Various approaches to high energy forward scattering in quantum gravity are compared using the eikonal approximation. The massless limit of the eikonal is shown to be equivalent to other approximations for the same process, specifically the semiclassical calculation due to G. 't Hooft and the topological field theory due to H. and E. Verlinde. This comparison clarifies these previous results, as it is seen that the amplitude arises purely from a linearized gravitational interaction. The interpretation of poles in the scattering amplitude is also clarified.

The quantization of a massless conformally coupled scalar field on the 2+1 dimensional Anti de Sitter black hole background is presented. The Green's function is calculated, using the fact that the black hole is Anti de Sitter space with points identified, and taking into account the fact that the black hole spacetime is not globally hyperbolic. It is shown that the Green's function calculated in this way is the Hartle-Hawking Green's function. The Green's function is used to compute T µ ν , which is regular on the black hole horizon, and diverges at the singularity. A particle detector response function outside the horizon is also calculated and shown to be a fermi type distribution. The back-reaction from T µν is calculated exactly and is shown to give rise to a curvature singularity at r = 0 and to shift the horizon outwards. For M = 0 a horizon develops, shielding the singularity. Some speculations about the endpoint of evaporation are discussed. 03.70.+k, 04.50.+h, 97.60.Lf

We calculate the density of states of the 2+1 dimensional BTZ black hole in the micro-and grand-canonical ensembles. Our starting point is the relation between 2+1 dimensional quantum gravity and quantised Chern-Simons theory. In the microcanonical ensemble, we find the Bekenstein-Hawking entropy by relating a Kac-Moody algebra of global gauge charges to a Virasoro algebra with a classical central charge via a twisted Sugawara construction. This construction is valid at all values of the black hole radius. At infinity it gives the asymptotic isometries of the black hole, and at the horizon it gives an explicit form for a set of deformations of the horizon whose algebra is the same Virasoro algebra. In the grand-canonical ensemble we define the partition function by using a surface term at infinity that is compatible with fixing the temperature and angular velocity of the black hole. We then compute the partition function directly in a boundary Wess-Zumino-Witten theory, and find that we obtain the correct result only after we include a source term at the horizon that induces a non-trivial spin-structure on the WZW partition function.

The three dimensional black hole solutions of Ba\~nados, Teitelboim and Zanelli (BTZ) are dimensionally reduced in various different ways. Solutions are obtained to the Jackiw-Teitelboim theory of two dimensional gravity for spinless BTZ black holes, and to a simple extension with a non-zero dilaton potential for black holes of fixed spin. Similar reductions are given for charged black holes. The resulting two dimensional solutions are themselves black holes, and are appropriate for investigating exact ``S-wave'' scattering in the BTZ metrics. Using a different dimensional reduction to the string inspired model of two dimensional gravity, the BTZ solutions are related to the familiar two dimensional black hole and the linear dilaton vacuum.Comment: 12 pages, CTP #2181, January 199

Background Zoonotically transmitted coronaviruses are responsible for three disease outbreaks since 2002, including the current COVID-19 pandemic, caused by SARS-CoV-2. Its efficient transmission and range of disease severity raise questions regarding the contributions of virus-receptor interactions. ACE2 is a host ectopeptidase and the receptor for SARS-CoV-2. Numerous reports describe ACE2 mRNA abundance and tissue distribution; however, mRNA abundance is not always representative of protein levels. Currently, there is limited data evaluating ACE2 protein and its correlation with other SARS-CoV-2 susceptibility factors. Materials and methods We systematically examined the human upper and lower respiratory tract using single-cell RNA sequencing and immunohistochemistry to determine receptor expression and evaluated its association with risk factors for severe COVID-19. Findings Our results reveal that ACE2 protein is highest within regions of the sinonasal cavity and pulmonary alveoli, sites of presumptive viral transmission and severe disease development, respectively. In the lung parenchyma, ACE2 protein was found on the apical surface of a small subset of alveolar type II cells and colocalized with TMPRSS2, a cofactor for SARS-CoV2 entry. ACE2 protein was not increased by pulmonary risk factors for severe COVID-19. Additionally, ACE2 protein was not reduced in children, a demographic with a lower incidence of severe COVID-19. Interpretation These results offer new insights into ACE2 protein localization in the human respiratory tract and its relationship with susceptibility factors to COVID-19.

This paper is concerned with the question of the existence of composition laws in the sum-overhistories approach to relativistic quantum mechanics and quantum cosmology, and its connection with the existence of a canonical formulation. In nonrelativistic quantum mechanics, the propagator is represented by a sum over histories in which the paths move forward in time. The composition law of the propagator then follows from the fact that the paths intersect an intermediate surface of constant time once and only once, and a partition of the paths according to their crossing position may be aftected. In relativistic quantum mechanics, by contrast, the propagators (or Green functions) may be represented by sums over histories in which the paths move backward and forward in time. They therefore intersect surfaces of constant time more than once, and the relativistic composition law, involving a normal derivative term, is not readily recovered. The principal technical aim of this paper is to show that the relativistic composition law may, in fact, be derived directly from a sum over histories by partitioning the paths according to their ftrst crossing position of an intermediate surface. We review the various Green functions of the Klein-Gordon equation, and derive their composition laws. We obtain path-integral representations for all Green functions except the causal one. We use the proper time representation, in which the path integral has the form of a nonrelativistic sum over histories but is integrated over time. The question of deriving the composition laws therefore reduces to the question of factoring the propagators of nonrelativistic quantum mechanics across an arbitrary surface in configuration space. This may be achieved using a known result called the path decomposition expansion (PDX). We give a proof of the PDX using a spacetime lattice definition of the Euclidean propagator. We use the PDX to derive the composition laws of relativistic quantum mechanics from the sum over histories. We also derive canonical representations of all of the Green functions of relativistic quantum mechanics, i, e. , express them in the form (x"~x'), where the [ ix ) I are a complete set of configuration-space eigenstates.These representations make it clear why the Hadamard Green function 6"' does not obey a standard composition law. They also give a hint as to why the causal Green function does not appear to possess a sum-over-histories representation.We discuss the broader implications of our methods and results for quantum cosmology, and parametrized theories generally. We show that there is a close parallel between the existence of a composition law and the existence of a canonical formulation, in that both are dependent on the presence of a timelike Killing vector. We also show why certain naive composition laws that have been proposed in the past for quantum cosmology are incorrect. Our results suggest that the propagation amplitude between three-metrics in quantum cosmology, as constructed from the sum over histories, do...

We compute the canonical partition function of 2+1 dimensional de Sitter space using the Euclidean SU (2) × SU (2) Chern-Simons formulation of 3d gravity with a positive cosmological constant. Firstly, we point out that one can work with a ChernSimons theory with level k = l/4G, and its representations are therefore unitary for integer values of k. We then compute explicitly the partition function using the standard character formulae for SU (2) WZW theory and find agreement, in the large k limit, with the semiclassical result. Finally, we note that the de Sitter entropy can also be obtained as the degeneracy of states of representations of a Virasoro algebra with c = 3l/2G.

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