like’ components of the structure have to be detected independently, even if their distance is lower than the conventional optical resolution limit. This goal can be achieved by exploiting various photo-physical properties of the fluorescence labeling (‘spectral signatures’). In first experiments, spectral precision distance microscopy/spectral position determination microscopy (SPDM) was limited to a relatively small number of components to be resolved within the observation volume. Recently, the introduction of photoconvertable molecules has dramatically increased the number of components which can be independently localized. Here, we present an extension of the SPDM concept, exploiting the novel spectral signature offered by reversible photobleaching of fluorescent proteins. In combination with spatially modulated illumination (SMI) microscopy, at the present stage, we have achieved an estimated effective optical resolution of approximately 20 nm in the lateral and 50 nm in the axial direction, or about 1/25th–1/10th of the exciting wavelength.     

Rank-Driven Markov Processes

We study a class of Markovian systems of N elements taking values in [0,1] that evolve in discrete time t via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak–Sneppen model of evolution, in which the system represents an evolutionary ‘fitness landscape’ and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-N asymptotics for the general model in which, at each time step, K randomly chosen elements are discarded and replaced by independent U[0,1] variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution κ _N on {1,2,…,N} ^K . Our main results are that, under appropriate conditions on κ _N , the system exhibits threshold behavior at s ^∗∈[0,1], where s ^∗ is a function of κ _N , and the marginal distribution of a randomly selected element converges to U[s ^∗,1] as t→∞ and N→∞. Of this class of models, results in the literature have previously been given for special cases only, namely the ‘mean-field’ or ‘random neighbor’ Bak–Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster–Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K=2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest.     

A remark on the motivic Galois group and the quantum coadjoint action

It has been suggested that the Grothendieck–Teichmüller group GT should act on the Duflo isomorphism of su(2), but the corresponding realization of GT turned out to be trivial. We show that a solvable quotient of the motivic Galois group – which is supposed to agree with GT – is closely related to the quantum coadjoint action on for q a root of unity, i.e. in the quantum group case one has a nontrivial realization of a quotient of the motivic Galois group. From a discussion of the algebraic properties of this realization we conclude that in more general cases than it should be related to a quantum version of the motivic Galois group. Finally, we discuss the relation of our construction to quantum field and string theory and explain what we believe to be the ‘physical reason’ behind this relation between the motivic Galois group and the quantum coadjoint action. This might be a starting point for the generalization of our construction to more involved examples.     

Conformally invariant pomeron interaction in perturbative QCD with large Nc

An effective non-local quantum field theory is constructed, which describes the interaction of pomerons in high-colored QCD. The theory includes both splitting and merging triple pomeron vertexes and diagrams with pomeronic loops. The Schwinger–Dyson equations for the ‘physical’ pomeron are written. Conformal invariance allows one to reduce the theory to the old-fashioned Gribov pomeron theory with an infinite number of pomerons, one of which is supercritical.     

Conformal Proper Times According to the Woodhouse Causal Axiomatics of Relativistic Spacetimes

On the basis of the Woodhouse causal axiomatics, we show that conformal proper times and an extra variable in addition to those of space and time, together give a physical justification for the ‘chronometric hypothesis’ of general relativity. Indeed, we show that, with a lack of these latter two ingredients and of this hypothesis, clock paradoxes exist for which the unparadoxical asymmetry cannot be recovered when using the ‘clock and message functions’ only. These proper times originate from a given conformal structure of the spacetime when ascribing different compatible projective structures to each Woodhouse particle, and then, each defines a specific Weylian ‘sheaf structure’. In addition, the proper time parameterizations are defined via path-dependent conformal scale factors, which act like sockets for any kind of physical interaction and also represent the values of the variable associated with the extra dimension.     

Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon

We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, , with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as ‘past terminal accumulation points’ of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's ‘Chronology Protection Conjecture’, according to which the laws of physics prevent one from manufacturing a ’time machine‘. Specifically, we prove: Theorem 1. There is no extension to of the usual field algebra on the initial globally hyperbolic region which satisfies the condition of F-locality at any base point. In other words, any extension of the field algebra must, in any globally hyperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globally hyperbolic spacetimes. Theorem 2. The two-point distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant Klein-Gordon equation on the full spacetime) be singular at every base point x in the sense that the difference between this two point distribution and a local Hadamard distribution cannot be given by a bounded function in any neighbourhood (in M × M) of (x,x). In consequence of Theorem 2, quantities such as the renormalized expectation value of φ² or of the stress-energy tensor are necessarily ill-defined or singular at any base point. The proof of these theorems relies on the ‘Propagation of Singularities’ theorems of Duistermaat and H?rmander. Received: 14 March 1996/Accepted: 11 June 1996     

Division Algebras and Quantum Theory

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex’ representations), those that are self-dual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure.     

A unified theory of weak interactions

A gauge model for the weak interactions of the leptons (v _e, e, μ, ν_μ) and the quarks (q _p, q_n,qλ,q _p′) is presented in which deviations from universality, such as the Cabibbo suppression, are explicitly and spontaneously generated. The gauge group is, to begin with SU(4). There are three quartets of Higgs scalars with suitable vacuum expectation values, sufficient and necessary to give masses to all gauge bosons. It turns out that this gauge group is too ‘large’ and fails to account for many observed symmetries of weak interactions, especially electron-muon symmetry. This symmetry corresponds to a discrete transformationR which is an element of SU(4). To accommodate it, the gauge group is restricted to the subgroup of SU(4) which commutes withR. There are now 7 gauge bosons, 4 charged and 3 neutral. One pair of charged bosons is necessarily heavier than the other pair (denotedW ^±) and two neutrals are necessarily heavier than the third (W ⁰). The electron and the muon become massive while the neutrinos and the quark fields remain massless. The dominant charged weak currents coupling toW ^± havee-μ universality and Cabibbo universality for both of whichR-symmetry is essential—the Cabibbo angle is a simple function of the vacuum expectation values. The same symmetry ensurese-μ symmetry and the absence of flavour-changing components in the neutral currents. The currents coupling to the heavier gauge bosons break all these symmetries but these bosons can be made arbitrarily heavy and so are relevant only in the domain of ‘ultraweak’ interactions. The Cabibbo angleϑ _c itself is determined by minimising a very general class of Higgs potentials, leading to a numerical valueϑ _c = ±π/8, | tanϑ _c | = √2 − 1 (an alternative solution | tanϑ _c | = (√2+1) is rejected), independent of the parameters and of the precise form of the potential. This is the ‘bare’ϑ _c; in low energy/momentum transfer processes, this value is renormalised by the structure of the hadrons. A model is given for this renormalisation which reduces the renormalised value of | tanϑ _c | to about 0.2–0.3 from the bare value 0.41. Recent data on highly inelastic neutrino interactions are shown to be not inconsistent with | tanϑ _c | = 0.4.     

On the lyman problem

This paper attempts to answer Lyman's question (1990) on the non-uniqueness in defining the 3D measure of the boundary vorticity-creation rate. Firstly, a straightforward analysis of the vorticity equation introduces a definition of a general vorticity flux-density tensor and its ‘effective’ part. The approach is strictly based on classical field theory and is independent of the constitutive structure of continuous medium. Secondly, the fundamental question posed by Lyman dealing with the ambiguity of the 3D measure of the boundary vorticity-creation rate for incompressible flow is discussed. It is shown that the original 3D measure (for an incompressible Newtonian fluid defined by Panton 1984), which is reminiscent of an analogy to Fourier's law, is in its character ‘effective’ and plays a crucial role in the prognostic vorticity transport equation. The alternative 3D measure proposed by Lyman includes, on the other hand, a ‘non-effective’ part, which plays a role in the local determination of the ‘effective’ measure as well as in a certain diagnostic integral boundary condition.     

Glafka–2004 ‘Iconoclasm’: The Soul and Spirit of the Meeting

The following is a brief talk that opened and attempted to set the atmosphere for the first ‘Glafka–2004: Iconoclastic Approaches to Quantum Gravity’ international theoretical physics conference. It aimed to capture the general spirit of the meeting, as well as to inspire and unite its participants under the following envisioned ‘cause’: to bring together and scrutinize certain important current quantum gravity research approaches in a fresh, unconventional, almost unorthodox, way.Introductory remarks to the 1st Glafka–2004: Iconoclastic Approaches to Quantum Gravity international theoretical physics conference, held in Athens, Greece (summer 2004).     

Two-state random walk model of lattice diffusion. 1. Self-correlation function

Diffusion with interruptions (arising from localized oscillations, or traps, or mixing between jump diffusion and fluid-like diffusion, etc.) is a very general phenomenon. Its manifestations range from superionic conductance to the behaviour of hydrogen in metals. Based on a continuous-time random walk approach, we present a comprehensive two-state random walk model for the diffusion of a particle on a lattice, incorporating arbitrary holding-time distributions for both localized residence at the sites and inter-site flights, and also the correct first-waiting-time distributions. A synthesis is thus achieved of the two extremes of jump diffusion (zero flight time) and fluid-like diffusion (zero residence time). Various earlier models emerge as special cases of our theory. Among the noteworthy results obtained are: closed-form solutions (ind dimensions, and with arbitrary directional bias) for temporally uncorrelated jump diffusion and for the ‘fluid diffusion’ counterpart; a compact, general formula for the mean square displacement; the effects of a continuous spectrum of time scales in the holding-time distributions, etc. The dynamic mobility and the structure factor for ‘oscillatory diffusion’ are taken up in part 2.     

Managing, profiling and analyzing a library of 2.6 million compounds gathered from 32 chemical providers

Summary The data for 3.8 million compounds from structural databases of 32 providers were gathered and stored in a single chemical database. Duplicates are removed using the IUPAC International Chemical Identifier. After this, 2.6 million compounds remain. Each database and the final one were studied in term of uniqueness, diversity, frameworks, ‘drug-like’ and ‘lead–like’ properties. This study also shows that there are more than 87 000 frameworks in the database. It contains 2.1 million ‘drug-like’ molecules among which, more than one million are ‘lead-like’. This study has been carried out using ‘ScreeningAssistant’, a software dedicated to chemical databases management and screening sets generation. Compounds are stored in a MySQL database and all the operations on this database are carried out by Java code. The druglikeness and leadlikeness are estimated with ‘in–house’ scores using functions to estimate convenience to properties; unicity using the InChI code and diversity using molecular frameworks and fingerprints. The software has been conceived in order to facilitate the update of the database. ‘ScreeningAssistant’ is freely available under the GPL license.     

Super <Emphasis Type="Italic">D</Emphasis>-Differentiation for <Emphasis Type="Italic">R</Emphasis><Superscript> ∞ </Superscript>-Supermanifolds

The concept of ‘D-Differentiation’, which, in the context of smooth manifolds, generalises Lie and covariant differentiation, is extended to R ^∞-supermanifolds under the name of ‘Super D-Differentiation’. This is done by defining new (non-linear) mappings, called ‘μ-mappings’ and by relating their non-linearity to the Leibniz rule that a derivation must satisfy when it acts on a tensor product. The resulting axiomatics, which is basis-independent and coordinate-free, is then expressed in a general basis (not necessarily holonomic). Super Lie and Super covariant differentiation are, amongst others, special cases of Super D-Differentiation. In particular, the transformation rules for the connection coefficients and the commutation coefficients of non-holonomic bases are obtained. These special cases are found to be in agreement with the DeWitt Super covariant and Super Lie derivatives.      

Optico-Mechanical Analogy: an Axiomatic Approach

An axiomatic characterization of a ‘two-level Hamiltonian structure’ is proposed, which expresses the optico-mechanical analogy by representing optics and mechanics as (disjoint) classes of models satisfying the axioms. There is the ‘Hamilton–Jacobi level’, which involves a differential manifold on which the characteristic function satisfying the Hamilton–Jacobi equation is defined; and the ‘symplectic level’, involving the Hamiltonian, defined on the cotangent bundle of the manifold. The two levels, with the (analogous) structures on them, concern both optics and mechanics.     

Cosmology with cosmic microwave background anisotropy

Measurements of CMB anisotropy and, more recently, polarization have played a very important role in allowing precise determination of various parameters of the ‘standard’ cosmological model. The expectation of the paradigm of inflation and the generic prediction of the simplest realization of inflationary scenario in the early Universe have also been established — ‘acausally’ correlated initial perturbations in a flat, statistically isotropic Universe, adiabatic nature of primordial density perturbations. Direct evidence for gravitational instability mechanism for structure formation from primordial perturbations has been established. In the next decade, future experiments promise to strengthen these deductions and uncover the remaining crucial signature of inflation — the primordial gravitational wave background.     

A geometrical framework for dyons in the presence of the dilaton and the axion in four dimensions

The action of the bosonic sector of the effective field theory induced by heterotic strings in four dimensions, which is relevant (for instance) to the study of dyons (Shapere et al. in Mod Phys Lett A6: 2677, 1991), is re-interpreted geometrically by using the new concept of ‘D-Differentiation’. This extends the authors’ geometrical unification of the Einstein–Maxwell theory.