A Geometrical Interpretation of ‘Supergauge’ Transformations Using D-Differentiation

D-transport is employed to construct, within the limited setting of a non-graded manifold, a geometrical framework that yields a generalisation of the ‘supergauge’ transformations of Supergravity. Killing’s equation is shown to be at the origin of the ‘gauged’ supersymmetry transformations. The presence of a field-dependent Lorentz transformation is traced to the fact that, for every given X, the difference of two D-differentiation operators and is a linear transformation that necessarily depends on X.      

Phase transitions in the adsorbed molecular chains

Rotational excitations of molecular adsorbed layers are studied theoretically. Nonlinear dynamical equations are obtained with accounting of quadrupolar interactions between molecules and freezing of translational degrees of freedom. The equilibrium positions of the molecules are found to be experimentally observed structures with alternating rotational ordering of planar rotors along the direction to the nearest neighbor (for linear or square structures) under low temperature. Dynamical analysis gives an integral of motion (energy) of the chain that in the long-wave limit leads consequently to the existence of four phases. The first one corresponds to oscillations near equilibrium ordered states. The second phase corresponds to low-energy rotational excitations along ‘valleys’ (easy directions in the effective potential) that do not destroy strong correlations between molecules while structural data can show rotational disorder (melting). The third phase corresponds to an energy that is enough to travel between ‘valleys’; only some ‘islands’ in the angle space are forbidden. Complete destruction of correlation when the energy is over the peaks of the effective potential corresponds to the fourth phase. Therefore rotational melting is a complex phenomenon that has several stages. Presented at the 2nd International Conference “Physics of Liquid Matter: Modern Problems” (September 2003, Kyiv, Ukraine).     

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.      

Stationary Flow Past a Semi-Infinite Flat Plate: Analytical and Numerical Evidence for a Symmetry-Breaking Solution

We consider the question of the existence of stationary solutions for the Navier Stokes equations describing the flow of a incompressible fluid past a semi-infinite flat plate at zero incidence angle. By using ideas from the theory of dynamical systems we analyze the vorticity equation for this problem and show that a symmetry-breaking term fits naturally into the downstream asymptotic expansion of a solution. Finally, in order to check that our asymptotic expressions can be completed to a symmetry-breaking solution of the Navier–Stokes equations we solve the problem numerically by using our asymptotic results to prescribe artificial boundary conditions for a sequence of truncated domains. The results of these numerical computations a clearly compatible with the existence of such a solution. Mathematics Subject Classification (2000). 76D05, 76D25, 76M10, 41A60, 35Q35 Supported in part by the Fonds National Suisse.     

B and collider physics: Working group report

This report summarises the work done during WHEPP-6 (Institute of Mathematical Sciences, Chennai, India, Jan 3–15, 2000) in Working group on ‘B and collider physics’.     

Coherence vs. decoherence in (some) problems of condensed matter physics

We present an ‘overview’ of coherence-to-decoherence transition in certain selected problems of condensed matter physics. Our treatment is based on a subsystem-plus-environment approach. All the examples chosen in this paper have one thing in common — the environmental degrees of freedom are taken to be bosonic and their spectral density of excitations is assumed to be ‘ohmic’. The examples are drawn from a variety of phenomena in condensed matter physics involving, for instance, quantum diffusion of hydrogen in metals, Landau diamagnetism and c-axis transport in high T _c superconductors.     

Some comments on the Coriolis attenuation problem

To explore the Coriolis attenuation problem we have carried out a schematici _13/2 rotor plus single quasi-particle band-mixing calculation. The results reveal that the calculations are largely insensitive towards the location of the Fermi energy near the low-K single particle states only, and therefore are incapable of taking into account the transition from ‘full’ decoupling to ‘partial’ decoupling as the Fermi level is increased. We trace the possible reasons for this insensitivity and find that this may be primarily due to thebcs approximation for calculating the quasiparticle energies.     

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.     

Solution of approximate gluon evolution equation for small-x at fixed ‘ρ’

A simple method has been applied to solve the approximate gluon evolution equation for small-x at fixedρ(≡√ln(x ₀/x)/ln[ln(Q ²/Λ²)/ln(Q ₀ ² /Λ²)]. Numerical comparison is made with the predictions from ‘double asymptotic scaling’ and fit. Better agreement is found between our solution and fit nearρ=1. The solution gives approximate double scaling in this region having ‘hard’ pomeron with small contamination.     

A visual description of the convective flow field around the head of a human

Mean velocity data obtained by PIV (Particle Image Velocimetry) around the head of a real-life size breathing thermal manikin are presented for two cases of ‘no breathing’ and ‘continuous exhalation through nose’. Experiments were conducted in a special chamber which provided stationary convective flows around the seated manikin. Results are limited to the plane of symmetry. The paper aims to describe the physical structure of the turbulent flow field by presenting velocity and vorticity data in color graphics.     

Approximate analytical lower bounds on ‘double asymptotic scaling’ variables and physics at HERA

We analytically examine the asymptotic solution of gluon evolution equation in terms of the ‘double scaling variables’ρ andσ of perturbative QCD and find the approximate lower bounds on these, above which the solution is considered to be valid. Comparison of this asymptotic solution is made with the fit obtained from data and the estimated lower bound onρ is nearly equal to our analytical finding. To analyze the data below the lower bound onρ, other analytical solutions of gluon evolution equation are to be used which depend highly on the inputx-distributions of gluon to study the physics at low-x of HERA range.     

Higher dimensional supersymmetric quantum mechanics and Dirac equation

We exhibit the supersymmetric quantum mechanical structure of the full 3+1 dimensional Dirac equation considering ‘mass’ as a function of coordinates. Its usefulness in solving potential problems is discussed with specific examples. We also discuss the ‘physical’ significance of the supersymmetric states in this formalism.     

Large Time Behavior for Vortex Evolution in the Half-Plane

 In this article we study the long-time behavior of incompressible ideal flow in a half plane from the point of view of vortex scattering. Our main result is that certain asymptotic states for half-plane vortex dynamics decompose naturally into a nonlinear superposition of soliton-like states. Our approach is to combine techniques developed in the study of vortex confinement with weak convergence tools in order to study the asymptotic behavior of a self-similar rescaling of a solution of the incompressible 2D Euler equations on a half plane with compactly supported, nonnegative initial vorticity. Received: 28 June 2002 / Accepted: 6 January 2003 Published online: 5 May 2003 RID="⋆" ID="⋆" Research supported in part by CNPq grant 300.962/91-6 RID="⋆⋆" ID="⋆⋆" Research supported in part by CNPq grant 300.158/93-9 Communicated by P. Constantin     

Isothermal fluid sphere: Uniqueness and conformal mapping

We prove the theorem: A necessary and sufficient condition for a spacetime to represent an isothermal fluid sphere (linear equation of state with density falling off as inverse square of the curvature radius) without boundary is that it is conformal to a spacetime of zero gravitational mass (‘minimally’ curved).     

Chiral soliton model vs. pentaquark structure for Θ(1540)

The exotic baryon Θ⁺(1540 MeV) is visualized as an expected (iso) rotational excitation in the chiral soliton model. It is also argued as a pentaquark baryon state in a constituent quark model with strong diquark correlations. I contrast these two points of view; observe the similarities and differences between the two pictures. Collective excitation, the characteristic of chiral soliton model, points toward small mixing of representations in the wake ofSU (3) breaking. In contrast, constituent quark models prefer near ‘ideal’ mixing, similar to ω-φ mixing.     

The Possibilist Transactional Interpretation and Relativity

A recent ontological variant of Cramer’s Transactional Interpretation, called “Possibilist Transactional Interpretation” or PTI, is extended to the relativistic domain. The present interpretation clarifies the concept of ‘absorption,’ which plays a crucial role in TI (and in PTI). In particular, in the relativistic domain, coupling amplitudes between fields are interpreted as amplitudes for the generation of confirmation waves (CW) by a potential absorber in response to offer waves (OW), whereas in the nonrelativistic context CW are taken as generated with certainty. It is pointed out that solving the measurement problem requires venturing into the relativistic domain in which emissions and absorptions take place; nonrelativistic quantum mechanics only applies to quanta considered as ‘already in existence’ (i.e., ‘free quanta’), and therefore cannot fully account for the phenomenon of measurement, in which quanta are tied to sources and sinks.     

Generalized Fock spaces,new forms of quantum statistics and their algebras

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ‘infinite’, Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are: new algebras for infinite statistics,q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ‘doubly-infinite’ statistics, many representations of orthostatistics, Hubbard statistics and its variations.     

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.     

Negative chemical pressure effects induced by Y substitution for Ca on the ‘exotic’ magnetic behavior of the spin-chain compound, Ca3Co2O6

The magnetic behavior of a solid solution, Ca_{3 x} Y_x CO₂ O₆, based on the ‘exotic’ spin-chain compound, Ca₃Co₂O₆, crystallizing in K₄CdCl₆-derived rhombohedral structure is investigated. Among the compositions investigated(x = 0.0, 0.3, 0.5, 0.75 and 1.0), single-phase formation persists up tox = 0.75, with the elongation of the c-axis. The present investigations reveal that the temperature at which the ‘so-called’ ‘partially disordered antiferromagnetic structure’ sets in (which occurs at 24 K for the parent compound,x = 0.0) undergoes gradual reduction with the substitution of Y for Ca, attaining the value of about 2.2 K for the nominalx = 1.0. The trend observed in this characteristic temperature is opposite to that reported under external pressure, thereby establishing that Y substitution exerts negative chemical pressure. Anomalous steps observed in the isothermal magnetization at very low temperatures (around 2 K) forx = 0.0, which have been proposed to arise from ‘quantum tunneling effects’ are found to vanish by a small substitution (x = 0.3) of Y for Ca. Systematics in AC and DC magnetic susceptibility behavior with Y substitution for Ca have also been probed. We believe that the present results involving the expansion of chain length without disrupting the magnetic chain may be useful to the overall understanding of the novel magnetism of the parent compound.     

The algebra and geometry ofSU(3) matrices

We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular groupSU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real linear vector space, are developed in anSU(3) covariant manner. Thef andd symbols ofSU(3) lead to two ways of ‘multiplying’ two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization ofSU(3) is developed as a generalization of that forSU(2), and the specifically new features are brought out. Application to the dynamics of three-level systems is outlined.