Immirzi Parameter Research Articles

Black hole interior quantization: a minimal uncertainty approach

In a previous work we studied the interior of the Schwarzschild black hole implementing an effective minimal length, by applying a modification to the Poisson brackets of the theory. In this work we perform a proper quantization of such a system. Specifically, we quantize the interior of the Schwarzschild black hole in two ways: once by using the standard quantum theory, and once by following a minimal uncertainty approach. Then, we compare the obtained results from the two approaches. We show that, as expected, the wave function in the standard approach diverges in the region where classical singularity is located and the expectation value of the Kretschmann scalar also blows up on this state in that region. On the other hand, by following a minimal uncertainty quantization approach, we obtain 5 new and important results as follows. (1) All the interior states remain well-defined and square-integrable. (2) The expectation value of the Kretschmann scalar on the states remains finite over the whole interior region, particularly where used to be the classical singularity, therefore signaling the resolution of the black hole singularity. (3) A new quantum number is found which plays a crucial role in determining the convergence of the norm of states, as well as the convergence and finiteness of the expectation value of the Kretschmann scalar. (4) A minimum for the radius of the (2-spheres in the) black holes is found (5) By demanding square-integrability of states in the whole interior region, an exact relation between the Barbero-Immirzi parameter and the minimal uncertainty scale is found.

Natural metric-affine inflation

We consider here natural inflation in the low energy (two-derivative) metric-affine theory containing only the minimal degrees of freedom in the inflationary sector, i.e. the massless graviton and the pseudo-Nambu-Goldstone boson (PNGB). This theory contains the Ricci-like and parity-odd Holst invariants together with non-minimal couplings between the PNGB and the above-mentioned invariants. The Palatini and Einstein-Cartan realizations of natural inflation are particular cases of our construction. Explicit models of this type featuring non-minimal couplings are shown to emerge from the microscopic dynamics of a QCD-like theory with an either sub-Planckian or trans-Planckian confining scale and that is renormalizable on Minkowski spacetime. Moreover, for these models, we find regions of the parameter space where the inflationary predictions agree with the most recent observations at the 2σ level. We find that in order to enter the 1σ region it is necessary (and sufficient) to have a finite value of the Barbero-Immirzi parameter and a sizable non-minimal coupling between the inflaton and the Holst invariant (with sign opposite to the Barbero-Immirzi parameter). Indeed, in this case the potential of the canonically normalized inflaton develops a plateau as shown analytically.

Dark torsion oscillations and bounces induced by Barbero-Immirzi and quantum correction on Einstein-Cartan gravity with very light inflatons

In this paper we present several examples of how to constrain loop quantum gravity Barbero-Immirzi (BI) parameter, either by using quantum corrections in Nieh–Yan (NY) modified Einstein-Cartan (EC) gravity with the Holst term. The reason why we introduce some extra terms in the actions, is that even when one adds both NY and Holst terms, being topological terms, they do not contribute to the dynamics of the problem and to the equations of motion (EOM). Examples run from quantum corrections and torsion mass in the dark universe with very light inflaton approximation to bouncing cosmologies in NY gravity. This follows similar model from Tukhashvili and Steinhardt (2023) of a bouncing cosmology induced by torsion condensates in EC gravity. The Lagrangian in this paper, is shown to be expressed in terms of NY polynomial form. The quantum one-loop correction is taken in NY Lagrangian as the inverse of BI parameter. This has been recently appeared in He et al. (2024) quantum corrections in Einstein-Cartan-Nieh–Yan one loop quantum gravity in Starobinsky inflation. We show that to have a cosmological NY dark universe one needs a real BI parameter. Inflation Hubble scale of the order of O(100eV) and very light inflaton of mass ∼10−12TeV are used due to its relation with dark matter. The very light mass character of inflatons and much heavier torsion are responsible to induced cosmological bounce in dark universe. When one introduces the quantum correction parameter b as second-order in front of the NY topological term in squared and introduce the NY term back into the ECH gravity, the EC gravity limit where BI parameter β→∞, yields a chiral fermion density ρ5. This result lies very close to the upper bound in the range of 0≤β≤1.185, previously obtained by Panza et al. Analysis of fermionic currents shows that Holst term can be eliminated by choosing β=0.288. Massive torsion oscillates in Einstein-Cartan gravity in the BI limit of β→∞.

Dirac Geometric Approach for the Unimodular Holst Action

We perform a Hamiltonian analysis of unimodular gravity in its first-order formulation, specifically a modification of the Holst action. In order to simplify the analysis, prior studies on this theory have introduced (for several reasons) additional elements, such as parametrization, complex fields, or considering the Barbero–Immirzi parameter as imaginary. We show that, by using a geometric implementation of the Dirac algorithm, a comprehensive analysis of the theory can be conducted without relying on these additional ingredients. The resulting theory reproduces the behavior of metric unimodular gravity.

Introduction to loop quantum gravity. The Holst’s action and the covariant formalism

We review Holst formalism and dynamical equivalence with standard GR (in dimension [Formula: see text]). Holst formalism is written for a spin coframe field [Formula: see text] and a Spin(3,1)-connection [Formula: see text] on spacetime [Formula: see text] and it depends on the Holst parameter [Formula: see text]. We show the model is dynamically equivalent to standard GR, in the sense that up to a pointwise Spin(3,1)-gauge transformation acting on (uppercase Latin) frame indices, solutions of the two models are in one-to-one correspondence. Hence, the two models are classically equivalent. One can also introduce new variables by splitting the spin connection into a pair of a Spin(3)-connection [Formula: see text] and a Spin(3)-valued 1-form [Formula: see text]. The construction of these new variables relies on a particular algebraic structure, called a reductive splitting. A weaker structure than requiring that the gauge group splits as the products of two sub-groups, as it happens in Euclidean signature in the selfdual formulation originally introduced in this context by Ashtekar, and it still allows to deal with the Lorentzian signature without resorting to complexifications. The reductive splitting of [Formula: see text] is not unique and it is parameterized by a real parameter [Formula: see text] which is called the Immirzi parameter. The splitting is here done on spacetime, not on space as it is usually done in the literature, to obtain a Spin(3)-connection [Formula: see text], which is called the Barbero–Immirzi connection on spacetime. One obtains a covariant model depending on the fields [Formula: see text] which is again dynamically equivalent to standard GR. Usually in the literature one sets [Formula: see text] for the sake of simplicity. Here, we keep the Holst and Immirzi parameters distinct to show that eventually only [Formula: see text] will survive in boundary field equations.

Hamiltonian formulation of gravity as a spontaneously-broken gauge theory of the Lorentz group

A number of approaches to gravitation have much in common with the gauge theories of the standard model of particle physics. In this paper, we develop the Hamiltonian formulation of a class of gravitational theories that may be regarded as spontaneously-broken gauge theories of the complexified Lorentz group SO(1,3)C with the gravitational field described entirely by a gauge field valued in the Lie algebra of SO(1,3)C and a ‘Higgs field’ valued in the group’s fundamental representation. The theories have one free parameter β which appears in a similar role to the inverse of the Barbero–Immirzi parameter of Einstein–Cartan theory. However, contrary to that parameter, it is shown that the number of degrees of freedom (DOFs) crucially depends on the value of β. For non-zero values of β, it is shown the theories possesses three complex DOFs, and for the specific values β=±i an extension to general relativity is recovered in a symmetry-broken regime. For the value β = 0, the theory possesses no local DOFs. A non-zero value of β corresponds to the self-dual and anti-self-dual gauge fields appearing asymmetrically in the action, therefore in these models, the existence of gravitational DOFs is tied to chiral asymmetry in the gravitational sector.

On the gravitational diagram technique in the discrete setup

This paper is in the spirit of our work on the consequences of the Regge calculus, where some edge length scale arises as an optimal initial point of the perturbative expansion after functional integration over connection. Now consider the perturbative expansion itself. To obtain an algorithmizable diagram technique, we consider the simplest periodic simplicial structure with a frozen part of the variables (“hypercubic”). After functional integration over connection, the system is described by the metric [Formula: see text] at the sites. We parameterize [Formula: see text] so that the functional measure becomes Lebesgue. The discrete diagrams are free from ultraviolet divergences and reproduce (for ordinary, non-Planck external momenta) those continuum counterparts that are finite. We give the parametrization of [Formula: see text] up to terms, providing, in particular, additional three-graviton and two-graviton-two-matter vertices, which can give additional one-loop corrections to the Newtonian potential. The edge length scale is [Formula: see text], where [Formula: see text] defines the free factor [Formula: see text] in the measure and should be a large parameter to ensure the true action after integration over connection. We verify the important fact that the perturbative expansion does not contain increasing powers of [Formula: see text] if its initial point is chosen close enough to the maximum point of the measure, thus justifying this choice. Discrete propagators depend on the Barbero–Immirzi parameter [Formula: see text], which determines the ratio of timelike and spacelike elementary length scales. The existing estimates of [Formula: see text] allow the propagator poles to have real energy for any (real) spatial momenta.

Matter-antimatter asymmetry induced by Barbero-Immirzi parameter

We review the baryon asymmetry generated in the early Universe in the framework of the metric-spinor gravity with the Holst term, which accounts for the Barbero-Immirzi parameter. For the generation of the matter-antimatter asymmetry, we have considered the model in which the time derivative of the Ricci scalar couples with matter currents (this model is known in the literature as gravitational baryogenesis). By using the current bounds on the parameter of the asymmetry, the ratio of baryon number to the entropy density, we infer a bound on the parameters characterizing the model under consideration (that is, the cosmological model in the presence of the torsion and Barbero-Immirzi parameter).

Primordial black holes in loop quantum cosmology: the effect on the threshold

Primordial black holes form in the early Universe and constitute one of the most viable candidates for dark matter. The study of their formation process requires the determination of a critical energy density perturbation threshold , which in general depends on the underlying gravity theory. Up to now, the majority of analytic and numerical techniques calculate within the framework of general relativity. In this work, using simple physical arguments we estimate semi-analytically the PBH formation threshold within the framework of quantum gravity, working for concreteness within loop quantum cosmology (LQC). In particular, for low mass PBHs formed close to the quantum bounce, we find a reduction in the value of up to compared to the general relativistic regime quantifying for the first time to the best of our knowledge how quantum effects can influence PBH formation within a quantum gravity framework. Finally, by varying the Barbero–Immirzi parameter γ of loop quantum gravity (LQG) we show its effect on the value of while using the observational/phenomenological signatures associated to ultra-light PBHs, namely the ones affected by LQG effects, we propose the PBH portal as a novel probe to constrain the potential quantum nature of gravity.

A diffeomorphism invariant family of metric-affine actions for loop cosmologies

In loop quantum cosmology (LQC) the big bang singularity is generically resolved by a big bounce. This feature holds even when modified quantization prescriptions of the Hamiltonian constraint are used such as in mLQC-I and mLQC-II. While the later describes an effective description qualitatively similar to that of standard LQC, the former describes an asymmetric evolution with an emergent Planckian de-Sitter pre-bounce phase even in the absence of a potential. We consider the potential relation of these canonically quantized non-singular models with effective actions based on a geometric description. We find a 3-parameter family of metric-affine f(ℛ) theories which accurately approximate the effective dynamics of LQC and mLQC-II in all regimes and mLQC-I in the post-bounce phase. Two of the parameters are fixed by enforcing equivalence at the bounce, and the background evolution of the relevant observables can be fitted with only one free parameter. It is seen that the non-perturbative effects of these loop cosmologies are universally encoded by a logarithmic correction that only depends on the bounce curvature of the model. In addition, we find that the best fit value of the free parameter can be very approximately written in terms of fundamental parameters of the underlying quantum description for the three models. The values of the best fits can be written in terms of the bounce density in a simple manner, and the values for each model are related to one another by a proportionality relation involving only the Barbero-Immirzi parameter.

Thermodynamics of isolated horizons in loop quantum gravity

The statistical mechanical calculation of the thermodynamical properties of nonrotating isolated horizons are studied in the loop quantum gravity framework. By employing the Hawking temperature and horizon mass of isolated horizons as physical inputs, the microcanonical ensemble associated with the system are well established. As a result, the black hole entropy and other thermodynamical quantities can be computed and are consistent with well-known Hawking's semiclassical analysis. Moreover, the value of the Immirzi parameter of loop quantum gravity for the higher-dimensional case and four-dimensional U(1) case are also obtained.

Constraining the Barbero-Immirzi parameter from the duration of inflation in loop quantum cosmology

We revisit the predictions for the duration of the inflationary phase after the bounce in loop quantum cosmology. We present our analysis for different classes of inflationary potentials that include the monomial power-law chaotic type of potentials, the Starobinsky and the Higgs-like symmetry breaking potential with different values for the vacuum expectation value. Our setup can easily be extended to other forms of primordial potentials than the ones we have considered. Independently on the details of the contracting phase, if the dynamics starts sufficiently in the far past, the kinetic energy will come to dominate at the bounce, uniquely determining the amplitude of the inflaton at this moment. This will be the initial condition for the further evolution that will provide us with results for the number of e-folds from the bounce to the beginning of the accelerated inflationary regime and the subsequent duration of inflation. We also discuss under which conditions each model considered could lead to observable signatures on the spectrum of the cosmic microwave background or else be excluded for not predicting a sufficient amount of accelerated expansion. A first analysis is performed considering the standard value for the Barbero-Immirzi parameter obtained from black hole entropy calculations. In a second analysis, we consider the possibility of varying the value of this parameter, which is motivated by the fact that the Barbero-Immirzi parameter can be considered a free parameter of the underlying quantum theory in the context of loop quantum gravity. From this analysis, we obtain a lower limit for this parameter by requiring the minimum amount of inflationary expansion that makes the model consistent with the CMB observations. [abridged]

The Barbero–Immirzi Parameter: An Enigmatic Parameter of Loop Quantum Gravity

The Barbero–Immirzi parameter, (γ), is introduced in loop quantum gravity (LQG), whose physical significance is still the biggest open question because of its profound traits. In some cases, it is real valued, while it is complex valued in other cases. This parameter emerges in the process of denoting a Lorentz connection with a non-compact group SO(3,1) in the form of a complex connection with values in a compact group of rotations, either SO(3) or SU(2). Initially, it appeared in the Ashtekar variables. Fernando Barbero proposed its possibility for inclusion within formalism. Its present value is fixed by counting micro states in loop quantum gravity and matching with the semi-classical black hole entropy computed by Stephen Hawking. This parameter is used to count the size of the quantum of area in Planck units. Until the discovery of the spectrum of the area operator in LQG, its significance remained unknown. However, its complete physical significance is yet to be explored. In the present paper, an introduction to the Barbero–Immirzi parameter in LQG, a timeline of this research area, and various proposals regarding its physical significance are given.

Einstein–Cartan–Holst–Proca dynamos massive photons as dark matter and GWs

In this paper, we extend Shaposhnikov et al.'s parity violating the Einstein–Cartan–Holst action by adding to Holst term H, a Proca photon–torsion coupling χpt = 10−24 as computed by De Sabbata, Garcia de Andrade, and Sivaram. The strength of Holst term is obtained, in the case of early universe torsion as, T ∼ 1 MeV. The gravity part is the Planck mass product with the Holst term. The photon mass in this case yields m2 γ H ∼ 10−69 GeV4. Therefore, in this case, the photon interacts very weakly with spin-torsion matter in Einstein–Cartan (EC) gravity and the strength of the Maxwell–Proca–Holst interaction is much weaker than in the pure gravity sector. For dark photons, this situation changes drastically since a dark photon mass reaches 1.5 GeV, and from this mass and T = 10−3 GeV, one obtains a Maxwell–Proca–Holst strength of 10−6 GeV4, which is now comparable with the gravity sector. Dynamo mechanism competes with the chirality and dissipation by the handness of the magnetic field with respect to the torsion trace vector to regenerate the magnetic field decay. Bombacigno and Mantovani recently found a sign of the Immirzi parameter in the Holst–Nieh–Yan extension of EC gravity. Here, we also find torsion waves for the Immirzi inverse parameter in the sense that this divergence of torsion trace is porportional to the Immirzi field.

Numerical analysis of the self-energy in covariant loop quantum gravity

We study numerically the first order radiative corrections to the self-energy, in covariant loop quantum gravity. We employ the recently developed 'sl2cfoam-next' spinfoam amplitudes library, and some original numerical methods. We analyze the scaling of the divergence with the infrared cutoff, for which previous analytical estimates provided widely different lower and upper bounds. Our findings suggest that the divergence is approximately linear in the cutoff. We also investigate the role of the Barbero-Immirzi parameter in the asymptotic behavior, the dependence of the scaling on some boundary data and the expectation values of boundary operators.

The Golden Mean Number System: Its Platonic Roots, Paradigmatic Symmetry and Quantum Parameters

The golden mean number system may well be the most powerful tool in the physicist/mathematician’s arsenal. It emerges naturally out of Plato’s two principles, the One and the Indefinite Dyad of the Greater (Φ) and the Lesser (ϕ or 1/Φ). Herein we unpack its structural backbone in the golden series of exponential powers, …, ϕ7, ϕ6, ϕ5, ϕ4, ϕ3, ϕ2, ϕ, 1, Φ, Φ2, Φ3, Φ4, Φ5, Φ6, Φ7, … , along with its perfect combinatorial properties of addition and subtraction in growth and diminution, as well as, through multiplication and division via application of the modular Φ. We unravel the underlying paradigmatic symmetry of any given golden power serving simultaneously as geometric, arithmetic and harmonic means. And in the process, we reveal how the quantum parameters, including the pre-quantum particle, pre-quantum wave, Einstein spacetime, Unruh temperature, Hardy entanglement and the Barbero-Immirzi parameter, emerge naturally within Plato’s famous Republic similes of the One, Divided Line and Cave. The golden mean number system has now reemerged most completely and successfully in the E-Infinity theory.

Semiclassical and quantum features of the Bianchi I cosmology in the polymer representation

We analyze the Bianchi I cosmology in the presence of a massless scalar field and describe its dynamics via a semiclassical and quantum polymer approach by adopting three different sets of configurational variables: the natural Ashtekar connections, a set of anisotropic volumelike coordinates and the Universe volume plus two anisotropy coordinates (the latter two sets of variables would coincide in the case of an isotropic Universe). In the semiclassical analysis we demonstrate that the big bounce clearly emerges in all the sets of variables. Moreover, when adopting the proper volume variables (i.e. when defining the Universe volume itself on the polymer lattice) we also derive the exact polymer-modified Friedmann equation for the Bianchi I model. This way, the expression of the critical energy density that includes the anisotropic energylike contribution is obtained, demonstrating that in this set of variables the big bounce has a universal nature (i.e. its critical energy density has a maximum value fixed by fundamental constants and the Immirzi parameter). Finally, a proposal is made to recover the equivalence between the dynamics in the Ashtekar set and in the anisotropic volumelike one, showing that this request is satisfied when a polymer parameter depending on the configurational coordinates is considered. Then, we apply the Arnowitt-Deser-Misner reduction of the variational principle and we quantize the system. We study the resulting Schr\"odinger-like dynamics only in the Ashtekar variables and in the proper volume ones, stressing that the behavior of the Universe wave packet over time singles out common features with the semiclassical trajectories. However, we show that the standard deviation for the Universe volume operator grows with time. This is a signal of the spreading of the Bianchi I wave packet and hence of the importance of the quantum fluctuations in order to properly study the big bounce picture.

Quasinormal modes and horizon area quantisation in Loop Quantum Gravity

It is argued that the quantum of area between consecutive, high overtones quasinormal modes of a black hole horizon coincides with the area gap predicted by Loop Quantum Gravity, as long as the horizon is isolated and the Barbero-Immirzi parameter is \(\gamma \approx \sqrt{3}/6\), in agreement with the value derived from the Bekenstein-Hawking horizon entropy.

On the role of Barrow's fractal entropy in the loop quantum gravity Barbero-Immirzi parameter

It is claimed that, using the Boltzmann-Gibbs statistical mechanics together with the Bekenstein-Hawking area entropy law, it is possible to determine a value for the Barbero-Immirzi parameter, . In this letter, we have shown that if we use Barrow's entropy rather than Bekenstein-Hawking's area entropy law then the Barbero-Immirzi parameter will depend on the Δ parameter. As a consequence, the loop quantum gravity area element will also depend on the Δ parameter.

Wick rotation for spin foam quantum gravity

Models of spin foam quantum gravity come in different spacetime signatures, either Euclidean or Lorentzian. The choice is reflected in the use of $Spin(4)$ or $SL(2,\mathbb{C})$ as gauge groups for the holonomies of the connection. In this work we show that a rotation of the Immirzi parameter to purely imaginary values maps the Euclidean Engle-Pereira-Rovelli-Livine spin foam model to its Lorentzian version and vice versa. Our methods provide a general recipe for relating spin foam models of different signature through analytic continuation of the gauge groups and their unitary irreducible representations.