Sandipan Sengupta, Romesh K. Kaul
A canonical formulation of the N=1 supergravity theory containing the topological Nieh-Yan term in its Lagrangian density is developed. The constraints are analysed without choosing any gauge. In the time gauge, the theory is shown to be described in terms of real SU(2) variables.
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A general argument provides the motivation to consider the Barbero--Immirzi parameter as a field. The specific form of the geometrical effective action allows to relate the value of the Barbero--Immirzi parameter to other quantum ambiguities through the analog of the Peccei--Quinn mechanism.
It has been shown that Sakharov's induced, from the fields entering the standard model, Barbero-Immirzi parameter assumes, in the framework of euclidean formalism, the UV cutoff-independent value, 1/9. The calculus uses the Schwinger's proper-time formalism, the Seeley-DeWitt heat-kernel expansion, and it is akin to the derivation of the ABJ chiral anomaly in space-time with torsion.
The Barbero-Immirzi (BI) parameter is promoted to a field and a canonical analysis is performed when it is coupled with a Nieh-Yan topological invariant. It is shown that, in the effective theory, the BI field is a canonical pseudoscalar minimally coupled with gravity. This framework is argued to be more natural than the one of the usual Holst action. Potential consequences in relation with inflation and the quantum theory are briefly discussed.
We set up a canonical Hamiltonian formulation for a theory of gravity based on a Lagrangian density made up of the Hilbert-Palatini term and, instead of the Holst term, the Nieh-Yan topological density. The resulting set of constraints in the time gauge are shown to lead to a theory in terms of a real SU(2) connection which is exactly the same as that of Barbero and Immirzi with the coefficient of the Nieh-Yan term identified as the inverse of Barbero-Immirzi parameter. This provides a topological interpretation for this parameter. Matter coupling can then be introduced in the usual manner, {\…
In this note, we review the canonical analysis of the Holst action in the time gauge, with a special emphasis on the Hamiltonian equations of motion and the fixation of the Lagrange multipliers. This enables us to identify at the Hamiltonian level the various components of the covariant torsion tensor, which have to be vanishing in order for the classical theory not to depend upon the Barbero-Immirzi parameter. We also introduce a formulation of three-dimensional gravity with an explicit phase space dependency on the Barbero-Immirzi parameter as a potential way to investigate its fate and rele…
We consider gravity in 2+1 space-time dimensions, with negative cosmological constant and a `Barbero-Immirzi' (B-I) like parameter, when the space-time topology is of the form $ T^2 \times \mathbbm{R}$. The phase space structure, both in covariant and canonical framework is analyzed. Full quantization of the theory in the 'constrain first' approach reveals a finite dimensional physical Hilbert space. An explicit construction of wave functions is presented. The dimension of the Hilbert space is found to depend on the `Barbero-Immirzi' like parameter in an interesting fashion. Comparative study …
The 1-parameter family of transformations identified by Barbero and Immirzi plays a significant role in non-perturbative approaches to quantum gravity, among them Loop Quantum Gravity and Spin Foams. It facilitates the loop quantization programme and subsequently the Barbero-Immirzi parameter (gamma) arises in both the spectra of geometrical operators and in the dynamics provided by Spin Foams. However, the debate continues as to whether quantum physics should be Barbero-Immirzi parameter dependent. Starting from a discrete SO(4)-BF theory phase space, we find two possible reductions with resp…
We embed the Loop Quantum Gravity Barbero-Immirzi parameter and field within an action describing 4D, $\cal N$ = 1 supergravity and thus within a Low Energy Effective Action of Superstring/M-Theory. We use the fully gauge-covariant description of supergravity in (curved) superspace. The gravitational constant is replaced with the vacuum expectation value of a scalar field, which in local supersymmetry is promoted to a complex, covariantly chiral scalar superfield. The imaginary part of this superfield couples to a supersymmetric Holst term. The Holst term also serves as a starting point in the…
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…
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…
In Loop Quantum Gravity the classical point of departure is the Einstein-Hilbert action modified by the addition of the so-called Holst term. Classically, this term does not affect the equations of motion, but it induces a well-known quantization ambiguity in the quantum theory, parametrized by the Barbero--Immirzi parameter. Recently, it has been suggested to promote the Barbero--Immirzi parameter to a field. The resulting theory, obtainable starting from the usual Holst action, is General Relativity coupled to a pseudo-scalar field. However, this theory turns out to have an unconventional ki…
We consider Regge calculus in the representation in terms of area tensors and self- and antiselfdual connections generalised to the case of Holst action that is standard Einstein action in the tetrad-connection variables plus topological (on equations of motion for connections) term with coefficient $1/γ$, $γ$ is Barbero-Immirzi parameter. The quantum measure is shown to exponentially decrease with areas with typical cut-off scales $4πG$ and $4πGγ$ in spacelike and timelike regions, respectively ($G$ is the Newton constant).
We discuss how: (i) a dilaton/axion superfield can play the role of a Barbero-Immirzi field in four-dimensional conformal quantum supergravity theories, (ii) a fermionic component of such a dilaton/axion superfield may play the role of a Goldstino in the low-energy effective action obtained from a superstring theory with F-type global supersymmetry breaking, (iii) this global supersymmetry breaking is communicated to the gravitational sector via the supergravity coupling of the Goldstino, and (iv) such a scenario may be realized explicitly in a D-foam model with D-particle defects fluctuating …
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.
We consider cosmological solution for Einstein gravity with massive fermions with a four-fermion coupling, which emerges from the Holst action and is related to the Barbero-Immirzi (BI) parameter. This gravitational action is an important object of investigation in a non-perturbative formalism of quantum gravity. We study the equation of motion for for the Dirac field within the standard Friedman-Robertson-Walker (FRW) metric. Finally, we show the theory with BI parameter and minimally coupling Dirac field, in the zero mass limit, is equivalent to an additional term which looks like a perfect …
In this paper, we analyse a curvature- and torsion-square quantum gravity action with an additional Holst term minimally coupled to a massive Dirac field in four dimensions. The main purpose here is to try to estimate and compare the value of the Barbero-Immirzi (BI) parameter with its currently known results. To do that, we work out the physical mass of the fermion as a function of this parameter in a perturbative one-loop calculation, assuming the scenario of a physics at the TeV-scale.
The Barbero-Immirzi parameter is a one parameter quantization ambiguity underpinning the loop approach to quantum gravity that bears tantalizing similarities to the theta parameter of gauge theories such as Yang-Mills and QCD. Despite the apparent semblance, the Barbero-Immirzi field has resisted a direct topological interpretation along the same lines as the theta-parameter. Here we offer such an interpretation. Our approach begins from the perspective of Einstein-Cartan gravity as the symmetry broken phase of a de Sitter gauge theory. From this angle, just as in ordinary gauge theories, a th…
We consider a loop-quantum gravity inspired modification of general relativity, where the Holst action is generalized by making the Barbero-Immirzi (BI) parameter a scalar field, whose value could be dynamically determined. The modified theory leads to a non-zero torsion tensor that corrects the field equations through quadratic first-derivatives of the BI field. Such a correction is equivalent to general relativity in the presence of a scalar field with non-trivial kinetic energy. This stress-energy of this field is automatically covariantly conserved by its own dynamical equations of motion,…
We study the semiclassical behavior of Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam model, by taking into account of the sum over spins in the large spin regime. The large spin parameter λand small Barbero-Immirzi parameter γare treated as two independent parameters for the asymptotic expansion of spinfoam state-sum (such an idea was firstly pointed out in arXiv:1105.0216). Interestingly, there are two different spin regimes: 1<<γ^{-1}<<λ<<γ^{-2} and λ>γ^{-2}. The model in two spin regimes has dramatically different number of effective degrees of freedom. In 1<<γ^{-1}<<λ<<γ^{-2}, th…
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…
The canonical analysis of the (anti-) self-dual action for gravity supplemented with the (anti-) self-dual Pontrjagin term is carried out. The effect of the topological term is to add a `magnetic' term to the original momentum variable associated with the self-dual action leaving the Ashtekar connection unmodified. In the new variables, the Gauss constraint retains its form, while both vector and Hamiltonian constraints are modified. This shows, the contribution of the Euler and Pontrjagin terms is not the same as that coming from the term associated with the Barbero-Immirzi parameter, and thu…
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…
Holst action containing Immirzi parameter for pure gravity is generalised to the supergravity theories. Supergravity equations of motion are not modified by such generalisations, thus preserving supersymmetry. Dependence on the Immirzi parameter does not emerge in the classical equations of motion. This is in contrast with the recent observation of Perez and Rovelli for gravity action containing original Holst term and a minimally coupled Dirac fermion where the classical equations of motion do develop a dependence on Immirzi parameter.
In four dimensional gravity theory, the Barbero-Immirzi parameter has a topological origin, and can be identified as the coefficient multiplying the Nieh-Yan topological density in the gravity Lagrangian, as proposed by Date et al.[1]. Based on this fact, a first order action formulation for spacetimes with boundaries is introduced. The bulk Lagrangian, containing the Nieh-Yan density, needs to be supplemented with suitable boundary terms so that it leads to a well-defined variational principle. Within this general framework, we analyse spacetimes with and without a cosmological constant. For …
We show that, in the first order gravity theory coupled to axions, the instanton number of the Giddings-Strominger wormhole can be interpreted as the Nieh-Yan topological index. The axion charge of the baby universes is quantized in terms of the Nieh-Yan integers. Tunneling between universes of different Nieh-Yan charges implies a nonperturbative vacuum state. The associated topological vacuum angle can be identified with the Barbero-Immirzi parameter.
Spinfoams, $γ$-duality and parity violation in primordial gravitational waves
研究通过γ-对称性探讨引力宇称破坏，分析有效场论中引力和标量场的耦合常数关系，揭示其在早期宇宙膨胀中的测量可能性。
ER = EPR in Loop Quantum Gravity: the Immirzi Parameter and the Continuum Limit
本文将 ER=EPR 的有限区域分析嵌入圈量子引力，建立纠缠与量子几何的精确字典，从中推导出 Immirzi 参数并使黑洞熵系数普适化，同时构造了抑制发散并给出连续极限的自旋泡沫重整化流。
In this article, the Cartan geometric approach toward (extended) supergravity in the presence of boundaries will be discussed. In particular, based on new developments in this field, we will derive the Holst variant of the MacDowell-Mansouri action for $\mathcal{N}=1$ and $\mathcal{N}=2$ pure AdS supergravity in $D=4$ for arbitrary Barbero-Immirzi parameters. This action turns out to play a crucial role in context of boundaries in the framework of supergravity if one imposes supersymmetry invariance at the boundary. For the $\mathcal{N}=2$ case, it follows that this amounts to the introduction…
Thesis provides an analysis of various aspects theory of gravity seen as a deformation of a topological SO(2,3) BF theory. Considered framework, originating from the 70's and known as MacDowell-Mansouri gravity, assures the most general form of action, containing the Einstein-Cartan action with a negative cosmological constant, the Holst term, and the topological Euler, Pontryagin and Nieh-Yan invariants. Main goal was to investigate the presence of Barbero-Immirzi parameter in the context of supergravity, as well as black hole thermodynamics arising from the gravitational Noether charges.
In Loop Quantum Gravity (LQG) the quantisation of General Relativity leads to precise predictions for the eigenvalues of geometrical observables like volume and area, up to the value of the only free parameter of the theory, the Barbero-Immirzi (BI) parameter. With the help of the eigenvalues equation for the area operator, LQG successfully derives the Bekenstein-Hawking entropy of large black holes with isolated horizons, fixing at this limit the BI parameter as $γ\approx 0.274$. In the present paper we show some evidence that a black hole with angular momentum $\hbar$ and Planck mass is an e…
We present the Hamiltonian analysis of the theory of gravity based on a Lagrangian density containing Hilbert-Palatini term along with three topological densities, Nieh-Yan, Pontryagin and Euler. The addition of these topological terms modifies the symplectic structure non-trivially. The resulting canonical theory develops a dependence on three parameters which are coefficients of these terms. In the time gauge, we obtain a real SU (2) gauge theoretic description with a set of seven first class constraints corresponding to three SU (2) rotations, three spatial diffeomorphism and one to evoluti…
Holst term represents an interesting addition to the Einstein-Cartan theory of gravity with torsion. When this term is present the contact interactions between vector and axial vector fermion currents gain an extra parity-violating component. We re-derive this interaction using a simple representation for the Holst term. The same representation serves as a useful basis for the calculation of one-loop divergences in the theory with external fermionic currents and cosmological constant. Furthermore, we explore the possibilities of the on-shell version of renormalization group and construct the e…
It has been shown that the old-fashioned idea of Sakharov's induced gravity and gauge interactions in the "one-loop dominance" version works astonishingly well yielding reasonable parameters. It appears that induced coupling constants of gauge interactions of the standard model assume qualitatively realistic values. Moreover, it is possible to induce the Barbero--Immirzi parameter of canonical gravity from the fields entering the standard model.
In this paper we formulate ${\cal N}=1$ supergravity as a constrained $BF$ theory with $OSp(4|1)$ gauge superalgebra. We derive the modified supergravity Lagrangian that, apart from the standard supergravity with negative cosmological constant, contains terms proportional to the (inverse of) Immirzi parameter. Although these terms do not change classical field equations, they might be relevant in quantum theory. We briefly discuss the perturbation theory around the supersymmetric topological vacuum.
Canonical Gravity in Degenerate Limit
本文构造了空间三标架行列式趋于零的哈密顿引力退化极限，在Barbero-Immirzi SU(2)表述下给出两种实现，使哈密顿约束简化并揭示卡罗尔引力作为特例，提供新几何解释。
One ambiguity in loop quantum gravity is the appearance of a free parameter which is called Immirzi parameter. Recently Dreyer has argued that this parameter may be fixed by considering the quasinormal mode spectrum of black holes, while at the price of changing the gauge group to SO(3) rather than the original one SU(2). Physically such a replacement is not quite natural or desirable. In this paper we study the relationship between the black hole entropy and the quasi normal mode spectrum in the loop quantization of N=1 supergravity. We find that a single value of the Immirzi parameter agrees…
We study the coupling of the gravitational action, which is a linear combination of the Hilbert-Palatini term and the quadratic torsion term, to the action of Dirac fermions. The system possesses local Poincare invariance and hence belongs to Poincare gauge theory with matter. The complete Hamiltonian analysis of the theory is carried out without gauge fixing but under certain ansatz on the coupling parameters, which leads to a consistent connection dynamics with second-class constraints and torsion. After performing a partial gauge fixing, all second-class constraints can be solved, and a SU(…
By identifying the recently introduced Barbero-Immirzi field with the QCD axion, the strong CP problem can be solved through the Peccei-Quinn mechanism. A specific energy scale for the Peccei-Quinn symmetry breaking is naturally predicted by this model. This provides a complete dynamical setting to evaluate the contribution of such an axion to the cold dark matter content of the Universe. Furthermore, a tight upper bound on the tensor-to-scalar ratio production of primordial gravitational waves can be fixed, representing a strong experimental test for this model.
In the Loop Quantum Gravity, black holes (or even more general Isolated Horizons) are described by a SU(2) Chern-Simons theory. There is an equivalent formulation of the horizon degrees of freedom in terms of a U(1) gauge theory which is just a gauged fixed version of the SU(2) theory. These developments will be surveyed here. Quantum theory based on either formulation can be used to count the horizon micro-states associated with quantum geometry fluctuations and from this the micro-canonical entropy can be obtained. We shall review the computation in SU(2) formulation. Leading term in the ent…
In a consistent heterotic string theory, the Kalb-Ramond field, which is the source of spacetime torsion, is augmented by Yang-Mills and gravitational Chern-Simons terms. When compactified to 4-dimensions and in the field theory limit, such additional terms give rise to interactions with interesting astrophysical predictions like rotation of plane of polarization for electromagnetic and gravitational waves. On the other hand, if one is also interested in coupling 2 or 3-form (Abelian or non-Abelian) gauge fields to torsion, one needs another class of interaction. In this paper, we shall study …
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