Diffeomorphism invariant eigenvalue problem for metric perturbations in a bounded region

We suggest a method of construction of general diffeomorphism invariant boundary conditions for metric fluctuations. The case of $d+1$ dimensional Euclidean disk is studied in detail. The eigenvalue problem for the Laplace operator on metric perturbations is reduced to that on $d$-dimensional vector, tensor and scalar fields. Explicit form of the eigenfunctions of the Laplace operator is derived. We also study restrictions on boundary conditions which are imposed by hermiticity of the Laplace operator.Comment: LATeX file, no figures, no special macro

The a3/2a_{3/2} heat kernel coefficient for oblique boundary conditions

We present a method for the calculation of the $a_{3/2}$ heat kernel coefficient of the heat operator trace for a partial differential operator of Laplace type on a compact Riemannian manifold with oblique boundary conditions. Using special case evaluations, restrictions are put on the general form of the coefficients, which, supplemented by conformal transformation techniques, allows the entire smeared coefficient to be determined.Comment: 30 pages, LaTe

Index-free Heat Kernel Coefficients

Using index-free notation, we present the diagonal values of the first five heat kernel coefficients associated with a general Laplace-type operator on a compact Riemannian space without boundary. The fifth coefficient appears here for the first time. For a flat space with a gauge connection, the sixth coefficient is given too. Also provided are the leading terms for any coefficient, both in ascending and descending powers of the Yang-Mills and Riemann curvatures, to the same order as required for the fourth coefficient. These results are obtained by directly solving the relevant recursion relations, working in Fock-Schwinger gauge and Riemann normal coordinates. Our procedure is thus noncovariant, but we show that for any coefficient the `gauged' respectively `curved' version is found from the corresponding `non-gauged' respectively `flat' coefficient by making some simple covariant substitutions. These substitutions being understood, the coefficients retain their `flat' form and size. In this sense the fifth and sixth coefficient have only 26 and 75 terms respectively, allowing us to write them down. Using index-free notation also clarifies the general structure of the heat kernel coefficients. In particular, in flat space we find that from the fifth coefficient onward, certain scalars are absent. This may be relevant for the anomalies of quantum field theories in ten or more dimensions.Comment: 38 pages, LaTe

The hybrid spectral problem and Robin boundary conditions

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. A hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat--kernel expansion which are evaluated explicitly.Comment: 24 pages. Typos and a few factors corrected. Minor comments added. Substantial Robin additions. Substantial revisio

Zeta function determinant of the Laplace operator on the DD-dimensional ball

We present a direct approach for the calculation of functional determinants of the Laplace operator on balls. Dirichlet and Robin boundary conditions are considered. Using this approach, formulas for any value of the dimension, $D$, of the ball, can be obtained quite easily. Explicit results are presented here for dimensions $D=2,3,4,5$ and $6$.Comment: 22 pages, one figure appended as uuencoded postscript fil

Bose-Einstein condensation in arbitrarily shaped cavities

We discuss the phenomenon of Bose-Einstein condensation of an ideal non-relativistic Bose gas in an arbitrarily shaped cavity. The influence of the finite extension of the cavity on all thermodynamical quantities, especially on the critical temperature of the system, is considered. We use two main methods which are shown to be equivalent. The first deals with the partition function as a sum over energy levels and uses a Mellin-Barnes integral representation to extract an asymptotic formula. The second method converts the sum over the energy levels to an integral with a suitable density of states factor obtained from spectral analysis. The application to some simple cavities is discussed.Comment: 10 pages, LaTeX, to appear in Physical Review

Fluctuations of quantum fields via zeta function regularization

Explicit expressions for the expectation values and the variances of some observables, which are bilinear quantities in the quantum fields on a D-dimensional manifold, are derived making use of zeta function regularization. It is found that the variance, related to the second functional variation of the effective action, requires a further regularization and that the relative regularized variance turns out to be 2/N, where N is the number of the fields, thus being independent on the dimension D. Some illustrating examples are worked through.Comment: 15 pages, latex, typographical mistakes correcte

A Study of 3CR Radio Galaxies from z = 0.15 to 0.65. II. Evidence for an Evolving Radio Structure

Radio structure parameters were measured from the highest quality radio maps available for a sample of 3CR radio galaxies in the redshift range 0.15 < z < 0.65. Combined with similar data for quasars in the same redshift range, these morphology data are used in conjunction with a quantification of the richness of the cluster environment around these objects (the amplitude of the galaxy-galaxy spatial covariance function, Bgg) to search for indirect evidence of a dense intracluster medium (ICM). This is done by searching for confinement and distortions of the radio structure that are correlated with Bgg. Correlations between physical size and hot spot placement with Bgg show evidence for an ICM only at z 0.4, suggesting an epoch of z ~ 0.4 for the formation of an ICM in these Abell richness class 0-1, FR2-selected clusters. X-ray selected clusters at comparable redshifts, which contain FR1 type sources exclusively, are demonstrably richer than the FR2-selected clusters found in this study. The majority of the radio sources with high Bgg values at z < 0.4 can be described as ``fat doubles'' or intermediate FR2/FR1s. The lack of correlation between Bgg and bending angle or Bgg and lobe length asymmetry suggests that these types of radio source distortion are caused by something other than interaction with a dense ICM. Thus, a large bending angle cannot be used as an unambiguous indicator of a rich cluster around powerful radio sources. These results support the hypothesis made in Paper 1 that cluster quasars fade to become FR2s, then FR1s, on a timescale of 0.9 Gyrs (for H0 = 50 km s^-1 Mpc^-1).Comment: 44 pages, 8 figures, 2 tables; to be published in the September 2002 issue of The Astronomical Journa

X-ray Spectral Survey of WGACAT Quasars, II: Optical and Radio Properties of Quasars with Low Energy X-ray Cut-offs

We have selected quasars with X-ray colors suggestive of a low energy cut-off, from the ROSAT PSPC pointed archive. We examine the radio and optical properties of these 13 quasars. Five out of the seven quasars with good optical spectra show associated optical absorption lines, with two having high delta-v candidate systems. Two other cut-off quasars show reddening associated with the quasar. We conclude that absorption is highly likely to be the cause of the X-ray cut-offs, and that the absorbing material associated with the quasars, not intervening along the line-of-sight. The suggestion that Gigahertz Peaked Sources are associated with X-ray cut-offs remains unclear with this expanded sample.Comment: 17 pages, LaTeX, including 2 Tables and 1 figure. Ap.J. in pres

The spin dependence of high energy proton scattering

Motivated by the need for an absolute polarimeter to determine the beam polarization for the forthcoming RHIC spin program, we study the spin dependence of the proton-proton elastic scattering amplitudes at high energy and small momentum transfer.We examine experimental evidence for the existence of an asymptotic part of the helicity-flip amplitude phi_5 which is not negligible relative to the largely imaginary average non-flip amplitude phi_+. We discuss theoretical estimates of r_5, essentially the ratio of phi_5 to phi_+, based upon extrapolation of low and medium energy Regge phenomenological results to high energies, models based on a hybrid of perturbative QCD and non-relativistic quark models, and models based on eikonalization techniques. We also apply the model-independent methods of analyticity and unitarity.The preponderence of evidence at available energy indicates that r_5 is small, probably less than 10%. The best available experimental limit comes from Fermilab E704:those data indicate that |r_5|<15%. These bounds are important because rigorous methods allow much larger values. In contradiction to a widely-held prejudice that r_5 decreases with energy, general principles allow it to grow as fast as ln(s) asymptotically, and some models show an even faster growth in the RHIC range. One needs a more precise measurement of r_5 or to bound it to be smaller than 5% in order to use the classical Coulomb-nuclear interference technique for RHIC polarimetry. As part of this study, we demonstrate the surprising result that proton-proton elastic scattering is self-analysing, in the sense that all the helicity amplitudes can, in principle, be determined experimentally at small momentum transfer without a knowledge of the magnitude of the beam and target polarization