Narrow Results

The formalism of space–time dependent Lagrangians developed in Ref. 1 is applied to the sine-Gordon and massive Thirring models. It is shown that the well-known equivalence of these models (in the context of weak–strong duality) can be understood in this approach from the same considerations as described in Ref. 1 for electromagnetic duality. A further new result is that all these can naturally be linked to the fact that the holographic principle has analogues at length scales much larger than quantum gravity. There is also the possibility of noncommuting coordinates residing on the boundaries.

We review a few topics in Planck-scale physics, with emphasis on possible manifestations in relatively low energy. The selected topics include quantum fluctuations of spacetime, their cumulative effects, uncertainties in energy–momentum measurements, and low energy quantum-gravity phenomenology. The focus is on quantum-gravity-induced uncertainties in some observable quantities. We consider four possible ways to probe Planck-scale physics experimentally: (i) looking for energy-dependent spreads in the arrival time of photons of the same energy from GRBs; (ii) examining spacetime fluctuation-induced phase incoherence of light from extragalactic sources; (iii) detecting spacetime foam with laser-based interferometry techniques; (iv) understanding the threshold anomalies in high energy cosmic ray and gamma ray events. Some other experiments are briefly discussed. We show how some physics behind black holes, simple clocks, simple computers, and the holographic principle is related to Planck-scale physics. We also discuss a formulation of the Dirac equation as a difference equation on a discrete Planck-scale spacetime lattice, and a possible interplay between Planck-scale and Hubble-scale physics encoded in the cosmological constant (dark energy).

We argue that the model of a quantum computer with N qubits on a quantum space background, which is a fuzzy sphere with n = 2^N elementary cells, can be viewed as the minimal model for quantum gravity. In fact, it is discrete, has no free parameters, is Lorentz-invariant, naturally realizes the holographic principle, and defines a subset of punctures of spin networks' edges of loop quantum gravity labelled by spins j = 2^N-1-½. In this model, the discrete area spectrum of the cells, which is not equally spaced, is given in units of the minimal area of loop quantum gravity (for j = 1/2), and provides a discrete emission spectrum for quantum black holes. When the black hole emits one string of N bits encoded in one of the n cells, its horizon area decreases of an amount equal to the area of one cell.

We show that two-dimensional anti-de Sitter spacetime (AdS₂) can be put in correspondence, holographically, both with the harmonic oscillator and the free particle. When AdS₂ has a horizon the corresponding mechanical system is a thermal harmonic oscillator at temperature given by the Hawking temperature of the horizon.

There has been, quite recently, a discussion on how holographic-inspired bounds might be used to encompass the present-day dark energy and early-universe inflation into a single paradigm. In the current treatment, we point out an inconsistency in the proposed framework and then provide a viable resolution. We also elaborate on some of the implications of this framework and further motivate the proposed holographic connection. The manuscript ends with a more speculative note on cosmic time as an emergent (holographically induced) construct.

Starting with a scalar field theory in Euclidean anti-de Sitter space constructed in an earlier paper, we examine the boundary limit of the quantized bulk field. Our AdS/CFT correspondence is generally valid for interacting fields, and is illustrated by a treatment of three-point function for scalar fields of arbitrary mass.

Starting from a critical analysis of recently reported surprisingly large uncertainties in length and position measurements deduced within the framework of quantum gravity, we embark on an investigation both of the correlation structure of Planck scale fluctuations and the role the holographic hypothesis is possibly playing in this context. While we prove the logical independence of the fluctuation results and the holographic hypothesis (in contrast to some recent statements in that direction) we show that by combining these two topics one can draw quite strong and interesting conclusions about the details of the fluctuation structure and the microscopic dynamics on the Planck scale. We further argue that these findings point to a possibly new and generalized form of quantum statistical mechanics of strongly (anti)correlated systems of degrees of freedom in this fundamental regime.

On the basis of postulates for the holographic description of gravity and the introduction of entropic force, for static sources we derive the universal law: the entropy of a holographic screen is equal to quarter of its area in the Planck system of units.

A permissible spectrum of transverse vibrations for the holographic screen modifies both a distribution of thermal energy over bits at low temperatures and the law of gravitation at small accelerations of free fall in agreement with observations of flat rotation curves in spiral galaxies. This modification relates holographic screen parameters in de Sitter spacetime with the Milgrom acceleration in MOND.

If the information transfer between test particle and holographic screen in entropic gravity respects both the uncertainty principle and causality, a lower limit on the number of bits in the universe relative to its mass may be derived. Furthermore, these limits indicate particles that putatively travel at the speed of light — the photon and/or graviton — have a nonzero mass m ≥10^-68kg. This result is found to be in excellent agreement with current experimental mass bounds on the graviton and photon, suggesting that entropic gravity may be the result of a (recent) softly-broken local symmetry. Stronger bounds emerge from consideration of ultradense matter such as neutron stars, yielding limits of m ≥10^-48–10^-50kg, barely within the experimental photon range and outside that of the graviton. We find that for black holes these criteria cannot be satisfied, and suggest some possible implications of this result.

According to the formal holographic principle, a modification to the assumption of holographic principle in Verlinder's investigation of entropy force is obtained. A more precise relation between entropy and area in the holographic system is proposed. With the entropy corrections to the area-relation, we derivate Newton's laws and Einstein equation with a static spherically symmetric holographic screen. Furthermore, we derived the correction terms to the modified Friedmann equation of the FRW universe starting from the holographic principle and the Debye model.

The entropy bound for the conventional quantum field theory is known as A^3/4. Thus, there is an entropy gap from A^3/4 to the holographic entropy A. In this paper, we show that the entropy gap can be overcome by introducing internal degrees of freedom to the system. Using a state counting method, we show explicitly how the familiar holographic thermodynamics with area entropy can be realized. It may help in the understanding of the microscopic origin of the holographic entropy.

We construct (m, n)-type holographic dark energy models at a phenomenological level, which can be viewed as a generalization of agegraphic models with the conformal-like age as the holographic characteristic size. For some values of (m, n) the holographic dark energy can automatically evolve across ω = -1 into a phantom phase even without introducing an interaction between the dark energy and background matter. Our construction is also applicable to the holographic dark energy with generalized future event horizon as the characteristic size. Finally, we address the issue on the stability of our model and show that they are generally stable under the scalar perturbation.

Assuming the equipartition of energy to hold on a holographic sphere, Erik Verlinde demonstrated that Newton’s gravity follows as an entropic force. Some comments are in place about Verlinde’s assumptions in his derivation. It is pointed out that the holographic principle allows for freedom up to a free scale factor in the choice of Planck scale area while leading to classical gravity. Similarity of this free parameter with the Immirzi parameter of loop quantum gravity is discussed. We point out that the equipartition of energy is inbuilt into the holographic principle and, therefore, need not be assumed from the outset.

We illustrate the entanglement mechanism of quantum spacetime itself. We consider a discrete, quantum version of de Sitter universe with a Planck time-foliation, to which is applied the quantum version of the holographic principle (a Planckian pixel encodes one qubit rather than a bit). This results in a quantum network, where the time steps label the nodes.

The quantum fluctuations of the vacuum are the connecting links of the quantum network, while the total number of pixels (qubits) of a spatial slice are the outgoing links from a node n.

At each node n there are a couple of quantum gates, the Hadamard gate (H) and the controlled-not (CNOT) gate, plus a projector P. The Hadamard gate transforms virtual states (bits) into qubits, the projector P measures a qubit at the antecedent node, giving rise to a new bit, and the CNOT gate entangles a qubit at node n with the new bit at node n$-$1.

We show that the above quantum-computational interpretation of spacetime entanglement has a geometrical counterpart. In fact, the quantum fluctuations of the metric on slice n are such that a tiny wormhole will connect one Planckian pixel of slice n with one of slice n$-$1. By the quantum holographic principle, such a geometrical connection is spacetime entanglement.

The holographic principle provides a deep insight into quantum gravity and resolves the fine-tuning crisis concerning the cosmological constant. Holographic dark energy introduces new ultraviolet (UV) and infrared (IR) cutoffs into quantum gravity which are necessarily strongly related. The equation of state for dark energy $\omega =p/\rho$ is discussed from the holographic point of view. The phantom option of $\omega <-1$ is resurrected, as in an earlier cyclic cosmology. Such a cyclic model can, however, equally use the cosmological constant with $\omega =-1$.

The modified gravity is considered in the framework of the holographic dark energy (DE). An analysis of the autonomous system, the critical points and their stability is presented. Unlike the DE models based on $f(R)$, it is found that working in the holographic frame enriches the possibility of accelerated and matter-type points for different cosmological scenarios, making viable trajectories of successful $f(R)$ models that are not allowed without the consideration of the holographic framework. The implications for the Hu–Sawicki model are analyzed.

We study the analogy between the Hawking radiation in Black-Holes and the quantum depletion process of a Bose–Einstein condensate by using the Bogoliubov transformations method. We find that the relation between the Bogoliubov coefficients is similar in both cases (in the appropriate regimes). We then connect the condensate variables with those associated to the Black-Hole, demonstrating then that the zero temperature regime of the condensate is equivalent to the existence of an event horizon in gravity.

A microscopic derivation of the Bekenstein–Hawking entropy for the Schwarzschild black hole was presented earlier by using a nontrivial phase space. It was argued that the Schwarzschild black hole behaves like a $1D$ quantum mechanical system. In this paper, we show that if we assume the phase space to obey the holographic principle and take the microscopic particles inside the quantum gravitational system to be ideal bosonic gas, we can derive the Bekenstein–Hawking entropy. The assumption of the phase space to follow the holographic principle such that the Schwarzschild black hole behaves as a $2D$ system is very much in the spirit of our understanding of black holes than their behavior as a $1D$ system. However, the argument suggests that the black hole be treated as a system with the equation of state $P=\rho$.

Information is intricately woven into the fabric of physical systems, implying a profound connection between the nature of information and the underlying physics governing computational processes. This deep-seated relationship was illuminated by Landauer in 1961, who revealed that erasing a mere bit of information demands an expenditure of energy equivalent to $k_{B}Tln2$, underscoring the thermodynamic costs associated with information processing. In this investigation, Landauer’s principle is applied within the framework of Bekenstein–Hawking entropy. The study demonstrates that the introduction of a single bit of information into a black hole is related to a corresponding tangible increase of one unit of the horizon, namely, $4l_p^{2}ln2$, aligning seamlessly with the holographic principle.