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We argue that the origin of unitarity violation and information loss paradox in our understanding of black holes (BH) lies in the standard way of doing quantum field theory in curved space-time (QFTCS), which is heavily biased on intuition borrowed from classical General Relativity. In this paper, with the quantum first approach, we formulate a so-called direct-sum QFT (DQFT) in BH space-time based on a novel formulation of discrete space-time transformations in gravity that potentially restores unitarity. By invoking the quantum effects associated with the gravitational backreaction, we show that the Hawking quanta emerging outside of the Schwarzschild radius ($r_S=2GM$) cannot be independent of the quanta that continue to be inside $r_S$. This enables the information to be carried by Hawking quanta, but in the BH DQFT formalism, we do not get any firewalls. Furthermore, DQFT leads to the BH evaporation involving only pure states. This means the quantum mechanical effects at the BH horizon produce two components of a maximally entangled pure state in geometric superselection sector Hilbert spaces. This construction enables pure states to evolve into pure states, restoring unitarity and observer complementarity. Finally, we discuss how our framework leaves important clues for formulating a scattering matrix and probing the nature of quantum gravity., Comment: 41 pages, 7 figures. Improved version with additional discussions and figures. Typos corrected

In this chapter we review the recent developments of realizing $R^2$-like inflation in the framework of a most general UV nonlocal extension of Einstein's general theory of relativity (GR). It is a well-motivated robust approach towards quantum gravity. In the past decades, nonlocal gravitational theories which are quadratic in curvature have been understood to be ghost-free and super-renormalizable around maximally symmetric spacetimes. However, in the context of early Universe cosmology we show that one must go beyond the quadratic curvature nonlocal gravity in order to achieve a consistent ghost-free framework of Universe evolution from quasi de Sitter to Minkowski spacetime. In this regard, we discuss a construction of a most general nonlocal gravity action that leads to $R^2$-like inflation and discuss the corresponding observational predictions for the scalar and tensor spectral tilts, tensor-to-scalar ratio, and the primordial non-Gaussianities. We present an analysis of how the nonlocal inflationary cosmology goes beyond the established notions of effective field theories of inflation. Finally, we comment on some open questions and prospects of higher curvature nonlocal gravity on its way of achieving the UV completion., Comment: 31 pages, 4 figures, Invited chapter of the Handbook of Quantum Gravity, C. Bambi, L. Modesto and I.L. Shapiro (Eds.), Springer, expected in 2023

Let $B, B'\subset \mathbb{R}^d$ with $d\geq 2$ be two balls such that $B'\subset \subset B$ and the position of $B'$ is varied within $B$. For $p\in (1, \infty ),$ $s\in (0,1)$, and $q \in [1, p^*_s)$ with $p^*_s=\frac{dp}{d-sp}$ if $sp < d$ and $p^*_s=\infty $ if $sp \geq d$, let $\lambda ^s_{p,q}(B\setminus \overline{B'})$ be the first $q$-eigenvalue of the fractional $p$-Laplace operator $(-\Delta _p)^s$ in $B\setminus \overline{B'}$ with the homogeneous nonlocal Dirichlet boundary conditions. We prove that $\lambda ^s_{p,q}(B\setminus \overline{B'})$ strictly decreases as the inner ball $B'$ moves towards the outer boundary $\partial B$. To obtain this strict monotonicity, we establish a strict Faber-Krahn type inequality for $\lambda _{p,q}^s(\cdot )$ under polarization. This extends some monotonicity results obtained by Djitte-Fall-Weth (Calc. Var. Partial Differential Equations, 60:231, 2021) in the case of $(-\Delta )^s$ and $q=1, 2$ to $(-\Delta _p)^s$ and $q\in [1, p^*_s).$ Additionally, we provide the strict monotonicity results for the general domains that are difference of Steiner symmetric or foliated Schwarz symmetric sets in $\mathbb{R}^d$., Comment: 14 pages, 2 figures

The presence of CMB Hemispherical Asymmetry (HPA) challenges the current understanding of inflationary cosmology which does not generically predict the parity violation in the primordial correlations. In this paper, we shall review the recently proposed resolution to this based on a new formulation of quantizing inflationary fluctuations by focusing on the discrete spacetime transformations in a gravitational context. The predictive power of this formulation is that one can generate a scale dependent HPA in the context of single field inflation for all the primordial modes including scalar and tensor fluctuations without introducing any additional parameters. This result can be seen as an indication of spontaneous breaking of $\mathcal{C}\mathcal{P}\mathcal{T}$ symmetry in an expanding Universe, if confirmed by future observations it would be a great leap in the subject of quantum field theory in curved spacetime., Comment: 13 pages, 2 figures, Based on the talk given at the workshop on "Tensions in Cosmology" in Corfu 2022. This manuscript is written for Proceedings of Science (PoS), Corfu 2022 and it is based on arXiv: 2209.03928 [gr-qc]

Before we ask what the quantum gravity theory is, it is a legitimate quest to formulate a robust quantum field theory in curved spacetime (QFTCS). Several conceptual problems, especially unitarity loss (pure states evolving into mixed states), have raised concerns over several decades. Unitarity in QFTCS is essential for formulating S-matrix and computing scattering amplitudes in curved spacetime. In this paper, acknowledging the fact that {time} is a parameter in quantum theory, which is different from its status in the context of General Relativity (GR), we start with a "quantum first approach" and propose a new formulation for QFTCS based on the discrete spacetime transformations ($\mathcal{C}\mathcal{P}\mathcal{T}$) which offer a way to achieve unitarity. Our framework for a QFTCS is built on a refined understanding of standard QFT S-matrix that explicitly reflects the \cpt\, invariance of scattering amplitudes. We rewrite the standard QFT equipped with a direct-sum Fock space structure that is explicitly $\mathcal{C}\mathcal{P}\mathcal{T}$, invariant. Applying this framework to QFTCS, in the context of de Sitter (dS) spacetime, we elucidate how this approach to quantization complies with unitarity and the observer complementarity principle. We then discuss defining an S-matrix in de Sitter spacetime that is analogous to Minkowski. Furthermore, we comment briefly the implications of our QFTCS approach for quantum gravity., Comment: 50 pages, 7 figures, title modified, new references added. Any comments are welcome