Implications of Conformal Invariance for Quantum Field Theories in d>2d>2

Recently obtained results for two and three point functions for quasi-primary operators in conformally invariant theories in arbitrary dimensions {\absit d} are described. As a consequence the three point function for the energy momentum tensor has three linearly independent forms for general {\absit d} compatible with conformal invariance. The corresponding coefficients may be regarded as possible generalisations of the Virasoro central charge to {\absit d} larger than 2. Ward identities which link two linear combinations of the coefficients to terms appearing in the energy momentum tensor trace anomaly on curved space are discussed. The requirement of positivity for expectation values of the energy density is also shown to lead to positivity conditions which are simple for a particular choice of the three coefficients. Renormalisation group like equations which express the constraints of broken conformal invariance for quantum field theories away from critical points are postulated and applied to two point functions.\hfill\break Talk presented at the XXVII Ahrenshoop International Symposium.Comment: 14pages, Damtp 93/6

Constraints on RG Flow for Four Dimensional Quantum Field Theories

The response of four dimensional quantum field theories to a Weyl rescaling of the metric in the presence of local couplings and which involve $a$, the coefficient of the Euler density in the energy momentum tensor trace on curved space, is reconsidered. Previous consistency conditions for the anomalous terms, which implicitly define a metric $G$ on the space of couplings and give rise to gradient flow like equations for $a$, are derived taking into account the role of lower dimension operators. The results for infinitesimal Weyl rescaling are integrated to finite rescalings $e^{2\sigma}$ to a form which involves running couplings $g_\sigma$ and which interpolates between IR and UV fixed points. The results are also restricted to flat space where they give rise to broken conformal Ward identities. Expressions for the three loop Yukawa $\beta$-functions for a general scalar/fermion theory are obtained and the three loop contribution to the metric $G$ for this theory are also calculated. These results are used to check the gradient flow equations to higher order than previously. It is shown that these are only valid when $\beta \to B$, a modified $\beta$-function, and that the equations provide strong constraints on the detailed form of the three loop Yukawa $\beta$-function. ${\cal N}=1$ supersymmetric Wess-Zumino theories are also considered as a special case. It is shown that the metric for the complex couplings in such theories may be restricted to a hermitian form.Comment: 86 pages, version 2, various corrections, section 3 significantly revised, version 3 further minor corrections, as to be published, version 4, some corrections and additional material in sections 2,

On minors of maximal determinant matrices

By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length asymptotically equal to n/2 is excluded from the allowable orders. We make a conjecture regarding a lower bound for sums of squares of minors of maximal determinant matrices, and give evidence in support of the conjecture. We give tables of the values taken by the minors of all maximal determinant matrices of orders up to and including 21 and make some observations on the data. Finally, we describe the algorithms that were used to compute the tables.Comment: 35 pages, 43 tables, added reference to Cohn in v

Habitable Zone Lifetime of Exoplanets around Main Sequence Stars

Funding: Dean's Scholarship at the University of East Anglia.The potential habitability of newly discovered exoplanets is initially assessed by determining whether their orbits fall within the circumstellar habitable zone of their star. However, the habitable zone (HZ) is not static in time or space, and its boundaries migrate outward at a rate proportional to the increase in luminosity of a star undergoing stellar evolution, possibly including or excluding planets over the course of the star’s main sequence lifetime. We describe the time that a planet spends within the HZ as its ‘‘habitable zone lifetime.’’ The HZ lifetime of a planet has strong astrobiological implications and is especially important when considering the evolution of complex life, which is likely to require a longer residence time within the HZ. Here, we present results from a simple model built to investigate the evolution of the ‘‘classic’’ HZ over time, while also providing estimates for the evolution of stellar luminosity over time in order to develop a ‘‘hybrid’’ HZ model. These models return estimates for the HZ lifetimes of Earth and 7 confirmed HZ exoplanets and 27 unconfirmed Kepler candidates. The HZ lifetime for Earth ranges between 6.29 and 7.79 · 109 years (Gyr). The 7 exoplanets fall in a range between ∼1 and 54.72 Gyr, while the 27 Kepler candidate planets’ HZ lifetimes range between 0.43 and 18.8 Gyr. Our results show that exoplanet HD 85512b is no longer within the HZ, assuming it has an Earth analog atmosphere. The HZ lifetime should be considered in future models of planetary habitability as setting an upper limit on the lifetime of any potential exoplanetary biosphere, and also for identifying planets of high astrobiological potential for continued observational or modeling campaigns.Publisher PDFPeer reviewe

Extending Greenstone for Institutional Repositories

We examine the problem of designing a generalized system for building institutional repositories. Widely used schemes such as DSpace are tailored to a particular set of requirements: fixed metadata set; standard view when searching and browsing; pre-determined sequence for depositing items; built-in workflow for vetting new items. In contrast, Fedora builds in flexibility: institutional repositories are just one possible instantiation—however generality incurs a high overhead and uptake has been sluggish. This paper shows how existing components of the Greenstone software can be repurposed to provide a generalized institutional repository that falls between these extremes