Random versus holographic fluctuations of the background metric. I. (Cosmological) running of space–time dimension

A profound quantum-gravitational effect of space–time dimension running with respect to the size of space–time region has been discovered a few years ago through the numerical simulations of lattice quantum gravity in the framework of causal dynamical triangulation [hep-th/0505113] as well as in renormalization group approach to quantum gravity [hep-th/0508202]. Unfortunately, along these approaches the interpretation and the physical meaning of the effective change of dimension at shorter scales is not clear. The aim of this Letter is twofold. First, we find that box-counting dimension in face of finite resolution of space–time (generally implied by quantum gravity) shows a simple way how both the qualitative and the quantitative features of this effect can be understood. Second, considering two most interesting cases of random and holographic fluctuations of the background space, we find that it is random fluctuations that gives running dimension resulting in modification of Newton's inverse square law in a perfect agreement with the modification coming from one-loop gravitational radiative corrections.     

Point-like sources and the scale of quantum gravity

We review the General Relativistic model of a (quasi) point-like particle represented by a massive shell of neutral matter which has vanishing total energy in the small-volume limit. We then show that, by assuming a Generalised Uncertainty Principle, which implies the existence of a minimum length of the order of the Planck scale, the total energy instead remains finite and equal to the shell's proper mass both for very heavy and very light particles. This suggests that the quantum structure of space–time might be related to the classical Equivalence Principle and possible implications for the late stage of evaporating black holes are briefly mentioned.     

Quantum Regge Calculus of Einstein–Cartan theory

We study the Quantum Regge Calculus of Einstein–Cartan theory to describe quantum dynamics of Euclidean space–time discretized as a 4-simplices complex. Tetrad field eμ(x)$e_{\mu }(x)$ and spin-connection field ωμ(x)${\omega }_{\mu }(x)$ are assigned to each 1-simplex. Applying the torsion-free Cartan structure equation to each 2-simplex, we discuss parallel transports and construct a diffeomorphism and local   gauge-invariant Einstein–Cartan action. Invariant holonomies of tetrad and spin-connection fields along large loops are also given. Quantization is defined by a bounded partition function with the measure of SO(4)$\mathit{SO}(4)$-group valued ωμ(x)${\omega }_{\mu }(x)$ fields and Dirac-matrix valued eμ(x)$e_{\mu }(x)$ fields over 4-simplices complex.     

The Mixmaster Universe in a generalized uncertainty principle framework

The Bianchi IX cosmological model is analyzed in a generalized uncertainty principle framework. The Arnowitt–Deser–Misner reduction of the dynamics is performed and a time-coordinate, namely the volume of the Universe, naturally arises. Such a variable is treated in the ordinary way while the anisotropies (the physical degrees of freedom) are described by a deformed Heisenberg algebra. The analysis of the model (passing through Bianchi I and II) is performed at classical level by studying the modifications induced on the symplectic geometry by the deformed algebra. We show that the Universe cannot isotropize because of the deformed Kasner dynamics, the triangular allowed domain is asymptotically stationary with respect to the particle (Universe) and its bounces against the walls are not interrupted by the deformed effects. Furthermore, no reflection law can be in general obtained since the Bianchi II model is no longer analytically integrable. This way, the deformed Mixmaster Universe can be still considered as a chaotic system.     

Quantum gravity, minimum length and Keplerian orbits

We conjecture that the modified commutation relations suggested in the context of quantum gravity (QG) persist also in the classical limit, if the momentum of the classical object is not too large, and calculate the corresponding perihelion precession rate for Keplerian orbits. The main result obtained in this Letter is not new. However the derivation is much simpler than the one proposed by Benczik et al. in [S. Benczik, L.N. Chang, D. Minic, N. Okamura, S. Rayyan, T. Takeuchi, Phys. Rev. D 66 (2002) 026003, arXiv:hep-th/0204049] where the corresponding precession rate was calculated for the first time. Our interpretation of the result is also quite different.     

The length operator in Loop Quantum Gravity

The dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, we introduce a new operator in Loop Quantum Gravity—the length operator. We describe its quantum geometrical meaning and derive some of its properties. In particular we show that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and an explicit check of its semiclassical properties is discussed.     

Operational definition of (brane-induced) space–time and constraints on the fundamental parameters

First we contemplate the operational definition of space–time in four dimensions in light of basic principles of quantum mechanics and general relativity and consider some of its phenomenological consequences. The quantum gravitational fluctuations of the background metric that comes through the operational definition of space–time are controlled by the Planck scale and are therefore strongly suppressed. Then we extend our analysis to the braneworld setup with low fundamental scale of gravity. It is observed that in this case the quantum gravitational fluctuations on the brane may become unacceptably large. The magnification of fluctuations is not linked directly to the low quantum gravity scale but rather to the higher-dimensional modification of Newton's inverse square law at relatively large distances. For models with compact extra dimensions the shape modulus of extra space can be used as a most natural and safe stabilization mechanism against these fluctuations.     

Real-time gauge theory simulations from stochastic quantization with optimized updating

We investigate simulations for gauge theories on a Minkowskian space–time lattice. We employ stochastic quantization with optimized updating using stochastic reweighting or gauge fixing, respectively. These procedures do not affect the underlying theory but strongly improve the stability properties of the stochastic dynamics, such that simulations on larger real-time lattices can be performed.     

The generalized uncertainty principle in (A)dS space and the modification of Hawking temperature from the minimal length

Recently, the Heisenberg's uncertainty principle has been extended to incorporate the existence of a large (cut-off) length scale in de Sitter or anti-de Sitter space, and the Hawking temperatures of the Schwarzshild–(anti) de Sitter black holes have been reproduced by using the extended uncertainty principle. I generalize the extended uncertainty to the case with an absolute minimum length and compute its modification to the Hawking temperature. I obtain a general trend that the generalized uncertainty principle due to the absolute minimum length “always” increases the Hawking temperature, implying “faster” decay, which is in conformity with the result in the asymptotically flat space. I also revisit the black hole-string phase transition, in the context of the generalized uncertainty principle.     

Semiclassical regime of Regge calculus and spin foams

Recent attempts to recover the graviton propagator from spin foam models involve the use of a boundary quantum state peaked on a classical geometry. The question arises whether beyond the case of a single simplex this suffices for peaking the interior geometry in a semiclassical configuration. In this paper we explore this issue in the context of quantum Regge calculus with a general triangulation. Via a stationary phase approximation, we show that the boundary state succeeds in peaking the interior in the appropriate configuration, and that boundary correlations can be computed order by order in an asymptotic expansion. Further, we show that if we replace at each simplex the exponential of the Regge action by its cosine—as expected from the semiclassical limit of spin foam models—then the contribution from the sign-reversed terms is suppressed in the semiclassical regime and the results match those of conventional Regge calculus.     

Yang's model as triply special relativity and the Snyder's model–de Sitter special relativity duality

We show that if Yang's quantized space–time model is completed at both classical and quantum level, it should contain both Snyder's model, the de Sitter special relativity and their duality.     

On the Green’s function and iterative solutions of loop quantum cosmology

Here we shall find the Green’s function of the difference equation of loop quantum cosmology. To illustrate how to use it, we shall obtain an iterative solution for closed model and evaluate its corresponding Bohmian trajectory.     

Space–time dimensionality from brane collisions

Collisions and subsequent decays of higher dimensional branes leave behind three-dimensional branes and anti-branes, one of which could play the rôle of our universe. This process also leads to the production of one-dimensional branes and anti-branes, however their number is expected to be suppressed. Brane collisions may also lead to the formation of bound states of branes. Their existence does not alter this result, it just allows for the existence of one-dimensional branes captured within the three-dimensional ones.     

Central charges in 2d reduced cosmological massive gravity

In the dimensionally reduced model of the (2+1)$(2+1)$-dimensional cosmological massive gravity, we obtain the central charges of the two types of the black hole based on the entropy function method. One is for the BTZ black hole and the other one is actually for the warped AdS₃ black hole.     

Entanglement of Area Quantums in Quantized Space

Employing the area quantum 1/2 excited by the Wilson loop, a possible origin of qubit is argued. At the same time, the existences of possible entanglement of area quantums, and non-local property of the entangled states are demonstrated in quantized space.     

Spin-0 and spin-1/2 particles in a constant scalar-curvature background

We study the Klein-Gordon and Dirac equations in the presence of a background metric ds²=−dt²+dx²+e^−2gx(dy²+dz²) in a semi-infinite lab (x>0). This metric has a constant scalar-curvature R=6g² and is produced by a perfect fluid with equation of state p=−ρ/3. The eigenfunctions of spin-0 and spin-1/2 particles are obtained exactly, and the quantized energy eigenvalues are compared. It is shown that both of these particles must have nonzero transverse momentum in this background. We show that there is a minimum energy E²_min=m²c⁴+g²c²?² for bosons (E_KG>E_min), while the fermions have no specific ground state (E_Dirac>mc²).     

Space-time curvature due to quantum vacuum fluctuations: An alternative to dark energy?

It is pointed out that quantum vacuum fluctuations may give rise to a curvature of space-time equivalent to the curvature currently attributed to dark energy. A simple calculation is made, involving plausible assumptions within the framework of quantized gravity, which suggests that the value of the dark energy density is roughly given by the product of Newton's constant times the quantity m⁶c⁴?−4, m being a typical mass of elementary particles. The estimate is compatible with observations.