arXiv subjects
Publications and source records attributed to Marcin Kisielowski.
The quantum dynamics of a self-gravitating thin matter shell in vacuum has been considered. Quantum Hamiltonian of the system is positive definite. Within chosen set of parameters, the quantum shell bounces above the horizon. Considered quantum system does not collapse to the gravitational singularity of the corresponding classical system.
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).
Recently, a new class of scalar constraint operators has been introduced in loop quantum gravity. They are defined on a space of solutions to the Gauss constraint and partial solutions to the vector constraint, called a vertex Hilbert space. We propose a subspace of the vertex Hilbert space formed by homogeneous-isotropic states, which is invariant under the action of the new scalar constraint operators. As a result, the operators can be reduced to our homogeneous-isotropic subspace. The (generalized) eigenstates of the reduced operator are eigenstates of the full operator. We discuss the feasibility of numerical diagonalization of the reduced scalar constraint operator.
We study some properties of the Rovelli-Smolin-DePietri volume operator in loop quantum gravity, which significantly simplify the diagonalization problem and shed some light on the pattern of degeneracy of the eigenstates. The operator is defined by its action in the spaces of tensor products $\mathcal{H}_{j_1}\otimes \ldots \otimes \mathcal{H}_{j_N}$ of the irreducible SU(2) representation spaces $\mathcal{H}_{j_i}, i=1,\ldots,N$, labelled with spins $j_i\in \frac{1}{2}\mathbb{N}$. We restrict to the spaces of SU(2) invariant tensors (intertwiners) with all spins equal $j_1=\ldots=j_N=j$. We call them spin $j$ monochromatic intertwiners. Such spaces are important in the study of SU(2) gauge invariant states that are isotropic and can be applied to extract the cosmological sector of the theory. In the case of spin $1/2$ we solve the eigenvalue problem completely: we show that the volume operator is proportional to identity and calculate the proportionality factor.
We perform the stationary phase analysis of the vertex amplitude for the EPRL spin foam model extended to include timelike tetrahedra. We analyse both, tetrahedra of signature $---$ (standard EPRL), as well as of signature $+--$ (Hnybida-Conrady extension), in a unified fashion. However, we assume all faces to be of signature $--$. The stationary points of the extended model are described again by $4$-simplices and the phase of the amplitude is equal to the Regge action. Interestingly, in addition to the Lorentzian and Euclidean sectors there appear also split signature $4$-simplices.
A spin-foam model is derived from the canonical model of Loop Quantum Gravity coupled to a massless scalar field. We generalized to the full theory the scheme first proposed in the context of Loop Quantum Cosmology by Ashtekar, Campiglia and Henderson, later developed by Henderson, Rovelli, Vidotto and Wilson-Ewing.
In Regge calculus the space-time manifold is approximated by certain abstract simplicial complex, called a pseudo-manifold, and the metric is approximated by an assignment of a length to each 1-simplex. In this paper for each pseudomanifold we construct a smooth manifold which we call a manifold with defects. This manifold emerges from the purely combinatorial simplicial complex as a result of gluing geometric realizations of its n-simplices followed by removing the simplices of dimension n-2. The Regge geometry is encoded in a boundary data of a BF-theory on this manifold. We construct an action functional which coincides with the standard BF action for suitably regular manifolds with defects and fields. On the other hand, the action evaluated at solutions of the field equations satisfying certain boundary conditions coincides with an evaluation of the Regge action at Regge geometries defined by the boundary data. As a result we trade the degrees of freedom of Regge calculus for discrete degrees of freedom of topological BF theory.
We have introduced Faddeev-Niemi type variables for static SU(3) Yang-Mills theory. The variables suggest that a non-linear sigma model whose sigma fields take values in SU(3)/(U(1)xU(1)) and SU(3)/(SU(2)xU(1)) may be relevant to infrared limit of the theory. Shabanov showed that the energy functional of the non-linear sigma model is bounded from below by certain functional. However, the Shabanov's functional is not homotopy invariant, and its value can be an arbitrary real number -- therefore it is not a topological charge. Since the third homotopy group of SU(3)/(U(1)xU(1)) is isomorphic to the group of integer numbers, there is a non-trivial topological charge (given by the isomorphism). We apply Novikov's procedure to obtain integral expression for this charge. The resulting formula is analogous to the Whitehead's realization of the Hopf invariant.
We propose new variables of Faddeev-Niemi type for static SU(3) Yang-Mills theory. These variables reveal a structure of a nonlinear sigma model, whose field variables are two chiral fields taking values in SU(3)/(U(1)xU(1)) and SU(3)/(SU(2)xU(1)). The nonlinear sigma model was introduced by Faddeev and Niemi as a natural extension of the Faddeev chiral model. Shabanov showed that the energy functional of the extended model is bounded from below by a topological invariant, and therefore may support knot-like excitations and a mass gap.
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.
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.
The simplicial framework of Engle-Pereira-Rovelli-Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity. The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams. The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations). In particular the notion of embedded spin-foam we use allows to consider knotting or linking spin-foam histories. Also the main tools as the vertex structure and the vertex amplitude are naturally generalized to arbitrary valency case. The correspondence between all the SU(2) intertwiners and the SU(2)$\times$SU(2) EPRL intertwiners is proved to be 1-1 in the case of the Barbero-Immirzi parameter $|γ|\ge 1$, unless the co-domain of the EPRL map is trivial and the domain is non-trivial.
In this paper we prove injectivity of the EPRL map for |γ|<1, filling the gap of our previous paper.
"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.
The goal of this paper is to introduce a systematic approach to spin foams. We define operator spin foams, that is foams labelled by group representations and operators, as the main tool. An equivalence relation we impose in the set of the operator spin foams allows to split the faces and the edges of the foams. The consistency with that relation requires introduction of the (familiar for the BF theory) face amplitude. The operator spin foam models are defined quite generally. Imposing a maximal symmetry leads to a family we call natural operator spin foam models. This symmetry, combined with demanding consistency with splitting the edges, determines a complete characterization of a general natural model. It can be obtained by applying arbitrary (quantum) constraints on an arbitrary BF spin foam model. In particular, imposing suitable constraints on Spin(4) BF spin foam model is exactly the way we tend to view 4d quantum gravity, starting with the BC model and continuing with the EPRL or FK models. That makes our framework directly applicable to those models. Specifically, our operator spin foam framework can be translated into the language of spin foams and partition functions. We discuss the examples: BF spin foam model, the BC model, and the model obtained by application of our framework to the EPRL intertwiners.