8,119 research outputs found
Implications of Conformal Invariance for Quantum Field Theories in
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 , 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 on the space of couplings and give
rise to gradient flow like equations for , are derived taking into account
the role of lower dimension operators. The results for infinitesimal Weyl
rescaling are integrated to finite rescalings to a form which
involves running couplings 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
-functions for a general scalar/fermion theory are obtained and the
three loop contribution to the metric 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 , a
modified -function, and that the equations provide strong constraints on
the detailed form of the three loop Yukawa -function.
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
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