Bianchi I model in terms of nonstandard loop quantum cosmology: Quantum dynamics

We analyze the quantum Bianchi I model in the setting of the nonstandard loop quantum cosmology. Elementary observables are used to quantize the volume operator. The spectrum of the volume operator is bounded from below and discrete. The discreteness may imply a foamy structure of spacetime at semiclassical level. The results are described in terms of a free parameter specifying loop geometry to be determined in astro-cosmo observations. An evolution of the quantum model is generated by the so-called true Hamiltonian, which enables an introduction of a time parameter valued in the set of all real numbers.Comment: 18 pages, version accepted for publication by Class. Quant. Gra

Degenerate Plebanski Sector and Spin Foam Quantization

We show that the degenerate sector of Spin(4) Plebanski formulation of four-dimensional gravity is exactly solvable and describes covariantly embedded SU(2) BF theory. This fact ensures that its spin foam quantization is given by the SU(2) Crane-Yetter model and allows to test various approaches of imposing the simplicity constraints. Our analysis strongly suggests that restricting representations and intertwiners in the state sum for Spin(4) BF theory is not sufficient to get the correct vertex amplitude. Instead, for a general theory of Plebanski type, we propose a quantization procedure which is by construction equivalent to the canonical path integral quantization and, being applied to our model, reproduces the SU(2) Crane-Yetter state sum. A characteristic feature of this procedure is the use of secondary second class constraints on an equal footing with the primary simplicity constraints, which leads to a new formula for the vertex amplitude.Comment: 34 pages; changes in the abstract and introduction, a few references adde

Unimodular Loop Quantum Cosmology

Unimodular gravity is based on a modification of the usual Einstein-Hilbert action that allows one to recover general relativity with a dynamical cosmological constant. It also has the interesting property of providing, as the momentum conjugate to the cosmological constant, an emergent clock variable. In this paper we investigate the cosmological reduction of unimodular gravity, and its quantization within the framework of flat homogeneous and isotropic loop quantum cosmology. It is shown that the unimodular clock can be used to construct the physical state space, and that the fundamental features of the previous models featuring scalar field clocks are reproduced. In particular, the classical singularity is replaced by a quantum bounce, which takes place in the same condition as obtained previously. We also find that requirement of semi-classicality demands the expectation value of the cosmological constant to be small (in Planck units). The relation to spin foam models is also studied, and we show that the use of the unimodular time variable leads to a unique vertex expansion.Comment: 26 pages. Revised version taking into account referee's comment

Lorentz-covariant Hamiltonian analysis of BF gravity with the Immirzi parameter

We perform the Lorentz-covariant Hamiltonian analysis of two Lagrangian action principles that describe general relativity as a constrained BF theory and that include the Immirzi parameter. The relation between these two Lagrangian actions has been already studied through a map among the fields involved. The main difference between these is the way the Immirzi parameter is included, since in one of them the Immirzi parameter is included explicitly in the BF terms, whereas in the other (the CMPR action) it is in the constraint on the B fields. In this work we continue the analysis of their relationship but at the Hamiltonian level. Particularly, we are interested in seeing how the above difference appears in the constraint structure of both action principles. We find that they both possess the same number of first-class and second-class constraints and satisfy a very similar (off-shell) Poisson-bracket algebra on account of the type of canonical variables employed. The two algebras can be transformed into each other by making a suitable change of variablesComment: LaTeX file, no figure

Loop quantum gravity without the Hamiltonian constraint

We show that under certain technical assumptions, including the existence of a constant mean curvature (CMC) slice and strict positivity of the scalar field, general relativity conformally coupled to a scalar field can be quantised on a partially reduced phase space, meaning reduced only with respect to the Hamiltonian constraint and a proper gauge fixing. More precisely, we introduce, in close analogy to shape dynamics, the generator of a local conformal transformation acting on both, the metric and the scalar field, which coincides with the CMC gauge condition. A new metric, which is invariant under this transformation, is constructed and used to define connection variables which can be quantised by standard loop quantum gravity methods. While it is hard to address dynamical problems in this framework (due to the complicated 'time' function), it seems, due to good accessibility properties of the CMC gauge, to be well suited for problems such as the computation of black hole entropy, where actual physical states can be counted and the dynamics is only of indirect importance. The corresponding calculation yields the surprising result that the usual prescription of fixing the Barbero-Immirzi parameter beta to a constant value in order to obtain the well-known formula S = a(Phi) A/(4G) does not work for the black holes under consideration, while a recently proposed prescription involving an analytic continuation of beta to the case of a self-dual space-time connection yields the correct result. Also, the interpretation of the geometric operators gets an interesting twist, which exemplifies the deep relationship between observables and the choice of a time function and has consequences for loop quantum cosmology.Comment: 8 pages. v2: Journal version. Black hole state counting based on physical states added. Applications to loop quantum cosmology discussed. Gauge condition used shown to coincide with CMC gauge. Minor clarifications. v3: Erroneous topology dependence of the entropy in journal version corrected, conclusions fixed accordingly. Main results unaffecte

Super-Group Field Cosmology

In this paper we construct a model for group field cosmology. The classical equations of motion for the non-interactive part of this model generate the Hamiltonian constraint of loop quantum gravity for a homogeneous isotropic universe filled with a scalar matter field. The interactions represent topology changing processes that occurs due to joining and splitting of universes. These universes in the multiverse are assumed to obey both bosonic and fermionic statistics, and so a supersymmetric multiverse is constructed using superspace formalism. We also introduce gauge symmetry in this model. The supersymmetry and gauge symmetry are introduced at the level of third quantized fields, and not the second quantized ones. This is the first time that supersymmetry has been discussed at the level of third quantized fields.Comment: 14 pages, 0 figures, accepted for publication in Class. Quant. Gra

Classical Setting and Effective Dynamics for Spinfoam Cosmology

We explore how to extract effective dynamics from loop quantum gravity and spinfoams truncated to a finite fixed graph, with the hope of modeling symmetry-reduced gravitational systems. We particularize our study to the 2-vertex graph with N links. We describe the canonical data using the recent formulation of the phase space in terms of spinors, and implement a symmetry-reduction to the homogeneous and isotropic sector. From the canonical point of view, we construct a consistent Hamiltonian for the model and discuss its relation with Friedmann-Robertson-Walker cosmologies. Then, we analyze the dynamics from the spinfoam approach. We compute exactly the transition amplitude between initial and final coherent spin networks states with support on the 2-vertex graph, for the choice of the simplest two-complex (with a single space-time vertex). The transition amplitude verifies an exact differential equation that agrees with the Hamiltonian constructed previously. Thus, in our simple setting we clarify the link between the canonical and the covariant formalisms.Comment: 38 pages, v2: Link with discretized loop quantum gravity made explicit and emphasize

From the discrete to the continuous - towards a cylindrically consistent dynamics

Discrete models usually represent approximations to continuum physics. Cylindrical consistency provides a framework in which discretizations mirror exactly the continuum limit. Being a standard tool for the kinematics of loop quantum gravity we propose a coarse graining procedure that aims at constructing a cylindrically consistent dynamics in the form of transition amplitudes and Hamilton's principal functions. The coarse graining procedure, which is motivated by tensor network renormalization methods, provides a systematic approximation scheme towards this end. A crucial role in this coarse graining scheme is played by embedding maps that allow the interpretation of discrete boundary data as continuum configurations. These embedding maps should be selected according to the dynamics of the system, as a choice of embedding maps will determine a truncation of the renormalization flow.Comment: 22 page

The Holst Spin Foam Model via Cubulations

Spin foam models are an attempt for a covariant, or path integral formulation of canonical loop quantum gravity. The construction of such models usually rely on the Plebanski formulation of general relativity as a constrained BF theory and is based on the discretization of the action on a simplicial triangulation, which may be viewed as an ultraviolet regulator. The triangulation dependence can be removed by means of group field theory techniques, which allows one to sum over all triangulations. The main tasks for these models are the correct quantum implementation of the Plebanski constraints, the existence of a semiclassical sector implementing additional "Regge-like" constraints arising from simplicial triangulations, and the definition of the physical inner product of loop quantum gravity via group field theory. Here we propose a new approach to tackle these issues stemming directly from the Holst action for general relativity, which is also a proper starting point for canonical loop quantum gravity. The discretization is performed by means of a "cubulation" of the manifold rather than a triangulation. We give a direct interpretation of the resulting spin foam model as a generating functional for the n-point functions on the physical Hilbert space at finite regulator. This paper focuses on ideas and tasks to be performed before the model can be taken seriously. However, our analysis reveals some interesting features of this model: first, the structure of its amplitudes differs from the standard spin foam models. Second, the tetrad n-point functions admit a "Wick-like" structure. Third, the restriction to simple representations does not automatically occur -- unless one makes use of the time gauge, just as in the classical theory.Comment: 25 pages, 1 figure; v3: published version. arXiv admin note: substantial text overlap with arXiv:0911.213