On relativistic spin network vertices

Barrett and Crane have proposed a model of simplicial Euclidean quantum gravity in which a central role is played by a class of Spin(4) spin networks called "relativistic spin networks" which satisfy a series of physically motivated constraints. Here a proof is presented that demonstrates that the intertwiner of a vertex of such a spin network is uniquely determined, up to normalization, by the representations on the incident edges and the constraints. Moreover, the constraints, which were formulated for four valent spin networks only, are extended to networks of arbitrary valence, and the generalized relativistic spin networks proposed by Yetter are shown to form the entire solution set (mod normalization) of the extended constraints. Finally, using the extended constraints, the Barrett-Crane model is generalized to arbitrary polyhedral complexes (instead of just simplicial complexes) representing spacetime. It is explained how this model, like the Barret-Crane model can be derived from BF theory by restricting the sum over histories to ones in which the left handed and right handed areas of any 2-surface are equal. It is known that the solutions of classical Euclidean GR form a branch of the stationary points of the BF action with respect to variations preserving this condition.Comment: 15 pages, one postscript figure (uses psfig

The Indefinite Logarithm, Logarithmic Units, and the Nature of Entropy

We define the indefinite logarithm [log x] of a real number x>0 to be a mathematical object representing the abstract concept of the logarithm of x with an indeterminate base (i.e., not specifically e, 10, 2, or any fixed number). The resulting indefinite logarithmic quantities naturally play a mathematical role that is closely analogous to that of dimensional physical quantities (such as length) in that, although these quantities have no definite interpretation as ordinary numbers, nevertheless the ratio of two of these entities is naturally well-defined as a specific, ordinary number, just like the ratio of two lengths. As a result, indefinite logarithm objects can serve as the basis for logarithmic spaces, which are natural systems of logarithmic units suitable for measuring any quantity defined on a logarithmic scale. We illustrate how logarithmic units provide a convenient language for explaining the complete conceptual unification of the disparate systems of units that are presently used for a variety of quantities that are conventionally considered distinct, such as, in particular, physical entropy and information-theoretic entropy.Comment: Manuscript of a 15 pp. review article. Suggestions for additional appropriate references to relevant prior work are solicited from the communit

Are some forecasters' probability assessments of macro variables better than those of others?

We apply the bootstrap test of DíAgostino et al. (2012) to determine whether some forecasters are able to make superior probability assessments to others. In contrast to the findings of DíAgostino et al. (2012) for point predictions, there is more evidence that some individuals really are better than others. The testing procedure controls for the different economic conditions the forecasters may face, given that each individual responds to only a subset of the surveys. One possible explanation for the different findings for point predictions and histograms is explored: that newcomers may make less accurate histogram forecasts than experienced respondents given the greater complexity of the task

Two Recent Results on B Physics from CDF

Preliminary results from two recent CDF b physics analysis are presented. The first obtains sin(2beta) = 0.79 + 0.41 -0.44 from a measurement of the asymmetry in B0, B0bar to J/psi K_short decays, providing the best direct indication so far that CP invariance is violated in the b sector. The second obtains new results on the parity even (A_0 and A_par) and odd (A_perp) polarization amplitudes from full angular analyses of B0 to J/psi K*0 and B_s to J/psi phi decays.Comment: 8 pages, 4 figures; presented at the 34th Recontres de Moriond, Les Arcs, 1800, France, 13-20 March 199

The Virasoro Algebra and Some Exceptional Lie and Finite Groups

We describe a number of relationships between properties of the vacuum Verma module of a Virasoro algebra and the automorphism group of certain vertex operator algebras. These groups include the Deligne exceptional series of simple Lie groups and some exceptional finite simple groups including the Monster and Baby Monster.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 2006, Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA