We study the role of the Barbero-Immirzi parameter $γ$ and the choice of connection in the construction of (a symmetry-reduced version of) loop quantum gravity. We start with the four-dimensional Lorentzian Holst action that we reduce to three dimensions in a way that preserves the presence of $γ$. In the time gauge, the phase space of the resulting three-dimensional theory mimics exactly that of the four-dimensional one. Its quantization can be performed, and on the kinematical Hilbert space spanned by SU(2) spin network states the spectra of geometric operators are discrete and $γ$-dependent. However, because of the three-dimensional nature of the theory, its SU(2) Ashtekar-Barbero Hamiltonian constraint can be traded for the flatness constraint of an sl(2,C) connection, and we show that this latter has to satisfy a linear simplicity-like condition analogous to the one used in the construction of spin foam models. The physically relevant solution to this constraint singles out the non-compact subgroup SU(1,1), which in turn leads to the disappearance of the Barbero-Immirzi parameter and to a continuous length spectrum, in agreement with what is expected from Lorentzian three-dimensional gravity.
We consider General Relativity as a limit case of the Scalar-Tensor theory with Barbero-Immirzi field when the field tends to a constant. We use Shapiro time delay experimental limit of $1/w = (2.1 \pm 2.3)10^{-5}$ provided by the Cassini spacecraft to find the Barbero-Immirzi parameter value.
We introduce a covariant formulation of Barbero-Immirzi connections, which are used in Loop Quantum Gravity to describe gravity. We show that Barbero-Immirzi connections can be uniquely defined out of a given spin connection for any $(n+1)$-dimensional lorentzian manifold which is spin. A remarkable result is that the presence of a real Barbero-Immirzi parameter is a feature unique to the $4$-dimensional case.
In this paper we naturally obtain the values of the Barbero-Immirzi parameter as the solution of the simplicity constraints rather than setting it a priori. Particularly the Main theorem shows that if $γ= \pm i$ then the simplicity constraints require that the corresponding Lorentz group representations be necessary finite dimensional and therefore non-unitary.
Starting from a constrained real $BF$-type action for general relativity that includes both the Immirzi parameter and the cosmological constant, we obtain the Ashtekar-Barbero variables used in the canonical approach to the quantization of the gravitational field. This is accomplished by explicitly solving the second-class constraints resulting from the Hamiltonian analysis of the considered action, and later imposing the time gauge. All throughout this work the tetrad formalism is left aside, obtaining the Ashtekar-Barbero variables entirely in terms of the $B$-fields that define the action.
In the context of the geometrical interpretation of the spin network states of Loop Quantum Gravity, we look at the holonomies of the Ashtekar-Barbero connection on loops embedded in space-like hyperboloids. We use this simple setting to illustrate two points. First, the Ashtekar-Barbero connection is not a space-time connection, its holonomies depend on the spacetime embedding of the canonical hypersurface. This fact is usually interpreted as an inconvenience, but we use it to extract the extrinsic curvature from the holonomy and separate it from the 3d intrinsic curvature. Second, we show th…
We carry out the complete variational analysis of the Barbero--Immirzi--Holst Lagrangian, which is the Holst Lagrangian expressed in terms of the triad of fields $(θ, A, κ)$, where $θ$ is the solder form/spin frame, $A$ is the spacetime Barbero--Immirzi connection, and $κ$ is the extrinsic spacetime field. The Holst Lagrangian depends on the choice of a real, non zero Holst parameter $γ\neq 0$ and constitutes the classical field theory which is then quantized in Loop Quantum Gravity. The choice of a real Immirzi parameter $β$ sets up a one-to-one correspondence between pairs $(A, κ)$ and spin …
Canonical description of Pontryagin and Euler classes with a Barbero-Immirzi parameter
Instituto de Física, Benemérita Universidad Autónoma de Puebla（普埃布拉自治大学物理研究所）；Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla（普埃布拉自治大学物理数学科学学院）
The Barbero-Immirzi parameter ($γ$) is introduced in loop quantum gravity (LQG) whose physical significance is still a 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 out in the process of denoting a Lorentz connection with 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 was appeared in the Ashtekar variables. Fernando Barbero proposed its possibility to include within formalism. Its pre…
It was showed by Perez and Rovelli in 2006 that the Holst action in gravity with torsion with massless and minimally coupling Dirac fermions gives rise to the four-fermion coupling term, whose coefficient is a function of the Barbero-Immirzi (BI) parameter. This parameter is present in the Holst action, which is an object of investigation in a non-perturbative formalism of quantum gravity. The key feature is the torsion because its absence implies no effect from Holst term in dynamical equations. In this paper we consider a spin fluid (also called Weyssenhoff fluid), which is a perfect fluid t…
The form of Hamiltonian and Diffeomorphism constraints in Sen-Ashtekar-Barbero-Immirzi variables is well known for the spacelike 3+1 ADM foliation. It is also known that Sen-Ashtekar-Barbero-Immirzi connection can be introduced only in 3 dimensional space and does not work for $D > 3$. The reason it works in $D = 3$ is due to existence of isomorphism between $so(3)$ algebra and $R^3$ space with vector product. It turns out that similar isomorphism exists between $so(2,1)$ algebra and $R^3_{2,1}$ space algebra with respect to its vector product. By using this isomorphism we find both analog of …
In this paper we research all possible finite-dimensional representations and corresponding values of the Barbero-Immirzi parameter contained in EPRL simplicity constraints by using Naimark's fundamental theorem of the Lorentz group representation theory. It turns out that for each non-zero pure imaginary with rational modulus value of the Barbero-Immirzi parameter $γ= i \frac{p}{q}, p, q \in Z, p, q \ne 0$, there is a solution of the simplicity constraints, such that the corresponding Lorentz representation is finite dimensional. The converse is also true - for each finite-dimensional Lorentz…
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 ene…
We consider spinfoam quantum gravity for general triangulations in the regime $l_P^2\ll a\ll a/γ$, namely in the combined classical limit of large areas $a$ and flipped limit of small Barbero-Immirzi parameter $γ$, where $l_P$ is the Planck length. Under few working hypotheses we find that the flipped limit enforces the constraints that turn the spinfoam theory into an effective Regge-like quantum theory with lengths as variables, while the classical limit selects among the possible geometries the ones satisfying the Einstein equations. Two kinds of quantum corrections appear in terms of power…
Action-level characterization of gravitational-wave propagation in dynamical Barbero--Immirzi gravity
State Key Laboratory of Radio Astronomy and Technology, Xinjiang Astronomical Observatory, Chinese Academy of Sciences（中国科学院新疆天文台射电天文与技术重点实验室）；Shanxi Province Intelligent Optoelectronic Sensing Application Technology Innovation Center, Yuncheng University（运城学院山西省智能光电传感应用技术创新中心）；Departamento de Física Teórica, IF-UERJ（里约热内卢州立大学理论物理研究所）；College of Mechanical and Electrical Engineering, Tarim University（塔里机械与电气工程学院）；Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University（北京师范大学天文与前沿物理研究所）；School of Physics and Astronomy, Beijing Normal University（北京师范大学物理与天文学院）
The Immirzi parameter is promoted to be a scalar field and the Hamiltonian analysis of the corresponding dynamical system is performed in the presence of gravity. We identified some SU(2) connections, generalizing Ashtekar-Barbero variables, and we rewrite the constraints in terms of them, setting the classical formulation suitable for loop quantization. Then, we consider the reduced system obtained when restricting to a flat isotropic cosmological model. By mimicking loop quantization via an effective semiclassical treatment, we outline how quantum effects are able to tame the initial singula…
Dynamical Barbero--Immirzi field coupled to quintessence: gravitational-wave propagation constraints and next-generation forecasts
State Key Laboratory of Radio Astronomy and Technology, Xinjiang Astronomical Observatory, Chinese Academy of Sciences（中国科学院新疆天文台射电天文与技术重点实验室）；Shanxi Province Intelligent Optoelectronic Sensing Application Technology Innovation Center, Yuncheng University（运城学院山西省智能光电传感应用技术创新中心）；Departamento de Física Teórica, IF-UERJ（里约热内卢州立大学理论物理系）；College of Mechanical and Electrical Engineering, Tarim University（塔里米大学机电工程学院）；Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University（北京师范大学天文与前沿物理研究所）
