A New Look at the Ashtekar-Magnon Energy Condition

In 1975, Ashtekar and Magnon showed that an energy condition selects a unique quantization procedure for certain observers in general, curved spacetimes. We generalize this result in two important ways, by eliminating the need to assume a particular form for the (quantum) Hamiltonian, and by considering the surprisingly nontrivial extension to nonminimal coupling.Comment: REVTeX, 10 page

Octonions, E6, and Particle Physics

In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes or quaternions. The remaining, exceptional Jordan algebra can be described by 3x3 Hermitian matrices over the octonions. We first review properties of the octonions and the exceptional Jordan algebra, including our previous work on the octonionic Jordan eigenvalue problem. We then examine a particular real, noncompact form of the Lie group E6, which preserves determinants in the exceptional Jordan algebra. Finally, we describe a possible symmetry-breaking scenario within E6: first choose one of the octonionic directions to be special, then choose one of the 2x2 submatrices inside the 3x3 matrices to be special. Making only these two choices, we are able to describe many properties of leptons in a natural way. We further speculate on the ways in which quarks might be similarly encoded.Comment: 13 pages; 6 figures; TonyFest plenary talk (York 2008

Quaternionic eigenvalue problem

We discuss the (right) eigenvalue equation for $\mathbb{H}$, $\mathbb{C}$ and $\mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the quaternionic problem into an {\em equivalent} real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.Comment: 13 pages, AMS-Te

How Black Holes Form in High Energy Collisions

We elucidate how black holes form in trans-Planckian collisions. In the rest frame of one of the incident particles, the gravitational field of the other, which is rapidly moving, looks like a gravitational shock wave. The shock wave focuses the target particle down to a much smaller impact parameter. In turn, the gravitational field of the target particle captures the projectile when the resultant impact parameter is smaller than its own Schwarzschild radius, forming a black hole. One can deduce this by referring to the original argument of escape velocities exceeding the speed of light, which Michell and Laplace used to discover the existence of black holes.Comment: 8 pages, 3 .eps figures, essa

General solutions of Einstein's spherically symmetric gravitational equations with junction conditions

Einstein's spherically symmetric interior gravitational equations are investigated. Following Synge's procedure, the most general solution of the equations is furnished in case $T^{1}_{1}$ and $T^{4}_{4}$ are prescribed. The existence of a total mass function, $M(r,t)$, is rigorously proved. Under suitable restrictions on the total mass function, the Schwarzschild mass $M(r,t)=m$, implicitly defines the boundary of the spherical body as $r=B(t)$. Both Synge's junction conditions as well as the continuity of the second fundamental form are examined and solved in a general manner. The weak energy conditions for an \emph{arbitrary boost} are also considered. The most general solution of the spherically symmetric anisotropic fluid model satisfying both junction conditions is furnished. In the final section, various exotic solutions are explored using the developed scheme including gravitational instantons, interior $T$-domains and $D$-dimensional generalizations.Comment: 23 pages, 1 figure, uses AMS packages. Updated version has corrected typos as well as added comments and extension regarding ISLD junction conditions. Accepted for publication in Journal of Mathematical Physic

Distributional Modes for Scalar Field Quantization

We propose a mode-sum formalism for the quantization of the scalar field based on distributional modes, which are naturally associated with a slight modification of the standard plane-wave modes. We show that this formalism leads to the standard Rindler temperature result, and that these modes can be canonically defined on any Cauchy surface.Comment: 15 pages, RevTe

Diffusion and social networks: revisiting medical innovation with agents

the classic study on diffusion of Tetracycline by Coleman, Katz and Menzel (1966). Medical Innovation articulates how different patterns of interpersonal communications can influence the diffusion process at different stages of adoption. In their pioneering study, individual network (discussion, friendship or advice) was perceived as a set of disjointed pairs, and the extent of influences were therefore, evaluated for pairs of individuals. Given the existence of overlapping networks and consequent influences on doctors’ adoption decisions, the complexity of actual events was not captured by pair analysis. Subsequent reanalyses (Burt 1987, Strang and Tuma 1993, Valente 1995, Van den Bulte and Lilien 2001) failed to capture the complexity involved in the diffusion process and had a static exposure of the network structure. In this paper, for the first time, we address these limitations by combining Agent-Based Modeling (ABM) and network analysis. Based on the findings of Coleman et. al. (1966) study, we develop a diffusion model, Gammanym. Using SMALLTALK programming language, Gammanym is developed with CORMAS platform under Visual Works environment. The medical community is portrayed in an 8 X 8 spatial grid. The unit cell captures three different locations for professional interactions: practices, hospitals, and conference centers, randomly located over the spatial grid. Two social agents- Doctor and Laboratory are depicted in the model. Doctors are the principal agents in the diffusion process and are initially located at their respective practices. A doctor’s adoption decision is influenced by a random friendship network, and a professional network created through discussions with office colleagues, or hospital visits or conference attendance. A communicating agent, Laboratory, on the other hand, influences doctors’ adoption decisions by sending information through multiple channels: medical representatives or detailman visiting practices, journals sent to doctors’ practices and commercial flyers available during conferences. Doctors’ decisions to adopt a new drug involve interdependent local interactions among different entities in Gammanym. The cumulative adoption curves (Figure 1) are derived for three sets of initial conditions, based on which network topology and evolution of uptake are analyzed. The three scenarios are specified to evaluate the degree of influences by different factors in the diffusion process: baseline scenario with one seed (initial adopter), one detailman and one journal; heavy media scenario with one seed but increasing degrees of external influence, with five detailman and four journals; and integration scenario with one seed, without any external influence from the laboratory

Diffusion and social networks: revisiting medical innovation with agents

the classic study on diffusion of Tetracycline by Coleman, Katz and Menzel (1966). Medical Innovation articulates how different patterns of interpersonal communications can influence the diffusion process at different stages of adoption. In their pioneering study, individual network (discussion, friendship or advice) was perceived as a set of disjointed pairs, and the extent of influences were therefore, evaluated for pairs of individuals. Given the existence of overlapping networks and consequent influences on doctors’ adoption decisions, the complexity of actual events was not captured by pair analysis. Subsequent reanalyses (Burt 1987, Strang and Tuma 1993, Valente 1995, Van den Bulte and Lilien 2001) failed to capture the complexity involved in the diffusion process and had a static exposure of the network structure. In this paper, for the first time, we address these limitations by combining Agent-Based Modeling (ABM) and network analysis. Based on the findings of Coleman et. al. (1966) study, we develop a diffusion model, Gammanym. Using SMALLTALK programming language, Gammanym is developed with CORMAS platform under Visual Works environment. The medical community is portrayed in an 8 X 8 spatial grid. The unit cell captures three different locations for professional interactions: practices, hospitals, and conference centers, randomly located over the spatial grid. Two social agents- Doctor and Laboratory are depicted in the model. Doctors are the principal agents in the diffusion process and are initially located at their respective practices. A doctor’s adoption decision is influenced by a random friendship network, and a professional network created through discussions with office colleagues, or hospital visits or conference attendance. A communicating agent, Laboratory, on the other hand, influences doctors’ adoption decisions by sending information through multiple channels: medical representatives or detailman visiting practices, journals sent to doctors’ practices and commercial flyers available during conferences. Doctors’ decisions to adopt a new drug involve interdependent local interactions among different entities in Gammanym. The cumulative adoption curves (Figure 1) are derived for three sets of initial conditions, based on which network topology and evolution of uptake are analyzed. The three scenarios are specified to evaluate the degree of influences by different factors in the diffusion process: baseline scenario with one seed (initial adopter), one detailman and one journal; heavy media scenario with one seed but increasing degrees of external influence, with five detailman and four journals; and integration scenario with one seed, without any external influence from the laboratory

Stochastic Tachyon Fluctuations, Marginal Deformations and Shock Waves in String Theory

Starting with exact solutions to string theory on curved spacetimes we obtain deformations that represent gravitational shock waves. These may exist in the presence or absence of sources. Sources are effectively induced by a tachyon field that randomly fluctuates around a zero condensate value. It is shown that at the level of the underlying conformal field theory (CFT) these deformations are marginal and moreover all \a'-corrections are taken into account. Explicit results are given when the original undeformed 4-dimensional backgrounds correspond to tensor products of combinations of 2-dimensional CFT's, for instance SL(2,R)/R \times SU(2)/U(1).Comment: 26 pages, harvmac, no figures. Very minor modifications, and in addition conditions (B.3) and (B.4) were also obtained using beta-function equations. Version to appear in Phys. Rev.