Narrow Results

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime (θ-Minkowski); on the other side, Connes' spectral distance in noncommutative geometry. Although in the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular, in the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d′_L, which coincides exactly with the spectral distance d_D on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator — together with their translations — d′_L and d_D coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d′_L and d_D as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.

A noncommutative spacetime admitting dilation symmetry was briefly mentioned in the seminal work [8] of Doplicher, Fredenhagen and Roberts. In this paper, we explicitly construct the model in detail and carry out an indepth analysis. The C*-algebra that describes this quantum spacetime is determined, and it is shown that it admits an action by _*-automorphisms of the dilation group, along with the expected Poincaré covariance. In order to study the main physical properties of this scale-covariant model, a free scalar neutral field is introduced as an investigation tool. Our key results are then the loss of locality and the irreducibility, or triviality, of special field algebras associated with regions of the ordinary Minkowski spacetime. It turns out, in the conclusions, that this analysis allows also to argue on viable ways of constructing a full conformally covariant model for quantum spacetime.

An approach to renormalization of scalar fields on the Doplicher–Fredenhagen–Roberts (DFR) quantum spacetime is presented. The effective nonlocal theory obtained through the use of states of optimal localization for the quantum spacetime is reformulated in the language of (perturbative) Algebraic Quantum Field Theory. The structure of the singularities associated to the nonlocal kernel that codifies the effects of non-commutativity is analyzed using the tools of microlocal analysis.

The area spectrum and entropy spectrum of the modified Schwarzschild black hole in gravity's rainbow are investigated via the quasinormal modes of the black hole. Using the modified Hod's method and Kunstatter's method that employ the proper frequency from the imaginary part other than the real part of the quasinormal modes, the area and entropy spacing of the black hole are calculated. The results obtained from these two methods agree with each other and the equally spaced area and entropy spectrum are derived. The obtained area and entropy spectrum are independent of the energies of test particles and are the same as from the usual Schwarzschild black hole.

We argue that the gravitational wave signal recently observed by the LIGO detectors provides a powerful tool to probe the fundamental structure of space and time. In particular, we properly model the inspiral phase of two merging black holes in a noncommutative spacetime and extract an upper bound on the scale of such quantum fuzziness at the order of the Planck scale. This improves previous constraints by $\sim 15$ orders of magnitude.

We generalize the idea of quantum clock time to quantum spacetime reference frame via physical realization of a reference system by quantum rulers and clocks. Omitting the internal degrees of freedom (such as spins) of the physical rulers and clocks, only considering their metric properties, the spacetime reference frame is described by a bosonic nonlinear sigma model (NLSM). We study the quantum behavior of the system under approximations, and obtain (1) a cosmological constant valued $(2/\pi ){\rho }_{c0}$ (${\rho }_{c0}$ the critical density at near current epoch) which is very close to the observations; (2) an effective Einstein–Hilbert term; (3) the ratio of variance to mean-squared of spacetime interval tends to a universal constant $2/\pi$ in the infrared region. This effect is testable by observing a linear dependence between the inherent quantum variance and mean-squared of the redshifts from cosmic distant spectral lines. The proportionality is expected to be the observed percentage of the dark energy. The equivalence principle is also generalized to the quantum level.

A generic implication of incorporating gravitational effects in the analysis of quantum measurements is the existence of a zero-point length of spacetime. This requires an inherently non-local description of spacetime, beyond the usual one based on metric $g_{ab}(x),$ etc. The quantum spacetime should instead be reconstructed from non-local bi-tensors of the form ${\mathcal{𝒢}}_{ab\dots i^{\prime }{j}^{\prime }\dots }(x,x^{\prime })$. A deeper look then reveals a subtle interplay between non-locality and the limit $G\hslash /c^3\to 0$. In particular, the so-called emergent gravity paradigm — in which gravitational dynamics/action/spacetime are emergent and characterized by an entropy functional — arises as the Cheshire grin of a fundamentally non-local quantum spacetime. This paper describes the flow of metric with respect to Planck length, and proposes a novel action for the same.

This article summarizes a new approach to quantum gravity based on the concepts of modular spacetime, Born geometry, and metastring theory and their applications to quantum gravity phenomenology. In particular, we discuss a new understanding of dark matter in terms of metaparticles (zero modes of the metastring) and its relation to dark energy (the curvature of dual spacetime) in view of the actual astronomical observations.

We present Friedmann flat spacetime uncertainty relations (STUR) together with some cosmological implications. An interesting link between the Principle of “gravitational stability against localization of events” (PGSL) and the holographic Bekenstein entropy bound (HEB) is also investigated. The same theorems leading to our STUR are used to calculate, thanks to the holographic principle, the entropy of the universe at its apparent horizon. The generalized entropy formula can be used to discuss interesting links with a quantum spacetime.