Holonomy spin foam models: Asymptotic geometry of the partition function

We study the asymptotic geometry of the spin foam partition function for a large class of models, including the models of Barrett and Crane, Engle, Pereira, Rovelli and Livine, and, Freidel and Krasnov. The asymptotics is taken with respect to the boundary spins only, no assumption of large spins is made in the interior. We give a sufficient criterion for the existence of the partition function. We find that geometric boundary data is suppressed unless its interior continuation satisfies certain accidental curvature constraints. This means in particular that most Regge manifolds are suppressed in the asymptotic regime. We discuss this explicitly for the case of the configurations arising in the 3-3 Pachner move. We identify the origin of these accidental curvature constraints as an incorrect twisting of the face amplitude upon introduction of the Immirzi parameter and propose a way to resolve this problem, albeit at the price of losing the connection to the SU(2) boundary Hilbert space. The key methodological innovation that enables these results is the introduction of the notion of wave front sets, and the adaptation of tools for their study from micro local analysis to the case of spin foam partition functions.Comment: 63 pages, 5 figures v2: Reference correcte

Geometric asymptotics for spin foam lattice gauge gravity on arbitrary triangulations

We study the behavior of holonomy spin foam partition functions, a form of lattice gauge gravity, on generic 4d-triangulations using micro local analysis. To do so we adapt tools from the renormalization theory of quantum field theory on curved space times. This allows us, for the first time, to study the partition function without taking any limits on the interior of the triangulation. We establish that for many of the most widely used models the geometricity constraints, which reduce the gauge theory to a geometric one, introduce strong accidental curvature constraints. These limit the curvature around each triangle of the triangulation to a finite set of values. We demonstrate how to modify the partition function to avoid this problem. Finally the new methods introduced provide a starting point for studying the regularization ambiguities and renormalization of the partition function.Comment: 4+6 pages, 1 figur

The Foster-Hart Measure of Riskiness for General Gambles

Foster and Hart proposed an operational measure of riskiness for discrete random variables. We show that their defining equation has no solution for many common continuous distributions including many uniform distributions, e.g. We show how to extend consistently the definition of riskiness to continuous random variables. For many continuous random variables, the risk measure is equal to the worst--case risk measure, i.e. the maximal possible loss incurred by that gamble. We also extend the Foster--Hart risk measure to dynamic environments for general distributions and probability spaces, and we show that the extended measure avoids bankruptcy in infinitely repeated gambles

Holonomy Spin Foam Models: Boundary Hilbert spaces and Time Evolution Operators

In this and the companion paper a novel holonomy formulation of so called Spin Foam models of lattice gauge gravity are explored. After giving a natural basis for the space of simplicity constraints we define a universal boundary Hilbert space, on which the imposition of different forms of the simplicity constraints can be studied. We detail under which conditions this Hilbert space can be mapped to a Hilbert space of projected spin networks or an ordinary spin network space. These considerations allow to derive the general form of the transfer operators which generates discrete time evolution. We will describe the transfer operators for some current models on the different boundary Hilbert spaces and highlight the role of the simplicity constraints determining the concrete form of the time evolution operators.Comment: 51 pages, 18 figure

Quantum collapse rules from the maximum relative entropy principle

We show that the von Neumann--Lueders collapse rules in quantum mechanics always select the unique state that maximises the quantum relative entropy with respect to the premeasurement state, subject to the constraint that the postmeasurement state has to be compatible with the knowledge gained in the measurement. This way we provide an information theoretic characterisation of quantum collapse rules by means of the maximum relative entropy principle.Comment: v2: some references added, improved presentation, result generalised to cover nonfaithful states; v3: cross-ref to arXiv:1408.3502 added, v4: some small corrections plus reference to a published version adde

Holonomy Spin Foam Models: Definition and Coarse Graining

We propose a new holonomy formulation for spin foams, which naturally extends the theory space of lattice gauge theories. This allows current spin foam models to be defined on arbitrary two-complexes as well as to generalize current spin foam models to arbitrary, in particular finite groups. The similarity with standard lattice gauge theories allows to apply standard coarse graining methods, which for finite groups can now be easily considered numerically. We will summarize other holonomy and spin network formulations of spin foams and group field theories and explain how the different representations arise through variable transformations in the partition function. A companion paper will provide a description of boundary Hilbert spaces as well as a canonical dynamic encoded in transfer operators.Comment: 36 pages, 12 figure

Bounds and Estimates for the Response to Correlated Fluctuations in Asymmetric Complex Networks

We study the spreading of correlated fluctuations through networks with asymmetric and weighted coupling. This can be found in many real systems such as renewable power grids. These systems have so far only been studied numerically. By formulating a network adapted linear response theory, we derive an analytic bound for the response. For colored we find that vulnerability patterns noise are linked to the left Laplacian eigenvectors of the overdamped modes. We show for a broad class of tree-like flow networks, that fluctuations are enhanced in the opposite direction of the flow. This novel mechanism explains vulnerability patterns that were observed in realistic simulations of renewable power grids