The Emergence of Anticommuting Coordinates and the Dirac-Ramond-Kostant operators

The history of anticommuting coordinates is decribed.Comment: 14 pages, Contribution to the Proceedings of The Gunnar Nordstr\"om Symposium on Theoretical Physics - The Physics of Extra Dimension

Spacetime Encodings IV - The Relationship between Weyl Curvature and Killing Tensors in Stationary Axisymmetric Vacuum Spacetimes

The problem of obtaining an explicit representation for the fourth invariant of geodesic motion (generalized Carter constant) of an arbitrary stationary axisymmetric vacuum spacetime generated from an Ernst Potential is considered. The coupling between the non-local curvature content of the spacetime as encoded in the Weyl tensor, and the existence of a Killing tensor is explored and a constructive, algebraic test for a fourth order Killing tensor suggested. The approach used exploits the variables defined for the B\"{a}ckland transformations to clarify the relationship between Weyl curvature, constants of geodesic motion, expressed as Killing tensors, and the solution generation techniques. A new symmetric non-covariant formulation of the Killing equations is given. This formulation transforms the problem of looking for fourth-order Killing tensors in 4D into one of looking for four interlocking two-manifolds admitting fourth-order Killing tensors in 2D.Comment: 15 page

Frontier Economics: Why Entrepreneurial Capitalism Is Needed Now More Than Ever

Examines the roles of imitative and innovative economic growth and public policy in developed and developing economies in the twentieth century. Outlines role of uncertainty and economic freedom as drivers of innovation, competition, and continued growth

Maximally Supersymmetric Yang-Mills Theory. The Story of N = 4 Yang-Mills Theory

This is a personally colored account of the history behind N=4 Yang-Mills Theory.Comment: Talk given at "60 years of Yang-Mills Theory", Singapore May 25-28, 2015. 15 page

Particle Physics as Representations of the Poincare Algebra

Eugene Wigner showed already in 1939 that the elementary particles are related to the irreducible representations of the Poincare algebra. In the light-cone frame formulation of quantum field theory one can extend these representations to depend also on a coupling constant. The representations then become non-linear and contain the interaction terms which are shown to have strong uniqueness. Extending the algebra to supersymmetry it is shown that two field theories stick out, N=4 Yang-Mills and N=8 Supergravity and their higher dimensional analogues. I also discuss string theory from this starting point.Comment: Lecture presented at the Poincare Symposium held in Brussels on October 8-9, 200

Maximal supersymmetry and exceptional groups

The article is a tribute to my old mentor, collaborator and friend Murray Gell-Mann. In it I describe work by Pierre Ramond, Sung-Soo Kim and myself where we describe the N = 8 Supergravity in the light-cone formalism. We show how the Cremmer-Julia E7(7) non-linear symmetry is implemented and how the full supermultiplet is a representation of the E7(7) symmetry. I also show how the E7(7) symmetry is a key to understand the higher order couplings in the theory and is very useful when we discuss possible counterterms for this theory.Comment: Proceedings of Conference in Honour of Murray Gell-Mann's 80th Birthda

Spacetime Encodings III - Second Order Killing Tensors

This paper explores the Petrov type D, stationary axisymmetric vacuum (SAV) spacetimes that were found by Carter to have separable Hamilton-Jacobi equations, and thus admit a second-order Killing tensor. The derivation of the spacetimes presented in this paper borrows from ideas about dynamical systems, and illustrates concepts that can be generalized to higher- order Killing tensors. The relationship between the components of the Killing equations and metric functions are given explicitly. The origin of the four separable coordinate systems found by Carter is explained and classified in terms of the analytic structure associated with the Killing equations. A geometric picture of what the orbital invariants may represent is built. Requiring that a SAV spacetime admits a second-order Killing tensor is very restrictive, selecting very few candidates from the group of all possible SAV spacetimes. This restriction arises due to the fact that the consistency conditions associated with the Killing equations require that the field variables obey a second-order differential equation, as opposed to a fourth-order differential equation that imposes the weaker condition that the spacetime be SAV. This paper introduces ideas that could lead to the explicit computation of more general orbital invariants in the form of higher-order Killing Tensors.Comment: 9 page

Balancing stress and recovery in sports

To reach elite level in sports, athletes have to start their intensive and time-consuming training at an early stage. The road to the top is a stressful one, not only due to the physiological stress of training, but also caused by psychological and social stress. In order to improve performance athletes continuously challenge their personal boundaries. This may lead to a local or general overload of the human body that results into injuries, illnesses and overtraining. Overtraining is characterized by an unexplained decrement in sport-specific performance and is often coupled with symptoms such as increased fatigue, poor concentration, disturbed mood, and altered eating and sleeping patterns. Full recovery may take months to years. Monitoring stress and recovery may help to optimize performance and prevent a local or general overload. Clinical measurements showed that mood state and hormonal responses to a double maximal exercise protocol provided valuable information to confirm the diagnosis of overtraining.

Orbital-Peierls State in NaTiSi2O6

Does the quasi one-dimensional titanium pyroxene NaTiSi2O6 exhibit the novel {\it orbital-Peierls} state? We calculate its groundstate properties by three methods: Monte Carlo simulations, a spin-orbital decoupling scheme and a mapping onto a classical model. The results show univocally that for the spin and orbital ordering to occur at the same temperature --an experimental observation-- the crystal field needs to be small and the orbitals are active. We also find that quantum fluctuations in the spin-orbital sector drive the transition, explaining why canonical bandstructure methods fail to find it. The conclusion that NaTiSi2O6 shows an orbital-Peierls transition is therefore inevitable.Comment: 4 pages, 3 figure