arXiv subjects
Publications and source records attributed to A. Mikovic.
We show that self-dual Bianchi IX and VIII cosmological models are described by the Nahm dynamical system for an appropriate type of matrices. We construct the general solutions in the case of the diagonal reductions of the corresponding Nahm equations and give the explicit expressions for the corresponding self-dual metrics in the Euclidean and in the Minkowski signature cases.
We study the state-sum models of quantum gravity based on a representation 2-category of the Poincare 2-group. We call them spin-cube models, since they are categorical generalizations of spin-foam models. A spin-cube state sum can be considered as a path integral for a constrained 2-BF theory, and depending on how the constraints are imposed, a spin-cube state sum can be reduced to a path integral for the area-Regge model with the edge-length constraints, or to a path integral for the Regge model. We also show that the effective actions for these spin-cube models have the correct classical limit.
We define an effective action for spin foam models of quantum gravity by adapting the background field method from quantum field theory. We show that the Regge action is the leading term in the semi-classical expansion of the spin foam effective action if the vertex amplitude has the large-spin asymptotics which is proportional to an exponential function of the vertex Regge action. In the case of the known three-dimensional and four-dimensional spin foam models this amounts to modifying the vertex amplitude such that the exponential asymptotics is obtained. In particular, we show that the ELPR/FK model vertex amplitude can be modified such that the new model is finite and has the Einstein-Hilbert action as its classical limit. We also calculate the first-order and some of the second-order quantum corrections in the semi-classical expansion of the effective action.
In this essay we describe a platonic metaphysics where time is a fundamental idea such that the passage of time is independent of observers and the laws of physics. Furthermore, time serves to distinguish between a real and an abstract universe, where a real universe is an appropriatelly ordered set of ideas in time, while an abstract universe is such a set outside of time. We allow for non-computable and non-mathematical ideas to be part of our model of the universe in order to include intelligent observers. The part of the universe which can be described by mathematics corresponds to a mathematical structure in Tegmark's sense, while the non-mathematical part can be described by a human language. We describe the mind-brain connection in our framework, and show that it resolves the problem of obtaining abstract knowledge in platonism. We briefly discuss how different interpretations of Quantum Mechanics can be implemented in our framework and point out some physics implications, most notably that the time travel would be impossible.
We show that if the flat-spacetime wavefunction in the spin network basis of Loop Quantum Gravity has a large-spin asymptotics given by Rovelli's ansatz then the corresponding graviton propagator has the correct large-distance asymptotics nonperturbatively and independently of the spin foam model used to describe the evolution operator. We also argue that even in the Rovelli approach the wavefunction should satisfy the Hamiltonian constraint and we give an explanation for the spin parameter appearing in Rovelli's ansatz.
We give the correct expressions for the spin network evaluations proposed in Class. Quant. Grav. 21 (2004) 3909 as the coefficients of the quantum gravity vacuum wavefunction in the spin network basis.
We describe a class of spin foam models of four-dimensional quantum gravity which is based on the integration of the tetrad one-forms in the path integral for the Palatini action of General Relativity. In the Euclidian gravity case this class of models can be understood as a modification of the Barrett-Crane spin foam model. Fermionic matter can be coupled by using the path integral with sources for the tetrads and the spin connection, and the corresponding state sum is based on a spin foam where both the edges and the faces are colored independently with the irreducible representations of the spacetime rotations group.
It is known that the Einstein-Hilbert action with a positive cosmological constant can be represented as a perturbation of the SO(4,1) BF theory by a symmetry-breaking term quadratic in the B field. Introducing fermionic matter generates additional terms in the action which are polynomial in the tetrads and the spin connection. We describe how to construct the generating functional in the spin foam formalism for a generic BF theory when the sources for the B and the gauge field are present. This functional can be used to obtain a path integral for General Relativity with matter as a perturbative series whose the lowest order term is a path integral for a topological gravity coupled to matter.
We study the time evolution of the expectation value of the anharmonic oscillator coordinate in a coherent state as a toy model for understanding the semiclassical solutions in quantum field theory. By using the deformation quantization techniques, we show that the coherent state expectation value can be expanded in powers of $\hbar$ such that the zeroth-order term is a classical solution while the first-order correction is given as a phase-space Laplacian acting on the classical solution. This is then compared to the effective action solution for the one-dimensional $\f^4$ perturbative quantum field theory. We find an agreement up to the order $ł\hbar$, where $ł$ is the coupling constant, while at the order $ł^2 \hbar$ there is a disagreement. Hence the coherent state expectation values define an alternative semiclassical dynamics to that of the effective action. The coherent state semiclassical trajectories are exactly computable and they can coincide with the effective action trajectories in the case of two-dimensional integrable field theories.
We propose a spin foam model of four-dimensional quantum gravity which is based on the integration of the tetrads in the path integral for the Palatini action of General Relativity. In the Euclidian gravity case we show that the model can be understood as a modification of the Barrett-Crane spin foam model. Fermionic matter can be coupled by using the path integral with sources for the tetrads and the spin connection, and the corresponding state sum is based on a spin foam where both the edges and the faces are colored independently with the irreducible representations of the spacetime rotations group.
We review briefly the spin foam formalism for constructing path integrals for the BF and related theories. Then we describe how the path integral for the string theory on a group manifold can be defined as a two-dimensional spin foam state sum.
We give a brief and a critical review of the Barret-Crane spin foam models of quantum gravity. Then we describe two new spin foam models which are obtained by direct quantization of General Relativity and do not have some of the drawbacks of the Barret-Crane models. These are the model of spin foam invariants for the embedded spin networks in loop quantum gravity and the spin foam model based on the integration of the tetrads in the path integral for the Palatini action.
We propose an approach to formulating string theory in a curved spacetime, which is based on the connection between the states of the WZW model for the isometry group of a background spacetime metric and the representations of the corresponding quantum group. In this approach the string states scattering amplitudes are defined by certain evaluations of the theta spin networks for the associated quantum group. We examine the evaluations given by the spin network invariants defined by the spin foam state sum model associated to the two-dimensional BF theory for the background isometry group. We show that the corresponding string amplitudes are well defined if the spacetime manifold is compact and admits a group metric. We compute the simplest scattering amplitudes in the case of the SU(2) background isometry group, and we provide arguments that these are the amplitudes of a topological string theory.
We give a short review of the spin foam models of quantum gravity, with an emphasis on the Barret-Crane model. After explaining the shortcomings of the Barret-Crane model, we briefly discuss two new approaches, one based on the 3d spin foam state sum invariants for the embedded spin networks, and the other based on representing the string scattering amplitudes as 2d spin foam state sum invariants.
We construct a state in the loop quantum gravity theory with zero cosmological constant, which should correspond to the flat spacetime vacuum solution. This is done by defining the loop transform coefficients of a flat connection wavefunction in the holomorphic representation which satisfies all the constraints of quantum General Relativity and it is peaked around the flat space triads. The loop transform coefficients are defined as spin foam state sum invariants of the spin networks embedded in the spatial manifold for the SU(2) quantum group. We also obtain an expression for the vacuum wavefunction in the triad represntation, by defining the corresponding spin networks functional integrals as SU(2) quantum group state sums.
We show that the path integral for the three-dimensional SU(2) BF theory with a Wilson loop or a spin network function inserted can be understood as the Rovelli-Smolin loop transform of a wavefunction in the Ashtekar connection representation, where the wavefunction satisfies the constraints of quantum general relativity with zero cosmological constant. This wavefunction is given as a product of the delta functions of the SU(2) field strength and therefore it can be naturally associated to a flat connection spacetime. The loop transform can be defined rigorously via the quantum SU(2) group, as a spin foam state sum model, so that one obtains invariants of spin networks embedded in a three-manifold. These invariants define a flat connection vacuum state in the q-deformed spin network basis. We then propose a modification of this construction in order to obtain a vacuum state corresponding to the flat metric spacetime.
We construct a spin foam model of Yang-Mills theory coupled to gravity by using a discretized path integral of the BF theory with polynomial interactions and the Barret-Crane ansatz. In the Euclidian gravity case we obtain a vertex amplitude which is determined by a vertex operator acting on a simple spin network function. The Euclidian gravity results can be straightforwardly extended to the Lorentzian case, so that we propose a Lorentzian spin foam model of Yang-Mills theory coupled to gravity.
We construct a class of spin foam models describing matter coupled to gravity, such that the gravitational sector is described by the unitary irreducible representations of the appropriate symmetry group, while the matter sector is described by the finite-dimensional irreducible representations of that group. The corresponding spin foam amplitudes in the four-dimensional gravity case are expressed in terms of the spin network amplitudes for pentagrams with additional external and internal matter edges. We also give a quantum field theory formulation of the model, where the matter degrees of freedom are described by spin network fields carrying the indices from the appropriate group representation. In the non-topological Lorentzian gravity case, we argue that the matter representations should be appropriate SO(3) or SO(2) representations contained in a given Lorentz matter representation, depending on whether one wants to describe a massive or a massless matter field. The corresponding spin network amplitudes are given as multiple integrals of propagators which are matrix spherical functions.