Bouncing Universe in Loop Quantum Gravity: full theory calculation

In Loop Quantum Gravity mathematically rigorous models of full quantum gravity were proposed. In this paper we study a cosmological sector of one of the models describing quantum gravity with positive cosmological constant coupled to massless scalar field. In our previous research we introduced a method to reduce the model to homogeneous-isotropic sector at the quantum level. In this paper we propose a method to restrict to the spatially flat sector. After this restriction the number of degrees of freedom gets substantially reduced. This allows us to make numerical calculations. Remarkably, the resulting model shares some structural similarities with the Loop Quantum Cosmological models and therefore sheds some new light on the relation between Loop Quantum Gravity and Loop Quantum Cosmology. According to our model the evolution of the Universe is periodic. The quantum gravity effects resolve the Big Bang singularity leading to a Big Bounce and cause the Universe to contract after a classical expansion phase (Big Crunch).Comment: 44 pages, 8 figure

Spin-foam dynamics of Loop Quantum Gravity states

This thesis studies the dynamics of the Loop Quantum Gravity states defined by the spin-foam models of Euclidean 4D Quantum Gravity. A link between the 4D spin-foam theory and the kinematics of the (3+1) Loop Quantum Gravity (LQG) was proposed by J. Engle, R. Pereira, C. Rovelli and E. Livine. Their model, called the EPRL spin-foam model, is a promising candidate for the spin-foam model of the dynamics of the Loop Quantum Gravity states. In the original formulation, the EPRL spin-foam model is defined for triangulations and is applicable to specific LQG states. A generalization of the model to all the LQG states was proposed by W. Kamiński, J. Lewandowski and myself. Some properties of the generalized model were studied. In particular, a general framework for studying symmetries of spin-foam models was proposed. The heart of the generalization is the generalized EPRL vertex amplitude. E. Bianchi, D. Regoli and C. Rovelli proposed another spin-foam model of 4D Quantum Gravity with the generalized EPRL vertex amplitude. E. Bianchi, C. Rovelli and F. Vidotto used the generalized EPRL spin-foam model to construct the first model of Quantum Cosmology based on the spin-foam formalism. They calculated a transition amplitude between coherent states peaked on homogeneous, isotropic geometries using certain approximations. The approximations were justified a posteriori by a correct semiclassical limit of the transition amplitude. One of them was a truncation of the transition amplitude to a contribution from a single foam with one internal vertex, four internal edges and a certain boundary, which we will call a BRV foam. F. Hellmann discussed contributions from other foams with these properties, which a priori cannot be discarded. All the possible foams were listed by J. Lewandowski, J. Puchta and myself. The class of the foams considered was defined by graph diagrams, which we introduced. We expect that the contributions from the foams we have found can be neglected in the limit of large universe.Przedmiotem badań prezentowych w mojej rozprawie doktorskiej jest dynamika stanów pętlowej kwantowej grawitacji zdefiniowana przez pianowo-spinowe modele euklidesowej czterowymiarowej kwantowej grawitacji. Związek pomiędzy czterowymiarowymi teoriami pian spinowych a kinematyką pętlowej kwantowej grawitacji został zaproponowany przez J. Engle’a, R. Pereirę, C. Rovelliego i E. Livine’a. Ich model, nazywany modelem EPRL, jest dobrze zapowiadającym się kandydatem na pianowo-spinowy model dynamiki stanów pętlowej kwantowej grawitacji. W pierwotnym sformułowaniu model EPRL jest zdefiniowany dla triangulacji czasoprzestrzeni i może być stosowany tylko dla pewnych stanów pętlowej kwantowej grawitacji. Uogólnienie modelu do wszystkich stanów zostało zaproponowane przez J. Lewandowskiego, W. Kamińskiego i przeze mnie. Pewne własności uogólnionego modelu zostały zbadane. W szczególności została zaproponowana ogólna metoda badania symetrii modeli pian spinowych. Głównym elementem zaproponowanego uogólnienia jest uogólniona amplituda wierzchołka EPRL. E. Bianchi, D. Regoli i C. Rovelli zaproponowali inny pianowo-spinowy model czterowymiarowej kwantowej grawitacji z uogólnioną amplitudą wierzchołka EPRL. E. Bianchi, C. Rovelli i F. Vidotto zastosowali model w celu skonstruowania pierwszego modelu kwantowej kosmologii opartego na formalizmie pian spinowych. Stosując pewne przybliżenia, obliczyli amplitudę przejścia pomiędzy stanami koherentnymi, skupionymi na jednorodnych, izotropowych geometriach. Zastosowane przybliżenia były uzasadnione a posteriori poprzez poprawną granicę semi-klasyczną amplitudy przejścia. Jednym z zastosowanych przybliżeń było obcięcie amplitudy przejścia do wkładu pochodzącego od jednej piany mającej jeden wierzchołek wewnętrzny, cztery wewnętrzne krawędzie i pewien brzeg, którą będziemy nazywać pianą BRV. F. Hellmann przedyskutował wkłady od innych pian, które nie mogą zostać odrzucone a priori. Wszystkie możliwe piany z tymi własnościami zostały znalezione przez J. Lewandowskiego, J. Puchtę i przeze mnie. Klasa rozważanych pian została zdefiniowana przez wprowadzone przez nas diagramy grafowe. Spodziewamy się, że wkłady od znalezionych pian mogą zostać zaniedbane w granicy dużych rozmiarów wszechświata

The EPRL intertwiners and corrected partition function

Do the SU(2) intertwiners parametrize the space of the EPRL solutions to the simplicity constraint? What is a complete form of the partition function written in terms of this parametrization? We prove that the EPRL map is injective for n-valent vertex in case when it is a map from SO(3) into SO(3)xSO(3) representations. We find, however, that the EPRL map is not isometric. In the consequence, in order to be written in a SU(2) amplitude form, the formula for the partition function has to be rederived. We do it and obtain a new, complete formula for the partition function. The result goes beyond the SU(2) spin-foam models framework.Comment: RevTex4, 15 pages, 5 figures; theorem of injectivity of EPRL map correcte

One vertex spin-foams with the Dipole Cosmology boundary

We find all the spin-foams contributing in the first order of the vertex expansion to the transition amplitude of the Bianchi-Rovelli-Vidotto Dipole Cosmology model. Our algorithm is general and provides spin-foams of arbitrarily given, fixed: boundary and, respectively, a number of internal vertices. We use the recently introduced Operator Spin-Network Diagrams framework.Comment: 23 pages, 30 figure

The kernel and the injectivity of the EPRL map

In this paper we prove injectivity of the EPRL map for |\gamma|<1, filling the gap of our previous paper.Comment: 17 pages, 3 figure

Feynman diagrammatic approach to spin foams

"The Spin Foams for People Without the 3d/4d Imagination" could be an alternative title of our work. We derive spin foams from operator spin network diagrams} we introduce. Our diagrams are the spin network analogy of the Feynman diagrams. Their framework is compatible with the framework of Loop Quantum Gravity. For every operator spin network diagram we construct a corresponding operator spin foam. Admitting all the spin networks of LQG and all possible diagrams leads to a clearly defined large class of operator spin foams. In this way our framework provides a proposal for a class of 2-cell complexes that should be used in the spin foam theories of LQG. Within this class, our diagrams are just equivalent to the spin foams. The advantage, however, in the diagram framework is, that it is self contained, all the amplitudes can be calculated directly from the diagrams without explicit visualization of the corresponding spin foams. The spin network diagram operators and amplitudes are consistently defined on their own. Each diagram encodes all the combinatorial information. We illustrate applications of our diagrams: we introduce a diagram definition of Rovelli's surface amplitudes as well as of the canonical transition amplitudes. Importantly, our operator spin network diagrams are defined in a sufficiently general way to accommodate all the versions of the EPRL or the FK model, as well as other possible models. The diagrams are also compatible with the structure of the LQG Hamiltonian operators, what is an additional advantage. Finally, a scheme for a complete definition of a spin foam theory by declaring a set of interaction vertices emerges from the examples presented at the end of the paper.Comment: 36 pages, 23 figure