Topological gauge theories and group cohomology

We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH ⁴(BG,Z). In a similar way, possible Wess-Zumino interactions of such a groupG are classified byH ³(G,Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map fromH ⁴(BG,Z) toH ³(G,Z). We generalize this correspondence to topological spin theories, which are defined on three manifolds with spin structure, and are related to what might be calledZ ₂ graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.     

On the self-adjointness of the (g(x)φ4)2 Hamiltonian

An alternate proof to that provided by Glimm and Jaffe of the essential self-adjointness of the HamiltonianH for a relativistic scalar quantum field in two dimensional space-time with a space cut-off quartic interactionH _I (g) is given. The proof depends mainly on the estimateH _I ² (g)const. (N+I)⁴ and on the semiboundedness ofH=H ₀+H _I (g).Supported in part by the National Research Council of Canada.     

Energy conditions in standard static spacetimes

Let (H, h) be a Riemannian manifold and letfH(0,) be a smooth function. The Lorentzian warped product (a,b) _f ×H, -a<b, with metricds ²=(–f ² dt ²)h is called a standard static spacetime. We investigate conditions on the warping functionf which guarantee that a standard static spacetime (a,b) _f ×H satisfies certain of the energy conditions from general relativity. Also, if (H, h) is a complete, finite volume Riemannian manifold with nonpositive Ricci curvature and the gradient off never vanishes onH, then (a, b) _f ×H cannot satisfy the null convergence condition.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Isolation Phenomenon for Complete Surfaces Arising from Yang–Mills Theory

Let M be a complete surface with parallel mean curvature in a complete simply connected space form F ^2+p (c) of constant curvature c. Denote by H and S the mean curvature and the squared length of the second fundamental form of M respectively. Motivated by L ²-isolation phenomenon in Yang–Mills theory, we prove that if , where c + H ² > 0, D(H,c) is an explicit positive constant depending on H and c, then , i.e., M is a totally umbilical sphere . Research supported by the Chinese NSF, Grant No. 10231010; Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.     

A model with persistent vacuum

It is shown that there exists a selfadjoint Hamilton operator corresponding to the interactionH ₀ ^(a) +H ₀ ^(b) + _b ⁺ (x) _a (x) _b ^– (x)d ³ x, wherea andb denote two types of scalar particles. We discuss the scattering theory of this operator.Work partially supported by the Swiss National Science Foundation.This paper contains results from the author's Ph. D. Thesis [3].     

A Particle in a Magnetic Field of an Infinite Rectilinear Current

We consider the Schrödinger operator H=(i+A)² in the space L ₂(R ³) with a magnetic potential A created by an infinite rectilinear current. We show that the operator H is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions exp(–i H t)f of the time dependent Schrödinger equation. Our main observation is that a quantum particle has always a preferable (depending on its charge) direction of propagation along the current. Similar result is true in classical mechanics.     

On characteristic initial-value and mixed problems

Existence and uniqueness are proved for certain initial-value problems for hyperbolic systems of second-order differential equations, each having the same principal partg ^ab _{a b} (whereg ^ab is indefinite). The initial data are given on two intersecting hypersurfaces H₁ andH ₂ one of which-sayH ₁-is a characteristic surface. The other surface,H ₂, is permitted to be spacelike, timelike, or characteristic. For Einstein's vacuum field equations we restrict ourselves to anH ₂ that is characteristic. Unlike the Cauchy problem, the data have to be necessarily of a considerably higher differentiability class (Sobolev classW ^2m–1) than the solution (Sobolev classW ^m ). On the other hand, in the mixed problem (where one of the surfaces is spacelike) corner conditions have to be fulfilled. The occurrence of constraint equations for Einstein's metric field and for harmonic coordinates can be prevented by solving certain ordinary differential propagation equations.     

Trace norm convergence of exponential product formula

LetH andK be lower-bounded self-adjoint operators whose form sum is denoted byH K. We show the norm inequality (e^–rH/2 e^–rK e^–rH/2)^1/r forr > 0 and any symmetric norm . WhenH +K is essentially self-adjoint and e^–K is of trace class, we prove that (e^–rH/2e^–rK e^–rH/2)^1/r converges asr 0 to e^–(H+K) in the trace norm.     

Time-resolved resonance-Raman study of porphyrins and metalloporphyrins in the electronic excited states

Current developments of our Time-Resolved Resonance-Raman (TR³) studies on the (*) excited states of free-base porphyrins and (dd) excited state of Ni-OctaEthylPorphyrin (NiOEP) in solutions are described. A device for quasi-simultaneous measurements has been developed to obtain correct difference spectra for pump/probe spectra of various delay times and probe-only spectrum. This technique is practically powerful to obtain TR³ spectra of porphyrins and metalloporphyrins. An orbital-symmetry marker band ( ₂) exhibited upshifts alongS ₀S ₁T ₁ transitions for free-base octaethylporphyrin (H₂OEP), while downshifts along the same transitions for free-base TetraPhenylPorphyrin (H₂TPP). This is interpreted by the symmetry differences in the highest occupied molecular orbitals of H₂OEP (a _u-orbital) and H₂TPP (b _1u-orbital). An intensity-borrowing mode ( ₁₉) was enhanced by the AlbrechtC-term in resonance with theS ₁S _n transition, but not in resonance with theT ₁T _n transition for H₂OEP. This fact can be considered to be due to different vibronic couplings between the singlet and triplet manifolds. For the (dd) excited state of NiOEP, our TR³ study suggests the excess energy for the ( ^*) to (dd) transition was vibrationally cooled prior to the electronic relaxation from the (dd) state to the ground state with a relaxation time of ca. 300 ps. Comparison between Stokes- and anti-Stokes resonance Raman spectra for the delay time of 0 ps indicated that the lower limit of vibrational temperature in 0–40 ps time range is 750 K.     

Dynamics of flux lines: thermal fluctuations

A relaxational dynamics for ad-dimensional model of flux lines is investigated. The flux lines are subject to thermal fluctuations such that they form a flux liquid rather than a flux lattice. We find that the density-density auto correlation function decays as (logt/t ²)² for large timest. In contrast, this correlation function decays for point-like vortices ast ^–2. At the transition to an empty lattice (Abrikosov-Meissner transition), the auto correlation function decays as a power law with dimension-dependent exponentsd+2 for flux lines andd for point-like vortices. Therefore, the fluctuations and the interactions of long flux lines have an accelerating effect on the relaxation.     

Zur Ganggeschwindigkeit von Uhren auf der rotierenden Erde

On the Rates of Advance of Clocks on the Rotating Earth In 1905 ALBERT EINSTEIN came from his Special Theory of Relativity to the conclusion that a “balance clock” located at the earth's equator is somewhat slowed down compared with a clock resting at one of the earth's poles P (see Fig. 2 below). But interestingly, this retardation of time is fully compensated by an opposing gravitational effect which (according to the General Theory of Relativity, discovered later by EINSTEIN, 1908 - 1915) causes all (atomic) clocks even at any point A on earth's surface at normal sea-level at arbitrary latitudes φ to advance at the same rate as the clock at the pole: The kinematic time-dilatation of a clock A moving with the velocity v of the earth's daily revolution around its axis turns out to be precisely equal in magnitude to the acceleration of the advance of the same clock A with respect to the clock at P due to the gravitational potential-difference between A and P, resulting from the ellipsoidal figure of the rotating earth. These two variations in the clock-rates, amounting relatively to 1/2 (v/c)², where c denotes the velocity of light in vacuum, exactly cancel each other. Furthermore the so called clock-paradox and the Hafele-Keating-Experiment are rediscussed in a somewhat different way than usual.     

Fourier transform of supermeasures

In the note, a Smolyanov-Shavgulidze model of infinite-dimensional superspace H _Λ = Λ₀ H ₀ ⊕ Λ₁ H ₁ corresponding to a Hilbert space H = H ₀ ⊕ H ₁ over a Hilbert superalgebra Λ = Λ₀ ⊕ Λ₁ is considered. Its relation to the Khrennikov superspace is discussed. Moreover, H _Λ is equipped with the structure of Hilbert superspace with an inner superproduct (·, ·)_Λ. The supermeasures are defined as Λ^ℂ (∧H ₁)-valued measures on H ₀ (Λ^ℂ stands for the complexification of Λ). The definition of the Fourier supertransform is similar to the ordinary one, (y) = μ(dz). It turns out that the values of the Fourier supertransform of a supermeasure on the subspace = H ₀ ⊕ Λ₁ H ₁ can be obtained by applying a certain operator to the values of the classical Fourier transform. The main result is the theorem claiming that the Fourier supertransform of supermeasures is isometric on superspaces with zero even part (H ₀ = {0}). As a corollary, we note that the operator of Fourier supertransform is injective. We also give necessary and sufficient conditions for the countable additivity of cylindrical supermeasures in terms of continuity of their Fourier supertransforms (an analog of the Minlos-Sazonov theorem).     

Computer simulation of some dynamical properties of the Lorentz gas

We carried out molecular dynamics simulations of a Lorentz gas, consisting of a lone hydrogen molecule moving in a sea of stationary argon atoms. A Lennard-Jones form was assumed for the H₂-Ar potential. The calculations were performed at a reduced temperatureK ^* =kT/_{H 2–Ar} = 4.64 and at reduced densities ^*= _{Ar Ar} ³ in the range 0.074–0.414. The placement of Ar atoms was assumed to be random rather than dictated by equilibrium considerations. We followed the trajectories of many H₂ molecules, each of which is assigned in turn a velocity given by the Maxwell-Boltzmann distribution at the temperature of the simulation. Solving the equations of motion classically, we obtained the translational part of the incoherent dynamic structure factor for the H₂ molecule,S _tr(q, ). This was convoluted with the rotational structure factorS _rot(q, ) calculated assuming unhindered rotation to obtain the total structure factorS(q, ). Our results agree well with experimental data on this function obtained by Egelstaffet al. At the highest density ( ^*=0.414) we studied the dependence ofS(q, ) on system size (number of Ar atoms), number of H₂ molecules for which trajectories are generated, and the length of time over which these trajectories are followed.     

Additivity of vector gleason measures

We study the degree of additivity of orthogonal Hilbert-space-valued measures on the latticeL(H) of all projections acting on a Hilbert spaceH. We present criteria for such measures to be completely additive and we establish the connection between the additivity of orthogonal measures and the size of almost disjoint families on dimH. [For example, we show that everyH-valued orthogonal measure is weakly-additive iff (dimH) > dim H]. As a corollary we see that finitely additive orthogonal measures distinguish dimensions of Hilbert spaces (this can be viewed as a generalization of a theorem by Kruszynski). As a further corollary, we obtain that, for cardinals, with >,3, there is no Jordan homomorphism from a typeI -factor into a typeI -factor. Finally, we show that every latticeL(H) with (dimH) = dimH admits a nonzero free orthogonal measure with values inH. Our results contribute to the noncommutative probability theory and also may find applications in the theory of the representation ofC ^*-algebras.     

Excited states of the molecular ion (H2μ)+

The rotation-vibration spectrum of (H₂ )⁺ is computed. Radiative lifetimes of the excited states are of order 10^–4 s or more. These times can be considered infinite compared to the lifetime of ⁺. For the ion in a crystal the lifetimes are significantly decreased by interaction with polarized molecules of the lattice. Transition rates to the ground state are calculated for (H₂ )⁺ in a hydrogen crystal. The results make it possible to interpret the experimental data from SR investigations of hydrogen, deuterium and hydrogen-deuterium mixtures.     

Essential self-adjointness of operators in ordered hilbert space

LetH ₀0 be a self-adjoint operator acting in a spaceL ²(M, ). It is assumed thatH ₀ e=0, wheree is strictly positive, and that exp(–tH ₀) is positivity preserving fort0. LetV be a real function onM such that its positive part is inL ²(M,e ²) and its negative part is relatively small with respect toH ₀. ThenH=H ₀+V is essentially self-adjoint on the intersection of the domains ofH ₀ andV. This result is applied to Schrödinger operators and to quantum field Hamiltonians.     

Local fluctuation of the spectrum of a multidimensional Anderson tight binding model

We consider the Anderson tight binding modelH=–+V acting inl ²(Z ^d ) and its restrictionH to finite hypercubes Z ^d . HereV={V _x ;xZ ^d } is a random potential consisting of independent identically distributed random variables. Let {E _j ()} _j be the eigenvalues ofH , and let _j (,E)=||(E _j ()–E),j1, be its rescaled eigenvalues. Then assuming that the exponential decay of the fractional moment of the Green function holds for complex energies nearE and that the density of statesn(E) exists atE, we shall prove that the random sequence { _j (,E)} _j , considered as a point process onR ¹, converges weakly to the stationary Poisson point process with intensity measuren(E)dx as gets large, thus extending the result of Molchanov proved for a one-dimensional continuum random Schrödinger operator. On the other hand, the exponential decay of the fractional moment of the Green function was established recently by Aizenman, Molchanov and Graf as a technical lemma for proving Anderson localization at large disorder or at extreme energy. Thus our result in this paper can be summarized as follows: near the energyE where Anderson localization is expected, there is no correlation between eigenvalues ofH if is large.     

Temperature dependence of SSW linewidth in metallic thin films

Measurements of the standing spin wave linewidth H _n at 17·6 GHz from 6 °K to 400 °K are presented for two permalloy films whose structure is either polycrystalline or epitaxial. It is shown that in both films, H _n is proportional ton ² (n is the mode number). Although H _n is little modified by temperature (20% variation maximum over the entire temperature range), we experimentally state that the slopep of H _n (n ²) is a linear function of the saturation magnetizationM which agrees with a planar inhomogeneity model.The authors wish to thank H. Pascard and F. Hoffmann who have prepared and selected the films as also E. Villain and A. Gatineau for invaluable technical assistance.     

UV excimer laser photochemistry of hybrid organometallic compounds of gallium

The gas phase ultraviolet (UV) excimer laser induced photolysis of the gallium-alkyls Ga(t-C₄H₉) _n ^– (CH₃)_3–n (n=0, 1, 2, 3) was studied, using photolysis wavelengths of 308, 248, and 193 nm. The photofragments Ga, GaH, and GaCH₃ were detected by laser ionization time-of-flight mass spectroscopy, while the hydrocarbon products CH₄, C₂H₆, HC(CH₃)₃ and H₂C=C(CH₃)₂ were identified using Fourier transform infrared (FTIR) spectroscopy. The formation of the GaH photofragment, and a high olefin-to-alkane product ratio, for Ga(t-C₄H₉)₂(CH₃) and Ga(t-C₄H₉)₃ are interpreted to indicate a -hydrogen elimination process. However, -hydrogen elimination only occurs after fission of the weakest Ga-C bond, thus no -hydride elimination is observed for Ga(t-C₄H₉)(CH₃)₂. Detection of C₂H₆ for Ga(CH₃)₃ and Ga(t-C₄H₉)(CH₃)₂, but not for Ga(t-C₄H₉)₂(CH₃), shows that under our experimental conditions the formation of ethane is as a result of the reductive elimination of the methyl groups, and is not due to the recombination of two free methyl radicals.     

The low-field galvanomagnetic transport coefficients of cubic metals: General formulae and experiments on the hall coefficient in Al(Ge)

The scattering of conduction electrons by defects is described by a low-field transport relaxation time _k ^x . It depends on the electron wavevectork and on the direction of the electric fieldE/E, but not on the magnetic fieldH. From a discussion of Kohler's rule written in terms of the exact relaxation time _k ^* it follows that this approximation is very good for low values ofH/, with being the resistivity forH=0. Assuming _k ^x to be known, the linearized Boltzmann equation is solved by a Jones-Zener expansion up to termsH ³. For metals with cubic symmetry we derive simple formulae for the coefficients of transverse and longitudinal magnetoresistance and for the two leading terms of the Hall coefficientR(H)=R ₀+R ₂(H/)². Simplifications occur for a _k with cubic symmetry and for metals with special Fermi surfaces.These formulae are used to interpret experimental results of the magnetic field dependence ofR(H) in Al(Ge) at 4 K. In this dilute aluminium alloyR ₀ is highly positive andR ₂ strongly negative. By irradiating a 3,000 ppm Al(Ge) sample with reactor neutrons at 4 K, an increasing concentration of self-interstitials and vacancies is added to the germanium impurities resulting in a decrease of bothR ₀ and |R ₂|. This is discussed in a three-group model of the Fermi surface of aluminium.This work was supported by the German Bundesministerium für Forschung und Technologie