Holonomies of the Ashtekar-Barbero connection can be considered as abstract elements of a Lie group exponentially mapped from their connections representation. This idea provides a possibility to compare the geometric and algebraic properties of these objects. The result allows to identify the next-to-the-leading-order terms in the geometric and algebraic expansion of a holonomy. This identification leads to the verification of the related Hilbert space formulation. If states are the representations of the holonomy's symmetry group, they preserve gauge transformations according to Wigner's theorem. Thus, the spin network in loop quantum gravity satisfies this theorem. Moreover, the considered identification of the different expansions ensures the reality of the Ashtekar connection. Only the holonomies of real connections lead to the formulation of states that satisfy Wigner's theorem.
The Ashtekar and Ashtekar-Barbero connection variable formulations of Kerr isolated horizons are derived. Using a regular Kinnersley tetrad in horizon-penetrating Kruskal-Szekeres-like coordinates, the spin coefficients of Kerr geometry are determined by solving the first Maurer-Cartan equation of structure. Isolated horizon conditions are imposed on the tetrad and the spin coefficients. A transformation into an orthonormal tetrad frame that is fixed in the time gauge is applied and explicit calculations of the spin connection, the Ashtekar and Ashtekar-Barbero connections, and the correspondi…
Boundary conditions for Ashtekar-Barbero variables in the context of asymptotically flat spacetimes which lead to supertranslations at spatial infinity
The Ashtekar-Barbero constraints for General Relativity with fermions are derived from the Einstein-Cartan canonical theory rescaling the state functional of the gravity-spinor coupled system by the exponential of the Nieh-Yan functional. A one parameter quantization ambiguity naturally appears and can be associated with the Immirzi parameter.
We describe the canonical phase space of asymptotically flat gravity in Ashtekar-Barbero variables. We show that the Gauss constraint multiplier must fall off slower than previously considered in order to recover ADM phase space. The generators of the asymptotic Poincare group are derived within the Ashtekar-Barbero phase space without reference to the ADM generators. The resulting expressions are shown to agree, modulo Gauss constraint terms, with those obtained from the ADM generators. A payoff of this procedure is a new expression for the generator of asymptotic rotations which is polynomia…
In order to introduce an interaction between gravity and fermions in the Ashtekar-Barbero-Immirzi formalism without affecting classical dynamics a non-minimal term is necessary. The non-minimal term together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant. As a consequence the Immirzi parameter, differently from the minimal coupling approach, does not affect the classical dynamics, which is described by the Einstein-Cartan action.
The Ashtekar-Barbero-Immirzi formulation of General Relativity is extended to include spinor matter fields. Our formulation applies to generic values of the Immirzi parameter and reduces to the Ashtekar-Romano-Tate approach when the Immirzi parameter is taken equal to the imaginary unit. The dynamics of the gravity-fermions coupled system is described by the Holst plus Dirac action with a non-minimal coupling term. The non-minimal interaction together with the Holst modification to the Hilbert-Palatini action reconstruct the Nieh-Yan invariant, so that the effective action coming out is the on…
An so(4,C)-covariant hamiltonian formulation of a family of generalized Hilbert-Palatini actions depending on a parameter (the so called Immirzi parameter) is developed. It encompasses the Ashtekar-Barbero gravity which serves as a basis of quantum loop gravity. Dirac quantization of this system is constructed. Next we study dependence of the quantum system on the Immirzi parameter. The path integral quantization shows no dependence on it. A way to modify the loop approach in the accordance with the formalism developed here is briefly outlined.
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…
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.
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 …
We show that within the Ashtekar formulation of General Relativity a considerably simple and compact form of the Lorentzian Hamiltonian constraint occurs for a particular value of the Barbero--Immirzi parameter.
This letter is a critique of Barbero's constrained Hamiltonian formulation of General Relativity on which current work in Loop Quantum Gravity is based. While we do not dispute the correctness of Barbero's formulation of general relativity, we offer some criticisms of an aesthetic nature. We point out that unlike Ashtekar's complex SU(2) connection, Barbero's real SO(3) connection does not admit an interpretation as a space-time gauge field. We show that if one tries to interpret Barbero's real SO(3) connection as a space-time gauge field, the theory is not diffeomorphism invariant. We conclud…
We formulate axisymmetric general relativity in terms of real Ashtekar--Barbero variables. We study the constraints and equations of motion and show how the Kerr, Schwarzschild and Minkowski solutions arise. We also discuss boundary conditions. This opens the possibility of a midisuperspace quantization using loop quantum gravity techniques for spacetimes with axial symmetry and time dependence.
We generalize the Lagrangian of N = 1 supergravity (SUGRA) by using an arbitrary parameter $ξ$, which corresponds to the inverse of Barbero's parameter $β$. This generalized Lagrangian involves the chiral one as a special case of the value $ξ= \pm i$. We show that the generalized Lagrangian gives the canonical formulation of N = 1 SUGRA with the real Ashtekar variable after the 3+1 decomposition of spacetime. This canonical formulation is also derived from those of the usual N = 1 SUGRA by performing Barbero's type canonical transformation with an arbitrary parameter $β(=ξ^{-1})$. We give some…
We analyze the semiclassical and quantum polymer dynamics of the isotropic Universe in terms of both the standard Ashtekar-Barbero-Immirzi connection and its conjugate momentum and also of the new generalized coordinate conjugate to the Universe volume. We study the properties of the resulting bouncing cosmology that emerges in both the representations and we show that the Big Bounce is an intrinsic cut-off on the cosmological dynamics only when the volume variable is implemented, while in terms of the standard connection the Universe Bounce energy density is fixed by the initial conditions on…
We study and compare the spectra of geometric operators (length and area) in the quantum kinematics of two formulations of three-dimensional Lorentzian loop quantum gravity. In the SU(2) Ashtekar-Barbero framework, the spectra are discrete and depend on the Barbero-Immirzi parameter $γ$ exactly like in the four-dimensional case. However, we show that when working with the self-dual variables and imposing the reality conditions the spectra become continuous and $γ$-independent.
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…
Starting from a Hamiltonian description of four dimensional general relativity in presence of a cosmological constant we perform the program of canonical quantisation. This is done using complex Ashtekar variables while keeping the Barbero--Immirzi parameter real. Introducing the SL(2,C) Kodama state formally solving all first class constraints we propose a spinfoam vertex amplitude. We construct SL(2,C) boundary spinnetwork functions coloured by finite dimensional representations of the group, and derive the skein relations needed to calculate the amplitude. The space of boundary states is sh…
We derive the Ashtekar-Barbero variables in Loop Quantum Gravity(LQG) starting from the Palatini action. We find that we need either the torsion-free condition or the compatibility condition for the Palatini action to describe General Relativity.
Using self dual Ashtekar variables, we investigate (at the effective level) the spherically symmetry reduced model of loop quantum gravity, both in vacuum and when coupled to a scalar field. Within the real Ashtekar-Barbero formulation, the system scalar field coupled to spherically symmetric gravity is known to possess a non closed (quantum) algebra of constraints once the holonomy corrections are introduced, which forbids the loop quantization of the model. Moreover, the vacuum case, while not anomalous, introduces modifications which are usually interpreted as an effective signature change …
We propose an unified approach to loop quantum gravity and Fedosov quantization of gravity following the geometry of double spacetime fibrations and their quantum deformations. There are considered pseudo-Riemannian manifolds enabled with 1) a nonholonomic 2+2 distribution defining a nonlinear connection (N-connection) structure and 2) an Arnowitt-Deser-Misner 3+1 decomposition. The Ashtekar-Barbero variables are generalized and adapted to the N-connection structure which allows us to write the general relativity theory equivalently in terms of Lagrange-Finsler variables and related canonical …
We introduce a holomorphic representation for the Lorentzian EPRL spinfoam on arbitrary 2-complexes. The representation is obtained via the Ashtekar-Lewandowski-Marolf-Mourão-Thiemann heat kernel coherent state transform. The new variables are classical holonomy-flux phase space variables $(h,X)\simeq \mathcal T^*SU(2)$ of Hamiltonian loop quantum gravity prescribing the holonomies of the Ashtekar connection $A=Γ+ γK$, and their conjugate gravitational fluxes. For small heat kernel `time' the spinfoam amplitude is peaked on classical space-time geometries, where at most countably many curvatur…
We introduce 3+1 timelike foliation of the four dimensional Lorentz manifold to derive the 3+1 Sen-Ashtekar-Barbero-Immirzi formalism in case of $SO(2,1)$ rotation gauge group, which is possible due to the existence of the $so(2,1)$ algebra isomorphism to $R^3_{2,1}$ algebra with respect to the vector product. We prove that the newly introduced flux and extrinsic curvature variables preserve the symplectic structure of the original variables. We then introduce the modified rotational constraint and succeed to write it as a Gauss constraint of a newly obtained connection. The newly obtained con…
Canonical gravity in real Ashtekar-Barbero variables is generalized to allow for fermionic matter. The resulting torsion changes several expressions in Holst's original vacuum analysis, which are summarized here. This in turn requires adaptations to the known loop quantization of gravity coupled to fermions, which is discussed on the basis of the classical analysis. As a result, parity invariance is not manifestly realized in loop quantum gravity.
We study the Geroch group in the framework of the Ashtekar formulation. In the case of the one-Killing-vector reduction, it turns out that the third column of the Ashtekar connection is essentially the gradient of the Ernst potential, which implies that the both quantities are based on the ``same'' complexification. In the two-Killing-vector reduction, we demonstrate Ehlers' and Matzner-Misner's SL(2,R) symmetries, respectively, by constructing two sets of canonical variables that realize either of the symmetries canonically, in terms of the Ashtekar variables. The conserved charges associated…
The Ashtekar-Barbero formulation of general relativity admits a one-parameter family of canonical transformations that preserves the expressions of the Gauss and diffeomorphism constraints. The loop quantization of the connection formalism based on each of these canonical sets leads to different predictions. This phenomenon is called the Immirzi ambiguity. It has been recently argued that this ambiguity could be generalized to the extent of a spatially dependent function, instead of a parameter. This would ruin the predictability of loop quantum gravity. We prove that such expectations are not…
We derive a canonical analysis of a double null 2+2 Hamiltonian description of General Relativity in terms of complex self-dual 2-forms and the associated SO(3) connection variables. The algebra of first class constraints is obtained and forms a Lie algebra that consists of two constraints that generate diffeomorphisms in the two surface, a constraint that generates diffeomorphisms along the null generators and a constraint that generates self-dual spin and boost transformations.
In quantum gravity the unitary evolution does not follow from the Wheeler-DeWitt dynamics equation as it follows from the Schrödinger equation in non-relativistic quantum mechanics. Therefore we can define a spin-foam model based on SL(2,C) spinor finite non-unitary representations. The recently discovered holomorphic Peter-Weyl theorem \cite{Huebschmann} made it possible to decompose the delta function of a non-compact Lorentz group into the convergent sum of the matrix coefficients. We calculate the vertex amplitude with the help of that theorem and obtain a simple expression for our model. …
By reintroducing Lorentz invariance in canonical loop quantum gravity, we define a geometrical notion of temperature for quantum isolated horizons. This is done by demanding that the horizon state satisfying the boundary conditions be a Kubo-Martin-Schwinger state. The exact formula for the temperature can be derived by imposing the reality conditions in the form of the linear simplicity constraints for an imaginary Barbero-Immirzi parameter. Thus, our analysis reveals the connection between the analytic continuation to the Ashtekar self-dual variables and the thermality of the horizon. The ho…
Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal $G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections on $P$ be a continuous function of the holonomies of $A$ along finitely many piecewise smoothly immersed curves in $M$, and let a generalized measure on $\A$ be a bounded linear functional on cylinder functions. We construct a generalized measure on the space of connections that extends the uniform measure of Ashtekar, Lewandowski and Baez to the smooth case, and prove it is invariant under all automorphisms of $P$, not necessarily the identity…
In attempts to combine non-commutative geometry and quantum gravity, Aastrup-Grimstrup-Nest construct a semi-finite spectral triple, modeling the space of G-connections for G=U(1) or SU(2). AGN show that the interaction between the algebra of holonomy loops and the Dirac-type operator D reproduces the Poisson structure of General Relativity in Ashtekar's loop variables. This article generalizes AGN's construction to any connected compact Lie group G. A construction of AGN's semi-finite spectral triple in terms of an inductive limit of spectral triples is formulated. The refined construction pe…
