The Wilson Loop Observables of Chern-Simons Theory on ℝ<Superscript>3</Superscript> in Axial Gauge

We define the Wilson loop observables (WLOs) for pure Chern-Simons models with base manifold M=³ rigorously as infinite dimensional oscillatory integrals by exploiting an axial gauge fixing and applying certain regularization techniques like loop-smearing and framing. The values of the WLOs can be computed explicitly. If the structure group G of the model considered is Abelian one obtains well-known linking number expressions for the WLOs. If G is Non-Abelian one obtains expressions which are similar but not identical to the state sum representations for the Homfly and Kauffman polynomials by Jones and Turaev.     

Asymptotic Completeness for Compton Scattering

Scattering in a model of a massive quantum-mechanical particle, an electron, interacting with massless, relativistic bosons, photons, is studied. The interaction term in the Hamiltonian of our model describes emission and absorption of photons by the electron; but electron-positron pair production is suppressed. An ultraviolet cutoff and an (arbitrarily small, but fixed) infrared cutoff are imposed on the interaction term. In a range of energies where the propagation speed of the dressed electron is strictly smaller than the speed of light, unitarity of the scattering matrix is proven, provided the coupling constant is small enough; (asymptotic completeness of Compton scattering). The proof combines a construction of dressed one–electron states with propagation estimates for the electron and the photons.Dedicated to Freeman Dyson on the occasion of his 80th birthdayWork partially supported by U.S. National Science Foundation grant DMS 01-00160.Acknowledgement. We thank V. Bach for his hospitality at the University of Mainz, where part of this work was done, and we are indebted to Gian Michele Graf for pointing out a serious gap in an earlier version of this paper. We also thank one of the referees for pointing out many typos and some small errors.     

Dual pairs of quantum spaces

We discuss the notion of metric dual pair corresponding to the algebraic condition in Alain Connes' definition of noncommutative Poincaré duality. We show that the DeRham complex of a metric dual pair is the homomorphic image of a skew tensor product, this leading to a natural definition of biconnections.     

On the relation between weak and strong invariance of differential equations

Some concepts of Lie algebra cohomology are used to systematize the search for differential equations invariant under a given Lie groupG. In particular, it is shown that if a strongly invariant equation exists, then all weakly invariant equations differ from it only by an arbitrary multiplicative factor. If no strongly invariant equation exists, then cohomology theory can be used to simplify the search for weakly invariant equations.     

Perturbed periodic Hamiltonians: Essential Spectrum and exponential decay of eigenfunctions

Spectral properties of – +V(x), whereV(x) lies in a neighbourhood of the periodic case and describes various models of disorder, are studied. We prove the exponential decay of generalized eigenfunctions corresponding to energies in the resolvent set of the unperturbed periodic Hamiltonian, as well as the stability of the essential spectrum for the dislocation disorder in two dimensions.     

Nanoscale Characterization for the Classroom

This report describes the development of a semester course in nano-scale characterization. The interdisciplinary course is opened to both advanced undergraduate and graduate students with a standard undergraduate preparation in Materials Science, Chemistry, or Physics. The approach is formal rather than the typical research seminar and has a laboratory component.     

Current controlled negative resistance phenomenon in SbPbSe system

Thin film Al/Sb₂Pb₁Se₇/Al metal-glass-metal sandwiched structures prepared using thermal evaporation technique have been studied. The I–V measurements showed that the devices switched from high resistance OFF state to a low resistance ON state when a particular voltage appeared across it. The OFF state I–V characteristics showed non-ohmic behaviour while in the ON state the devices displayed purely linear characteristics. The switching voltage (V _th) was found to depend on film thickness and temperature of the device. A linear relation between V_th and temperature was observed.     

Analysis of signal-to-noise ratio of the laser Doppler velocimeter

The signal-to-shot-noise ratio of the photocurrent of a laser Doppler anemometer is calculated as a function of the parameters which describe the system. It is found that the S/N is generally a growing function of receiver area, that few large particles are better than many small ones, and that generally the fringe or differential mode configuration is equal to, or better than, the reference beam mode.     

Ghost lines in Mössbauer relaxation spectra

The appearance in Mössbauer relaxation spectra of ghost lines, which are narrow lines that do not correspond to transitions between real hyperfine energy levels of the resonant system, is examined. It is shown that in many cases of interest, the appearance of these ghost lines can be interpreted in terms of the relaxational averaging of one or more of the static interactions of the ion.     

On Average Properties of Inhomogeneous Fluids in General Relativity: Dust Cosmologies

For general relativistic spacetimes filled with irrotational dust a generalized form of Friedmann's equations for an effective expansion factor a _D of inhomogeneous cosmologies is derived. Contrary to the standard Friedmann equations, which hold for homogeneous-isotropic cosmologies, the new equations include the backreaction effect of inhomogeneities on the average expansion of the model. A universal relation between backreaction and average scalar curvature is also given. For cosmologies whose averaged spatial scalar curvature is proportional to a _D ^-2, the expansion law governing a generic domain can be found. However, as the general equations show, backreaction acts as to produce average curvature in the course of structure formation, even when starting with space sections that are spatially flat on average.     

Sampling in holography of diffused objects or with diffused illumination

Differentiation between a diffused hologram and a specular hologram is determined by criteria of the wavelength and the relief of the object. In a hologram where the object beam is diffused (due to diffused object illumination, or because the object surface is reflecting diffusively), every part in the holographic plane contains information about the whole object, but from a different angle of view. In a specular hologram, the recording is somewhere between the image recording and the diffused holography. The latter part is contributed mainly by the edges of the object.This paper describes properties of sampled diffused holograms in terms of resolution,S/N (contrast), angle of view, power spectrum and the ability of determination of the object.Different sampling functions are compared and experimentally examined, while the total diffracting areas are preserved constant.It was found that when the sampling areas were distributed over the entire plane and the observation distance was preserved, the resolution, the angle of view and the ability of determination of the object was close to the unsampled hologram. At the same conditions theS/N and the power spectrum were reduced.It seems that these techniques of sampling may be applied for multiplexing, subtracting and correlating between two adjacent recordings.     

Crossing relations derived from (Extended) Relativity

Recently, Special Relativity has been straightforwardly extended to Superluminal inertial frames and faster-than-light objects. The Extended Relativity theory not only allowed building up a self-consistent classical theory of tachyons, but reveals itself useful also for the understanding of standard (subluminal) physics, i.e. of usual particles. In this paper, it is shown that Extended Relativity allows: (i) deriving the usual Crossing Relations of elementary particle (high-energy) physics; and (ii) deriving the CPT-covariance theorem as a particular case of G-covariance (i.e., covariance under the new group of Generalised Lorentz transformations, both subluminal and Superluminal).In this framework, the Analyticity postulate is unnecessary: it is better substituted by the G-covariance requirement.Moreover, new crossing-type relations are predicted on the basis of mere Extended Relativity. They may well serve as a test for relativistic covariance of force fields like strong interactions and, particularly, weak interactions, and possible new interaction fields (whicha priori are not relativistically covariant).     

On Essential Incompleteness of Hertz’s Experiments on Propagation of Electromagnetic Interactions

The historical background of the 19th century electromagnetic theory is revisited from the standpoint of the opposition between alternative approaches in respect to the problem of interactions. The 19th century electrodynamics became the battle-field of a paramount importance to test existing conceptions of interactions. Hertzs experiments were designed to bring a solid experimental evidence in favor of one of them. The modern scientific method applied to analyze Hertzs experimental approach as well as the analysis of his laboratory notes, dairy and private letters show that Hertzs crucial experiments cannot be considered as conclusive at many points as it is generally implied. We found that alternative Helmholtzs electrodynamics did not contradict any of Hertzs experimental observations of transverse components as Maxwells theory predicted. Moreover, as we now know from recently published Hertzs dairy and private notes, his first experimental results indicated clearly on infinite rate of propagation. Nevertheless, Hertzs experiments provided no further explicit information on non-local longitudinal components which were such an essential feature of Helmholtzs theory. Necessary and sufficient conditions for a decisive choice on the adequate account of electromagnetic interactions are discussed from the position of modern scientific method.     

What is a hidden variable theory of quantum phenomena?

The purpose of this paper is to clarify the relationship between existing so-called hidden variable theories of quantum phenomena and some well-known proofs, such as those of von Neumann, Jauch and Piron, and Kochen and Specker, which purport to establish that no such theory is possible. The proof of Kochen and Specker, which is a stronger version of von Neumann's result, demonstrates the impossibility of embedding the algebraic structure of physical parameters of the quantum theory, represented by the self-adjoint Hubert space operators, into the commutative algebra of real-valued functions on a phase space of hidden states. This is a necessary condition for a hidden variable extension of the quantum theory in the usual sense of a statistical mechanical derivation of the statistical theorems of the quantum theory in the classical manner. No existing so-called hidden variable theory is a counter-example to von Neumann's proof. The early 1951 hidden variable theory of Bohm and the recent theory of Bohm and Bub are not in fact hidden variable theories in the usual sense of the term. Since the term hidden variable theory is justifiably used to denote the kind of theory rejected by von Neumann, Jauch and Piron, and Kochen and Specker, it is suggested that the term should not be used as a label for the theories considered by Bohm and other workers in this field. Such theories could be regarded as fundamentally compatible with the original Copenhagen interpretation of the quantum theory, as expressed by Bohr.Supported by the National Science Foundation.     

Surface topology investigation for ancient coinage assessment using optical interferometry

In order to demonstrate the capabilities of white-light interferometry depth profiling (WLI-DP) for ancient coinage assessment, we investigated a series of notorious 1786 gold coins, bearing Louis XVIs horned effigy, and allegedly minted in Strasbourg. Scanning electron microscopy as well as WLI-DP observations unambiguously indicate that both previously differentiated single- and double-horned varieties originated from a unique minting tool. Moreover, from topological measurements, we infer that single-horned coins, rather than wearing out into double-horned coins, proceeded from the latter variety during minting by progressive failure of an already altered die. Whereas present observations do not exclude initial forgery, they suggest that protrusions resulted from progressive incidental in-service die deterioration. PACS 81.70.Fy; 07.60.Ly; 61.16.Bg; 81.40.Pq     

The many faces of a character

We prove an identity between three infinite families of polynomials which are defined in terms of bosonic, fermionic, and one-dimensional configuration sums. In the limit where the polynomials become infinite series, they give different-looking expressions for the characters of the two integrable representations of the affine su(2) algebra at level one. We conjecture yet another fermionic sum representation for the polynomials which is constructed directly from the Bethe-ansatz solution of the Heisenberg spin chain.     

Differential geometry on Grassmann algebras

H. C. Lee [1] developed the analogue of Riemannian geometry on a real symplectic manifold — the fundamental skew two-form taking the place of the symmetric tensor. The usual Riemannian concepts do not adapt themselves very well, thus curvature is represented by a tensor of the third rank and Killing's equations now involve this curvature tensor. The immediate reason for this is that otherwise familiar terms appear with the wrong sign. We have found that these unaesthetic features disappear, and formal elegance is marvellously restored, when the manifold is replaced by a Grassmann algebra. The connection with supersymmetry is explained but applications are not reported here.A part of this paper was presented to the International Symposium on Mathematical Physics, Mexico City, January 5–10, 1976.Supported in part by the National Science Foundation.     

The stress tensor in QED

The gauge-invariant point-splitting version of the energy-momentum tensor in QED is discussed with respect to Poincaré identities. In one-loop approximation an anomaly in the conservation law has been found for the triangle graph.This is part of the author's thesis, Tubingen 1974.     

Gauge-invariant variations in the redshift and cosmic background radiation anisotropies

The gauge-invariance of the calculations determining anisotropics in the cosmic microwave background radiation (the Sachs-Wolfe effect) is re-examined. It is shown that the results obtained are gauge-invariant only if a physically-based definition of the surface of last scattering is implemented, in a context where perturbations of the surface of last scattering as well as of the space-time are taken into account. Any physical interpretation of the results based on their splitting into scalar, vector, and tensor parts, is unique only if non-local (unverifiable) conditions are imposed; locally, any such interpretation is non-unique. The physical meaning of the Sachs-Wolfe potentialB and associated redshift formula depends on implementing a very specific gauge, without a clear physical or geometric meaning; its implications do not extend to other, more usual, gauges.     

Ultrageneralised complex analysis

The zero rest mass Euclidean Dirac equations in 2 (4) dimensions may be regarded as square roots of the second order harmonic equation, and give rise to the crucial integral theorem and integral formula of complex (quaternionic) analysis. Recently discovered 2rth root equations for the 2rth order harmonic equations are here shown to give rise to a similar integral theorem and integral formula.