Path Integral Approach for Spaces of Non-constant Curvature in Three Dimensions

In this contribution I show that it is possible to construct three-dimensional spaces of non-constant curvature, i.e. three-dimensional Darboux-spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that in the two three-dimensional Darboux spaces, which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In \threedDI we find seven coordinate systems which separate the Schr\"odinger equation. For the second space, \threedDII, all coordinate systems of flat three-dimensional Euclidean space which separate the Schr\"odinger equation also separate the Schr\"odinger equation in \threedDII. I solve the path integral on \threedDI in the $(u,v,w)$-system, and on \threedDII in the $(u,v,w)$-system and in spherical coordinates

On the Path Integral in Imaginary Lobachevsky Space

The path integral on the single-sheeted hyperboloid, i.e.\ in $D$-dimensional imaginary Lobachevsky space, is evaluated. A potential problem which we call ``Kepler-problem'', and the case of a constant magnetic field are also discussed.Comment: 16 pages, LATEX, DESY 93-14

On the Green function of linear evolution equations for a region with a boundary

We derive a closed-form expression for the Green function of linear evolution equations with the Dirichlet boundary condition for an arbitrary region, based on the singular perturbation approach to boundary problems.Comment: 9 page

Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: II. Darboux Spaces DIII and DIV

This is the second paper on the path integral approach of superintegrable systems on Darboux spaces, spaces of non-constant curvature. We analyze in the spaces \DIII and \DIV five respectively four superintegrable potentials, which were first given by Kalnins et al. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is determined by a higher order polynomial equation. We show that also the free motion in Darboux space of type III can contain bound states, provided the boundary conditions are appropriate. We state the energy spectrum and the wave-functions, respectively

Brillouin amplification supports 1×10−201\times10^{-20} accuracy in optical frequency transfer over 1400~km of underground fibre

We investigate optical frequency transfer over a 1400~km loop of underground fibre connecting Braunschweig and Strasbourg. Largely autonomous fibre Brillouin amplifiers (FBA) are the only means of intermediate amplification, allowing phase-continuous measurements over periods up to several days. Over a measurement period of about three weeks we find a weighted mean of the transferred frequency's fractional offset of $(1.1\pm0.4)\times10^{-20}$. In the best case we find an instability of $6.9\times10^{-21}$ and a fractional frequency offset of $4.4\times10^{-21}$ at an averaging time of around 30~000~s. These results represent an upper limit for the achievable uncertainty over 1400 km when using a chain of remote Brillouin amplifiers, and allow us to investigate systematic effects at the $10^{-20}$-level

Long-distance remote comparison of ultrastable optical frequencies with 1e-15 instability in fractions of a second

We demonstrate a fully optical, long-distance remote comparison of independent ultrastable optical frequencies reaching a short term stability that is superior to any reported remote comparison of optical frequencies. We use two ultrastable lasers, which are separated by a geographical distance of more than 50 km, and compare them via a 73 km long phase-stabilized fiber in a commercial telecommunication network. The remote characterization spans more than one optical octave and reaches a fractional frequency instability between the independent ultrastable laser systems of 3e-15 in 0.1 s. The achieved performance at 100 ms represents an improvement by one order of magnitude to any previously reported remote comparison of optical frequencies and enables future remote dissemination of the stability of 100 mHz linewidth lasers within seconds.Comment: 7 pages, 4 figure

Path Integrals in Polar Field Variables in QFT

We show how to transform a $d$-dimensional Euclidean path integral in terms of two (Cartesian) fields to a path integral in terms of polar field variables. First we present a conjecture that states how this transformation should be done. Then we show that this conjecture is correct in the case of two toy models. Finally the conjecture will be proven for a general QFT model with two fields

Signatures of pressure induced superconductivity in insulating Bi2212

We have performed several high pressure electrical resistance experiments on Bi1.98Sr2.06Y0.68Cu2O8, an insulating parent compound of the high-Tc Bi2212 family of copper oxide superconductors. We find a resistive anomaly, a downturn at low temperature, that onsets with applied pressure in the 20-40 kbar range. Through both resistance and magnetoresistance measurements, we identify this anomaly as a signature of induced superconductivity. Resistance to higher pressures decreases Tc, giving a maximum of 10 K. The higher pressure measurements exhibit a strong sensitivity to the hydrostaticity of the pressure environment. We make comparisons to the pressure induced superconductivity now ubiquitous in the iron arsenides.Comment: 5 pages, 4 figures, submitted to Phys. Rev.