In the present work we show that the widely believed pathology of the non-unitarity of anisotropic quantum cosmological models cannot be a generic problem. We exhibit a non trivial example, a Bianchi-I model with an ultrarelativistic fluid, that has a well behaved time independent norm. We also show that a suitable operator ordering should produce time independent norms for the wave packets in the case of other more realistic fluids as well.In the absence of a generally accepted quantum theory of general relativity, quantum mechanical principles are applied to many individual gravitational systems. Cosmological models are certainly amongst the fields where this kind of quantization finds an application. The universe at its early stage of evolution, at an energy scale where classical general relativity loses its viability, requires a quantum description. For some excellent reviews, we refer to [1,2]. Quantum Cosmology has, however, many issues yet to be resolved. As time is a coordinate in a relativistic theory of gravity, there is a problem of the identification of a suitable time parameter against which the evolution of the universe would be described [3][4][5][6]. Moreover, the interpretation of the wave function faces a challenge in quantum cosmology. The Copenhagen interpretation fails as there is no exterior observer for the system. There are attempts in this directi on with a many world interpretation and with Bohmian trajectories [7]. Problems regarding the imposition of proper boundary conditions are there as well [1].The present work deals with another widely known problem, the alleged non-unitarity of the anisotropic quantum cosmological models.The corresponding Hamiltonian, although hermitian, is not self-adjoint. The norm of the wave packet is thus time dependent and hence there is a non-conservation of probability. It may be argued that the observed universe is isotropic but one is not really certain whether the very early universe, beyond the Planck scale, is actually isotropic or not! Furthermore, this feature definitely makes the scheme of quantization unreliable. For a very recent review, we refer to the work of Pinto-Neto and Fabris [7].There is a scheme of quantization of a cosmological model with a matter field, namely a perfect fluid. Following Schutz's formalism, where the fluid variables are given dynamical degrees of freedom [8,9], the relevant action can be written in terms of the metric tensor components representing the gravity sector and some *

The Eigenstate Thermalization Hypothesis (ETH) provides a way to understand how an isolated quantum mechanical system can be approximated by a thermal density matrix. We find a class of operators in (1+1)-d conformal field theories, consisting of quasi-primaries of the identity module, which satisfy the hypothesis only at the leading order in large central charge. In the context of subsystem ETH, this plays a role in the deviation of the reduced density matrices, corresponding to a finite energy density eigenstate from its hypothesized thermal approximation. The universal deviation in terms of the square of the trace-square distance goes as the 8th power of the subsystem fraction and is suppressed by powers of inverse central charge (c). Furthermore, the non-universal deviations from subsystem ETH are found to be proportional to the heavy-light-heavy structure constants which are typically exponentially suppressed in h/c, where h is the conformal scaling dimension of the finite energy density state. We also examine the effects of the leading finite size corrections.

We derive a universal formula for the average heavy-heavy-light structure constants for 2d CFTs with non-vanishing u(1) charge. The derivation utilizes the modular properties of one-point functions on the torus. Refinements in N = 2 SCFTs, show that the resulting Cardy-like formula for the structure constants has precisely the same shifts in the central charge as that of the thermodynamic entropy found earlier. This analysis generalizes the recent results by Kraus and Maloney for CFTs with an additional global u(1) symmetry [1]. Our results at large central charge are also shown to match with computations from the holographic dual, which suggest that the averaged CFT three-point coefficient also serves as a useful probe of detecting black hole hair.

In (1 þ 1)-d CFTs, the 4-point function on the plane can be mapped to the pillow geometry and thereby crossing symmetry gets translated into a modular property. This feature arises from the Virasoro blocks in the elliptic representation. We use these modular features to derive a universal asymptotic formula for OPE coefficients in which one of the operators is averaged over heavy primaries. As an application, we demonstrate that the coarse-grained heavy channel then reproduces features of the holographic 2 → 2 S-matrix which has black holes as their intermediate states.

We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight (h,h) → (∞, ∞) for any arbitrary spin is derived using the theorem. We further rigorously show that the error term is controlled by the twist parameter and insensitive to spin. The sensitivity of the leading piece towards spin is discussed. We identify a universal piece in microcanonical entropy when the averaging window is large. An asymptotic spectral gap on (h,h) plane, hence the asymptotic twist gap is derived. We prove an universal inequality stating that in a compact unitary 2D CFT without any conserved current Ag ≤ π(c−1)r 2 24 is satisfied, where g is the twist gap over vacuum and A is the minimal "areal gap", generalizing the minimal gap in dimension to (h ,h ) plane and r = 4 √ 3 π 2.21. We investigate density of states in the regime where spin is parametrically larger than twist with both going to infinity. Moreover, the large central charge regime is studied. We also probe finite twist, large spin behavior of density of states.