Aleksandar Mikovic

Spin foam models of matter 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.

Spin foam models of Yang–Mills theory coupled to gravity

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 Barrett–Crane ansatz. In the Euclidean gravity case, we obtain a vertex amplitude which is determined by a vertex operator acting on a simple spin network function. The Euclidean 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.

Effective action and semi-classical limit of spin-foam models

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.

Quantum gravity vacuum and invariants of embedded spin networks

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 with 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.

Quantum field theory of spin networks

We study the transition amplitudes in the state-sum models of quantum gravity in D = 2-4 spacetime dimensions by using the field theory over the GD formulation, where G is the relevant Lie group. By promoting the group theory Fourier modes into creation and annihilation operators we construct a Fock space for the quantum field theory whose Feynman diagrams give the transition amplitudes. By making products of the Fourier modes we construct operators and states representing the spin networks associated with triangulations of spatial boundaries of a triangulated spacetime manifold. The corresponding spin network amplitudes give the state-sum amplitudes for triangulated manifolds with boundaries. We also show that one can introduce a discrete time evolution operator, where the time is given by the number of D-simplices in a triangulation, or equivalently by the number of vertices of the Feynman diagram. The corresponding transition amplitude is a finite sum of Feynman diagrams, and in this way one avoids the problem of infinite amplitudes caused by summing over all possible triangulations.

Poincaré 2-group and quantum gravity

We show that general relativity can be formulated as a constrained topological theory for flat 2-connections associated with the Poincare 2-group. Matter can be consistently coupled to gravity in this formulation. We also show that the edge lengths of the spacetime manifold triangulation arise as the basic variables in the path-integral quantization, while the state-sum amplitude is an evaluation of a colored 3-complex, in agreement with the category theory results. A 3-complex amplitude for Euclidean quantum gravity is proposed.

Coherent states expectation values as semiclassical trajectories

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 ℏ 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 ϕ4 perturbative quantum field theory. We find an agreement up to the order λℏ, where λ is the coupling constant, while at the order λ2ℏ 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 integrab...

Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

We formulate the spin foam perturbation theory for three-dimensional Euclidean Quantum Gravity with a cosmological constant. We analyse the perturbative expansion of the partition function in the dilute-gas limit and we argue that the Baez conjecture stating that the number of possible distinct topological classes of perturbative configurations is finite for the set of all triangulations of a manifold is not true. However, the conjecture is true for a special class of triangulations which are based on subdivisions of certain 3-manifold cubulations. In this case we calculate the partition function and show that the dilute-gas correction vanishes for the simplest choice of the volume operator. By slightly modifying the dilute-gas limit, we obtain a nonvanishing correction which is related to the second order perturbative correction. By assuming that the dilute-gas limit coupling constant is a function of the cosmological constant, we obtain a value for the partition function which is independent of the choice of the volume operator.

A finiteness bound for the EPRL/FK spin foam model

We show that the EPRL/FK spin foam model of quantum gravity has an absolutely convergent partition function if the vertex amplitude is divided by an appropriate power p of the product of dimensions of the vertex spins. This power is independent of the spin foam 2-complex and we find that p > 2 ensures the convergence of the state sum. Determining the convergence of the state sum for the values 0 ≤ p ≤ 2 requires the knowledge of the large-spin asymptotics of the vertex amplitude in the cases when some of the vertex spins are large and other are small.

Invariants of Spin Networks Embedded in Three-Manifolds

We study the invariants of spin networks embedded in a three-dimensional manifold which are based on the path integral for SU(2) BF-Theory. These invariants appear naturally in Loop Quantum Gravity, and have been defined as spin-foam state sums. By using the Chain-Mail technique, we give a more general definition of these invariants, and show that the state-sum definition is a special case. This provides a rigorous proof that the state-sum invariants of spin networks are topological invariants. We derive various results about the BF-Theory spin network invariants, and we find a relation with the corresponding invariants defined from Chern-Simons Theory, i.e. the Witten-Reshetikhin-Turaev invariants. We also prove that the BF-Theory spin network invariants coincide with V. Turaev’s definition of invariants of coloured graphs embedded in 3-manifolds and thick surfaces, constructed by using shadow-world evaluations. Our framework therefore provides a unified view of these invariants.