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Abstract. Spin foam vertex amplitudes are the key ingredient of spin foam models for quantum gravity. These fall into the realm of discretized path integral, and can be seen as generalized lattice gauge theories. They can be seen as an attempt at a 4-dimensional generalization of the Ponzano–Regge model for 3d quantum gravity. We motivate and review the construction of the vertex amplitudes of recent spin foam models, giving two different and complementary perspectives of this construction. The first proceeds by extracting geometric configurations from a topological theory of the BF type, and can be seen to be in the tradition of the work of Barrett, Crane, Freidel and Krasnov. The second keeps closer contact to the structure of Loop Quantum Gravity and tries to identify an appropriate set of constraints to define a Lorentz-invariant interaction of its quanta of space. This approach is in the tradition of the work of Smolin, Markopoulous, Engle, Pereira, Rovelli and Livine. Key words: spin foam models; discrete quantum gravity; generalized lattice gauge theory 2010 Mathematics Subject Classification: 81T25; 81T45 1

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Abstract: If one is willing to consider the current cosmic microwave back ground temperature as a quantum gravitational effect of the evolving primordial cosmic black hole general theory of relativity and quantum mechanics can be combined into a ‘scale independent ’ true unified model of quantum gravity. It can be suggested that, universe may or may not be a black hole, Hubble volume can be considered as a growing and light speed rotating black hole at 99 % confidence level. With this proposal galactic rotational curves, cosmic dynamic stability can be understood. By considering the ‘Stoney mass’ as the initial mass of the baby Hubble volume, past and current physical and thermal parameters of the cosmic black hole can be understood. Isotropy and anisotropy of CMB can be understood. If rate of decrease in current ‘Hubble’s constant is very small and is beyond the scope of current experimental verification, then the two possible states are: a) current ‘Hub-ble’s constant is decreasing at a very slow rate and current universe is expanding at a very slow rate and b) at present there is no ‘observable ’ cosmic expansion or acceleration. To understand the ground reality of current cosmic rate of expansion, sensitivity and accuracy of current methods of estimating the magnitudes of current CMBR temperature and current Hubble constant must be improved and alternative methods must be developed. If it is true that galaxy constitutes so many stars, each star constitutes so many hydrogen atoms and light is coming from the excited electron of galactic hydrogen atom, then considering redshift as an index of 'whole galaxy ' receding may not be reasonable. During cosmic evolution, at any time in the past, in hydrogen atom emitted photon energy was always inversely proportional to the CMBR temperature. Thus past

a double–scaling limit for tensor models:

Abstract By considering the strength of Schwarzschild interaction as ‘unity ’ and by considering squared Avogadro number as a suitable scaling factor, in the previously published papers the authors made an attempt to understand the basics of nuclear physics and strong interaction. In this paper an attempt is made fit the magnitude of the gravitational constant with nuclear binding energy data of naturally occurring stable atomic nuclides starting from Z=30 to 92. Characteristic binding potential can be taken as ()0 19.5 to 19.7 MeVB ≅. Stable atomic nuclides can be selected in such a way that, ratio of binding energy of the nuclide and characteristic binding potential is close to the proton number of that nuclide. Accuracy of the gravitational constant mainly depends on the selected number of stable atomic nuclides which in turn depends on the accuracy of the assumed binding potential. Very interesting observation is that, () [] ()20 0 01 4sB e Rα α πε ≅ − + where sα is the strong coupling constant, α is the fine structure ratio and 0R is the characteristic nuclear size (1.20 to 1.25) fm. If 0 19.6 MeV, B ≅ 3-1-26.68541 11 m.kg.secG E ≅ − and if 10