Communication presented to the Second International Conference on the Ontology of Spacetime, 9-11, June 2006, Concordia University, Montreal. Latex. 22 pages
We present some modern theories on the structure of spacetime that can be classified as relational theories in the direction of Leibniz's ontology. In order to analyze the nature of spacetime we consider three levels of knowledge -observational, theoretical and ontological- to which the different models can be adscribed. Following similar approach to these models mentioned at the first sections, we present our theoretical model of the structure of spacetime, some physical applications, and the ontological interpretation of the model.
We compute Teitelboim's causal propagator in the context of canonical loop quantum gravity. For the Lorentzian signature, we find that the resultant power series can be expressed as a sum over branched, colored two-surfaces with an intrinsic causal structure. This leads us to define a general structure which we call a ``causal spin foam''. We also demonstrate that the causal evolution models for spin networks fall in the general class of causal spin foams.
We describe how the Barrett-Crane spin foam model defines transition amplitudes for quantum gravity states and how causality can be consistently implemented in it.
We discuss the notion of causality in Quantum Gravity in the context of sum-over-histories approaches, in the absence therefore of any background time parameter. In the spin foam formulation of Quantum Gravity, we identify the appropriate causal structure in the orientation of the spin foam 2-complex and the data that characterize it; we construct a generalised version of spin foam models introducing an extra variable with the interpretation of proper time and show that different ranges of integration for this proper time give two separate classes of spin foam models: one corresponds to the sp…
I discuss how to impose causality on spin-foam models, separating forward and backward propagation, turning a given triangulation to a 'causal set', and giving asymptotically the exponential of the Regge action, not a cosine. I show the equivalence of the prescriptions which have been proposed to achieve this. Essential to the argument is the closure condition for the 4-simplices, all made of space-like tetrahedra.
Energetic causal sets are causal sets endowed by a flow of energy-momentum between causally related events. These incorporate a novel mechanism for the emergence of space-time from causal relations. Here we construct a spin foam model which is also an energetic causal set model. This model is closely related to the model introduced in parallel by Wolfgang Wieland in arXiv:1407.0025, and this construction makes use of results used there. What makes a spin foam model also an energetic causal set is Wieland's identification of new momenta, conserved at events (or four-simplices), whose norms are …
In this paper we tackle the issue of causality in quantum gravity, in the context of 3d spin foam models. We identify the correct procedure for implementing the causality/orientation dependence restriction that reduces the path integral for BF theory to that of quantum gravity in first order form. We construct explicitly the resulting causal spin foam model. We then add matter degrees of freedom to it and construct a causal spin foam model for 3d quantum gravity coupled to matter fields. Finally, we show that the corresponding spin foam amplitudes admit a natural approximation as the Feynman a…
We analyse the classical and quantum geometry of the Barrett-Crane spin foam model for four dimensional quantum gravity, explaining why it has to be considering as a covariant realization of the projector operator onto physical quantum gravity states. We discuss how causality requirements can be consistently implemented in this framework, and construct causal transiton amplitudes between quantum gravity states, i.e. realising in the spin foam context the Feynman propagator between states. The resulting causal spin foam model can be seen as a path integral quantization of Lorentzian first order…
Spin foam models are a new approach to a formulation of quantum gravity which is fully background independent, non-perturbative, and covariant, in the spirit of path integral formulations of quantum field theory. In this thesis we describe in details the general ideas and formalism of spin foam models, and review many of the results obtained recently in this approach. We concentrate, for the case of 3-dimensional quantum gravity, on the Turaev-Viro model, and, in the 4-dimensional case, which is our main concern, on the Barrett-Crane model. In particular, for the Barrett-Crane model: we descri…
We link the notion causality with the orientation of the 2-complex on which spin foam models are based. We show that all current spin foam models are orientation-independent, pointing out the mathematical structure behind this independence. Using the technology of evolution kernels for quantum fields/particles on Lie groups/homogeneous spaces, we construct a generalised version of spin foam models, introducing an extra proper time variable and prove that different ranges of integration for this variable lead to different classes of spin foam models: the usual ones, interpreted as the quantum g…
Current theories of particle physics, including the standard model, are dominated by the paradigm that nature is basically translation invariant. Deviations from translation invariance are described by the action of forces. General relativity is based on a different paradigm: There is no translation invariance in general. Interaction is a consequence of the geometry of spacetime, formed by the presence of matter, rather than of forces. In recent years the formation of spacetime on a quantum mechanical level, has been intensively studied within the framework of spin foams, following an old idea…
We show that the simplicity constraints, which define the dynamics of spin foam models, imply, and are implied by, the first law of thermodynamics, when the latter is applied to causal diamonds in the quantum spacetime. This result reveals an intimate connection between the holographic nature of gravity, as reflected by the Bekenstein entropy, and the fact that general relativity and other gravitational theories can be understood as constrained topological field theories. To state and derive this correspondence we describe causal diamonds in the causal structure of spin foam histories and gene…
We construct a generalised formalism for group field theories, in which the domain of the field is extended to include additional proper time variables, as well as their conjugate mass variables. This formalism allows for different types of quantum gravity transition amplitudes in perturbative expansion, and we show how both causal spin foam models and the usual a-causal ones can be derived from it, within a sum over triangulations of all topologies. We also highlight the relation of the so-derived causal transition amplitudes with simplicial gravity actions.
We study the (generalized) transition probability spaces, in the sense of Mielnik and Cantoni, for spacetime quantum states in loop quantum gravity. First, we show that loop quantum gravity admits the structures of transition probability spaces. This is exemplified by first checking such structures in covariant quantum mechanics, and then identify the transition probability spaces in spin foam models via a simplified discrete version of general boundary formulation. The transition probability space thus defined gives a simple way to reconstruct the discrete analog of the Hilbert space of the c…
The causal structure is a quintessential element of continuum spacetime physics and needs to be properly encoded in a theory of Lorentzian quantum gravity. Established spin foam (and tensorial group field theory (TGFT)) models mostly work with relatively special classes of Lorentzian triangulations (e.g. built from spacelike tetrahedra only), obscuring the explicit implementation of the local causal structure at the microscopic level. We overcome this limitation and construct a full-fledged model for Lorentzian quantum geometry the building blocks of which include spacelike, lightlike and time…
I review the formalism of loop quantum gravity, in both its real and complex formulations, and spin foam theory which is its path integral counterpart. Spin networks for non-compact groups are introduced (following hep-th/0205268) to deal with gauge invariant structures based on the Lorentz group. The whole formalism is studied in details in three dimensions in both its canonical formulation (loop gravity) and its spin foam formulation. The main output (following gr-qc/0212077) is the discreteness of timelike intervals and the continuous character of spacelike distances even at the quantum lev…
A statistical measure of dimension is used to compute the effective average space dimension for the Internet and other graphs, based on typed edges (links) from an ensemble of starting points. The method is applied to CAIDA's ITDK data for the Internet. The effective dimension at different scales is calibrated to the conventional Euclidean dimension using low dimensional hypercubes. Internet spacetime has a 'foamy' multi-scale containment hierarchy, with interleaving semantic types. There is an emergent scale for approximate long range order in the device node spectrum, but this is not evident…
We use two simple independent gedankan experiments to show that the holographic principle can be understood intuitively as having its origin in the quantum fluctuations of spacetime. Applied to cosmology, this consideration leads to a dynamical cosmological constant of the observed magnitude, a result that can also be obtained for the present and recent cosmic eras by using unimodular gravity and causal set theory. Next we generalize the concept of gravitational thermodynamics to a spacetime with positive cosmological constant (like ours) to reveal the natural emergence, in galactic dynamics, …
Suppose the usual description of spacetime as a 4-dimensional manifold with a Lorentzian metric breaks down at Planck energies. Can we still construct sensible theoretical models of the universe? Are they testable? Do they lead to a consistent quantum cosmology? Is this cosmology different than the standard one? The answer is yes, to all these questions, assuming that quantum theory is still valid at this scale. I describe the basic features of such models, mainly quantum causal histories and spin foams. They are given by a partition function, similar to spin systems and lattice gauge theories…
