Lorentzian spin foam amplitudes: graphical calculus and asymptotics

The amplitude for the 4-simplex in a spin foam model for quantum gravity is defined using a graphical calculus for the unitary representations of the Lorentz group. The asymptotics of this amplitude are studied in the limit when the representation parameters are large, for various cases of boundary data. It is shown that for boundary data corresponding to a Lorentzian simplex, the asymptotic formula has two terms, with phase plus or minus the Lorentzian signature Regge action for the 4-simplex geometry, multiplied by an Immirzi parameter. Other cases of boundary data are also considered, including a surprising contribution from Euclidean signature metrics.Comment: 30 pages. v2: references now appear. v3: presentation greatly improved (particularly diagrammatic calculus). Definition of "Regge state" now the same as in previous work; signs change in final formula as a result. v4: two references adde

Effective action and semiclassical 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.Comment: Improved presentation, 2 references added. 15 pages, no figure

Spin foams with timelike surfaces

Spin foams of 4d gravity were recently extended from complexes with purely spacelike surfaces to complexes that also contain timelike surfaces. In this article, we express the associated partition function in terms of vertex amplitudes and integrals over coherent states. The coherent states are characterized by unit 3--vectors which represent normals to surfaces and lie either in the 2--sphere or the 2d hyperboloids. In the case of timelike surfaces, a new type of coherent state is used and the associated completeness relation is derived. It is also shown that the quantum simplicity constraints can be deduced by three different methods: by weak imposition of the constraints, by restriction of coherent state bases and by the master constraint.Comment: 22 pages, no figures; v2: remarks on operator formalism added in discussion; correction: the spin 1/2 irrep of the discrete series does not appear in the Plancherel decompositio

Asymptotics of 4d spin foam models

We study the asymptotic properties of four-simplex amplitudes for various four-dimensional spin foam models. We investigate the semi-classical limit of the Ooguri, Euclidean and Lorentzian EPRL models using coherent states for the boundary data. For some classes of geometrical boundary data, the asymptotic formulae are given, in all three cases, by simple functions of the Regge action for the four-simplex geometry.Comment: 10 pages, Proceedings for the 2nd Corfu summer school and workshop on quantum gravity and quantum geometry, talk given by Winston J. Fairbair

Euclidean three-point function in loop and perturbative gravity

We compute the leading order of the three-point function in loop quantum gravity, using the vertex expansion of the Euclidean version of the new spin foam dynamics, in the region of gamma<1. We find results consistent with Regge calculus in the limit gamma->0 and j->infinity. We also compute the tree-level three-point function of perturbative quantum general relativity in position space, and discuss the possibility of directly comparing the two results.Comment: 16 page

A new look at loop quantum gravity

I describe a possible perspective on the current state of loop quantum gravity, at the light of the developments of the last years. I point out that a theory is now available, having a well-defined background-independent kinematics and a dynamics allowing transition amplitudes to be computed explicitly in different regimes. I underline the fact that the dynamics can be given in terms of a simple vertex function, largely determined by locality, diffeomorphism invariance and local Lorentz invariance. I emphasize the importance of approximations. I list open problems.Comment: 15 pages, 5 figure

A spin foam model for general Lorentzian 4-geometries

We derive simplicity constraints for the quantization of general Lorentzian 4-geometries. Our method is based on the correspondence between coherent states and classical bivectors and the minimization of associated uncertainties. For spacelike geometries, this scheme agrees with the master constraint method of the model by Engle, Pereira, Rovelli and Livine (EPRL). When it is applied to general Lorentzian geometries, we obtain new constraints that include the EPRL constraints as a special case. They imply a discrete area spectrum for both spacelike and timelike surfaces. We use these constraints to define a spin foam model for general Lorentzian 4-geometries.Comment: 27 pages, 1 figure; v4: published versio

Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Lorentzian Theory

The present paper studies the large-j asymptotics of the Lorentzian EPRL spinfoam amplitude on a 4d simplicial complex with an arbitrary number of simplices. The asymptotics of the spinfoam amplitude is determined by the critical configurations. Here we show that, given a critical configuration in general, there exists a partition of the simplicial complex into three type of regions R_{Nondeg}, R_{Deg-A}, R_{Deg-B}, where the three regions are simplicial sub-complexes with boundaries. The critical configuration implies different types of geometries in different types of regions, i.e. (1) the critical configuration restricted into R_{Nondeg}$implies a nondegenerate discrete Lorentzian geometry, (2) the critical configuration restricted into R_{Deg-A}$ is degenerate of type-A in our definition of degeneracy, but implies a nondegenerate discrete Euclidean geometry on R_{Deg-A}, (3) the critical configuration restricted into R_{Deg-B} is degenerate of type-B, and implies a vector geometry on R_{Deg-B}. With the critical configuration, we further make a subdivision of the regions R_{Nondeg} and R_{Deg-A} into sub-complexes (with boundary) according to their Lorentzian/Euclidean oriented 4-simplex volume V_4(v), such that sgn(V_4(v)) is a constant sign on each sub-complex. Then in the each sub-complex, the spinfoam amplitude at the critical configuration gives the Regge action in Lorentzian or Euclidean signature respectively on R_{Nondeg} or R_{Deg-A}. The Regge action reproduced here contains a sign factor sgn(V_4(v)) of the oriented 4-simplex volume. Therefore the Regge action reproduced here can be viewed a discretized Palatini action with on-shell connection. Finally the asymptotic formula of the spinfoam amplitude is given by a sum of the amplitudes evaluated at all possible critical configurations, which are the products of the amplitudes associated to different type of geometries.Comment: 54 pages, 2 figures, reference adde

Canonical path integral measures for Holst and Plebanski gravity. I. Reduced Phase Space Derivation

An important aspect in defining a path integral quantum theory is the determination of the correct measure. For interacting theories and theories with constraints, this is non-trivial, and is normally not the heuristic "Lebesgue measure" usually used. There have been many determinations of a measure for gravity in the literature, but none for the Palatini or Holst formulations of gravity. Furthermore, the relations between different resulting measures for different formulations of gravity are usually not discussed. In this paper we use the reduced phase technique in order to derive the path-integral measure for the Palatini and Holst formulation of gravity, which is different from the Lebesgue measure up to local measure factors which depend on the spacetime volume element and spatial volume element. From this path integral for the Holst formulation of GR we can also give a new derivation of the Plebanski path integral and discover a discrepancy with the result due to Buffenoir, Henneaux, Noui and Roche (BHNR) whose origin we resolve. This paper is the first in a series that aims at better understanding the relation between canonical LQG and the spin foam approach.Comment: 27 pages, minor correction

Generating Functions for Coherent Intertwiners

We study generating functions for the scalar products of SU(2) coherent intertwiners, which can be interpreted as coherent spin network evaluations on a 2-vertex graph. We show that these generating functions are exactly summable for different choices of combinatorial weights. Moreover, we identify one choice of weight distinguished thanks to its geometric interpretation. As an example of dynamics, we consider the simple case of SU(2) flatness and describe the corresponding Hamiltonian constraint whose quantization on coherent intertwiners leads to partial differential equations that we solve. Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2) flatness on coherent spin networks for arbitrary graphs.Comment: 31 page