Induced Gravity II: Grand Unification

As an illustration of a renormalizable, asymptotically-free model of induced gravity, we consider an $SO(10)$ gauge theory interacting with a real scalar multiplet in the adjoint representation. We show that dimensional transmutation can occur, spontaneously breaking $SO(10)$ to $SU(5){\otimes}U(1),$ while inducing the Planck mass and a positive cosmological constant, all proportional to the same scale $v$. All mass ratios are functions of the values of coupling constants at that scale. Below this scale (at which the Big Bang may occur), the model takes the usual form of Einstein-Hilbert gravity in de Sitter space plus calculable corrections. We show that there exist regions of parameter space in which the breaking results in a local minimum of the effective action, and a {\bf positive} dilaton $(\hbox{mass})^2$ from two-loop corrections associated with the conformal anomaly. Furthermore, unlike the singlet case we considered previously, some minima lie within the basin of attraction of the ultraviolet fixed point. Moreover, the asymptotic behavior of the coupling constants also lie within the range of convergence of the Euclidean path integral, so there is hope that there will be candidates for sensible vacua. Although open questions remain concerning unitarity of all such renormalizable models of gravity, it is not obvious that, in curved backgrounds such as those considered here, unitarity is violated. In any case, any violation that may remain will be suppressed by inverse powers of the reduced Planck mass.Comment: 44 pages, 5 figures, 2 tables. v2 has new discussion concerning stability of SSB plus related appendix. Additional references added. v3 is version to be published; contains minor revision

Zero modes in de Sitter background

There are five well-known zero modes among the fluctuations of the metric of de~Sitter (dS) spacetime. For Euclidean signature, they can be associated with certain spherical harmonics on the $S^4$ sphere, viz., the vector representation $\bf5$ of the global $SO(5)$ isometry. They appear, for example, in the perturbative calculation of the on-shell effective action of dS space, as well as in models containing matter fields. These modes are shown to be associated with collective modes of $S^4$ corresponding to certain coherent fluctuations. When dS space is embedded in flat five dimensions $E^5,$ they may be seen as a legacy of translation of the center of the $S^4$ sphere. Rigid translations of the $S^4$-sphere on $E^5$ leave the classical action invariant but are unobservable displacements from the point of view of gravitational dynamics on $S^4.$ Thus, unlike similar moduli, the center of the sphere is not promoted to a dynamical degree of freedom. As a result, these zero modes do not signify the possibility of physically realizable fluctuations or flat directions for the metric of dS space. They are not associated with Killing vectors on $S^4$ but can be with certain non-isometric, conformal Killing forms that locally correspond to a rescaling of the volume element $dV_4.$ For convenience, we frame our discussion in the context of renormalizable gravity, but the conclusions apply equally to the corresponding zero modes in Einstein gravity. We expect that these zero modes will be present to all orders in perturbation theory. They will occur for Lorentzian signature as well, so long as the hyperboloid $H^4$ is locally stable, but there remain certain infrared issues that need to be clarified. We conjecture that they will appear in any gravitational theory having dS background as a locally stable solution of the effective action, regardless of whether additional matter is included.Comment: v4, 28pages, no figures; final journal form, minor changes in text and refs from v

A study of wide bandwidth directional patterns from circular arrays

Passive radar warning receivers use amplitude comparison direction-finding to determine the angle of arrival of incoming signals. This type of direction-finding system requires wideband azimuth beams that do not change in shape with frequency. The subject of this thesis is the synthesis of wideband beams using a circular array. In the first part of this thesis the excitation of the circular array is analysed using the concept of 'phase modes': the orthogonal terms of a spatial Fourier series. If the variations in phase and amplitude of these modes with frequency are corrected, azimuth patterns formed from these modes are instantaneously wideband. Pattern synthesis uses the principle of linear array equivalence, allowing us to apply low sidelobe techniques developed for linear arrays to the phase modes. The design of the experimental system, operating over the frequency range 8 to 12 GHz, is subsequently presented. The characteristics of the phase modes excited on a four element monopoles array were evaluated, showing that the array could be used to form a low sidelobe beams. The initial beamformer design used a Butler matrix constructed from microwave directional couplers to excite the phase modes. By utilising microstrip compensation networks, the variation in phase and amplitude of these modes with frequency is corrected. Multiple beams are formed from the compensated modes by a second matrix. Both theoretical and measured results showed that the phase and amplitude errors introduced by this complicated network were unacceptable. Multiple beams were not demonstrated in this study. A simplified matrix design was developed to demonstrate a low sidelobe (-28 dB) beam at a single fi*equency. A weighted corporate feed was developed to demonstrate instantaneously wideband pattern synthesis of a single beam. The element excitation required to form a low sidelobe pattern was calculated using phase mode theory. The frequency-dependent element excitation was practically realised using the microstrip networks. Anechoic chamber measurements of the synthesised wideband beam showed that the variation in the -3 dB beamwidth across the band 8 to 12 GHz was less than ±3°. The sidelobe level was below -20 dB. Theoretical calculations limit the frequency bandwidth for this synthesis technique to about one octave. A wideband sin(Nx)/Nsin(x) pattern was also produced to demonstrate the versatility of this synthesis technique

Mixed mode oscillations in a conceptual climate model

Much work has been done on relaxation oscillations and other simple oscillators in conceptual climate models. However, the oscillatory patterns in climate data are often more complicated than what can be described by such mechanisms. This paper examines complex oscillatory behavior in climate data through the lens of mixed-mode oscillations. As a case study, a conceptual climate model with governing equations for global mean temperature, atmospheric carbon, and oceanic carbon is analyzed. The nondimensionalized model is a fast/slow system with one fast variable (corresponding to ice volume) and two slow variables (corresponding to the two carbon stores). Geometric singular perturbation theory is used to demonstrate the existence of a folded node singularity. A parameter regime is found in which (singular) trajectories that pass through the folded node are returned to the singular funnel in the limiting case where $\epsilon = 0$. In this parameter regime, the model has a stable periodic orbit of type $1^s$ for some $s>0$. To our knowledge, it is the first conceptual climate model demonstrated to have the capability to produce an MMO pattern.Comment: 28 pages, 11 figure

The non-linear transient behavior of second, third and fourth order phase-locked loops

Non-linear transient behavior of second, third, and fourth order phase-locked loop

Representation-theoretic derivation of the Temperley-Lieb-Martin algebras

Explicit expressions for the Temperley-Lieb-Martin algebras, i.e., the quotients of the Hecke algebra that admit only representations corresponding to Young diagrams with a given maximum number of columns (or rows), are obtained, making explicit use of the Hecke algebra representation theory. Similar techniques are used to construct the algebras whose representations do not contain rectangular subdiagrams of a given size.Comment: 12 pages, LaTeX, to appear in J. Phys.

Algebras in Higher Dimensional Statistical Mechanics - the Exceptional Partition (MEAN Field) Algebras

We determine the structure of the partition algebra $P_n(Q)$ (a generalized Temperley-Lieb algebra) for specific values of Q \in \C, focusing on the quotient which gives rise to the partition function of $n$ site $Q$-state Potts models (in the continuous $Q$ formulation) in arbitrarily high lattice dimensions (the mean field case). The algebra is non-semi-simple iff $Q$ is a non-negative integer less than $n$. We determine the dimension of the key irreducible representation in every specialization.Comment: 4 page