In our model of quantum gravity the quantum development of a Cauchy hypersurface is governed by a wave equation derived as the result of a canonical quantization process. To find physically interesting solutions of the wave equation we employ the separation of variables by considering a temporal eigenvalue problem which has a complete countable set of eigenfunctions with positive eigenvalues and also a spatial eigenvalue problem which has a complete set of eigendistributions. Assuming that the Cauchy hypersurface is asymtotically Euclidean we prove that the temporal eigenvalues are also spatial eigenvalues and the product of corresponding eigenfunctions and eigendistributions, which will be smooth functions with polynomial growth, are the physically interesting solutions of the wave equation. We consider these solutions to describe the quantum development of the Cauchy hypersurface.

We derive the alternating arm exponents of the critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions (⊖⊕), (⊖ free) and (free free), and the alternating interior arm exponents.

We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p∈Z_+, is equal to the category O_M(p) of C_1-cofinite M(p)-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since O_M(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)-modules. We also introduce a tensor subcategory O_M(p)^T, graded by an algebraic torus T, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl_2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)-modules, and we prove that both tensor categories O_M(p) and O_M(p)^T are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.

Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum$J$. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from J to $${K=J+\frac{1}{2}, J+1, ... ,}$$ with $${K=\infty}$$ corresponding to the Wehrl map to classical densities. These channels were later recognized as the optimal quantum cloning channels. For each$J$and $${J < K \leqslant \infty}$$ we show that the minimal output entropy for the channels occurs for a J coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.

The superradiant scattering of a scalar field with frequency and angular momentum (ω, m) by a near-extreme Kerr black hole with mass and spin (M, J) was derived in the seventies by Starobinsky, Churilov, Press and Teukolsky. In this paper we show that for frequencies scaled to the superradiant bound the full functional dependence on (ω, m, M, J) of the scattering amplitudes is precisely reproduced by a dual two-dimensional conformal field theory in which the black hole corresponds to a specific thermal state and the scalar field to a specific operator. This striking agreement corroborates a conjectured Kerr/CFT correspondence.