The Construction of Spinor Fields on Manifolds with Smooth Degenerate Metrics

We examine some of the subtleties inherent in formulating a theory of spinors on a manifold with a smooth degenerate metric. We concentrate on the case where the metric is singular on a hypersurface that partitions the manifold into Lorentzian and Euclidean domains. We introduce the notion of a complex spinor fibration to make precise the meaning of continuity of a spinor field and give an expression for the components of a local spinor connection that is valid in the absence of a frame of local orthonormal vectors. These considerations enable one to construct a Dirac equation for the discussion of the behavior of spinors in the vicinity of the metric degeneracy. We conclude that the theory contains more freedom than the spacetime Dirac theory and we discuss some of the implications of this for the continuity of conserved currents.Comment: 24 pages, LaTeX (RevTeX 3.0, no figures), To appear in J. Math. Phy

Finite action Yang-Mills solutions on the group manifold

We demonstrate that the left (and right) invariant Maurer-Cartan forms for any semi-simple Lie group enable one to construct solutions of the Yang-Mills equations on the group manifold equipped with the natural Cartan-Killing metric. For the unitary unimodular groups the Yang-Mills action integral is finite for such solutions. This is explicitly exhibited for the case of SU(3)

Casinos in context : the impacts of stand-alone casino development on urban neighborhoods

Thesis (M.C.P.)--Massachusetts Institute of Technology, Dept. of Urban Studies and Planning, 2007.Includes bibliographical references (p. 169-174).As the stigma of gambling fades and governments seek more sources of revenue the urban casino is becoming more common. Many of these are legalized to operate with limited competition in their state, standing alone in their respective city. while the general, diffuse impacts of casinos have been well studied, the paucity of casinos in urban settings has left a gap in the understanding of how stand-alone casinos affect their urban context. This thesis seeks to fill this gap by analyzing three of the oldest stand-alone urban casinos in the United States: Harrah's Casino in Joliet, Harrah's Casino in New Orleans, and Greektown Casino in Detroit. The thesis analyzes the impact of each of these casinos through the lens of five categories of impact: urban design of the neighborhood, city investment, real estate development, street-level commerce, and the residential community. The resulting analysis finds that casinos tend to be insular entities that do not impact much outside their own footprint. Their greatest potential for external impacts is an ability to act as an anchor for street-level commerce in a neighborhood if designed properly, a trend that has been observed for some time in resort communities with fake interior streets. The essay concludes with recommendations on how a city might engender a similar phenomenon.by Luke J. Schray.M.C.P

Quaternionic Spin

We rewrite the standard 4-dimensional Dirac equation in terms of quaternionic 2-component spinors, leading to a formalism which treats both massive and massless particles on an equal footing. The resulting unified description has the correct particle spectrum to be a generation of leptons, with the correct number of spin/helicity states. Furthermore, precisely three such generations naturally combine into an octonionic description of the 10-dimensional massless Dirac equation, as discussed in previous work.Comment: LaTeX2e, 15 pages, 1 PS figure; to appear in Clifford '99 proceeding

Formulation of polylactide-co-glycolic acid nanospheres for encapsulation and sustained release of poly(ethylene imine)-poly(ethylene glycol) copolymers complexed to oligonucleotides

Antisense oligonucleotides (AOs) have been shown to induce dystrophin expression in muscles cells of patients with Duchenne Muscular Dystrophy (DMD) and in the mdx mouse, the murine model of DMD. However, ineffective delivery of AOs limits their therapeutic potential. Copolymers of cationic poly(ethylene imine) (PEI) and non-ionic poly(ethylene glycol) (PEG) form stable nanoparticles when complexed with AOs, but the positive surface charge on the resultant PEG-PEI-AO nanoparticles limits their biodistribution. We adapted a modified double emulsion procedure for encapsulating PEG-PEI-AO polyplexes into degradable polylactide-co-glycolic acid (PLGA) nanospheres. Formulation parameters were varied including PLGA molecular weight, ester end-capping, and sonication energy/volume. Our results showed successful encapsulation of PEG-PEI-AO within PLGA nanospheres with average diameters ranging from 215 to 240 nm. Encapsulation efficiency ranged from 60 to 100%, and zeta potential measurements confirmed shielding of the PEG-PEI-AO cationic charge. Kinetic measurements of 17 kDa PLGA showed a rapid burst release of about 20% of the PEG-PEI-AO, followed by sustained release of up to 65% over three weeks. To evaluate functionality, PEG-PEI-AO polyplexes were loaded into PLGA nanospheres using an AO that is known to induce dystrophin expression in dystrophic mdx mice. Intramuscular injections of this compound into mdx mice resulted in over 300 dystrophin-positive muscle fibers distributed throughout the muscle cross-sections, approximately 3.4 times greater than for injections of AO alone. We conclude that PLGA nanospheres are effective compounds for the sustained release of PEG-PEI-AO polyplexes in skeletal muscle and concomitant expression of dystrophin, and may have translational potential in treating DMD

The general classical solution of the superparticle

The theory of vectors and spinors in 9+1 dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra \Cl(9,1). The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a $3 \times 3$ Grassmann, octonionic, Jordan matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in terms of Jordan products.Comment: 11 pages, REVTe

Non-Riemannian Gravity and the Einstein-Proca System

We argue that all Einstein-Maxwell or Einstein-Proca solutions to general relativity may be used to construct a large class of solutions (involving torsion and non-metricity) to theories of non-Riemannian gravitation that have been recently discussed in the literature.Comment: 9 pages Plain Tex (No Figures), Letter to Editor Classical and Quantum Gravit

Black Holes with Weyl Charge and Non-Riemannian Waves

A simple modification to Einstein's theory of gravity in terms of a non-Riemannian connection is examined. A new tensor-variational approach yields field equations that possess a covariance similar to the gauge covariance of electromagnetism. These equations are shown to possess solutions analogous to those found in the Einstein-Maxwell system. In particular one finds gravi-electric and gravi-magnetic charges contributing to a spherically symmetric static Reissner-Nordstr\"om metric. Such Weyl ``charges'' provide a source for the non-Riemannian torsion and metric gradient fields instead of the electromagnetic field. The theory suggests that matter may be endowed with gravitational charges that couple to gravity in a manner analogous to electromagnetic couplings in an electromagnetic field. The nature of gravitational coupling to spinor matter in this theory is also investigated and a solution exhibiting a plane-symmetric gravitational metric wave coupled via non-Riemannian waves to a propagating spinor field is presented.Comment: 18 pages Plain Tex (No Figures), Classical and Quantum Gravit

Octonionic representations of Clifford algebras and triality

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest \perm_3 \times SO(8) structure in this framework.Comment: 33 page