FROM DIMENSIONAL REDUCTION OF 4d SPIN FOAM MODEL TO ADDING NON-GRAVITATIONAL FIELDS TO 3d SPIN FOAM MODEL

A Kaluza–Klein-like approach for a 4d spin foam model is considered. By applying this approach to a model based on group field theory in 4d (TOCY model), and using the Peter–Weyl expansion of the gravitational field, reconstruction of new non-gravitational fields and interactions in the action are found. The perturbative expansion of the partition function produces graphs colored with SU(2) algebraic data, from which one can reconstruct a 3d simplicial complex representing space-time and its geometry (like in the Ponzano–Regge formulation of pure 3d quantum gravity), as well as the Feynman graph for typical matter fields. Thus a mechanism for generation of matter and construction of new dimensions are found from pure gravity.

A NOTE ON LONGITUDINAL FIELDS IN THE WEYL EXPANSION OF THE ELECTROMAGNETIC GREEN TENSOR

Many calculations of dispersion interactions between atoms and macroscopic bodies or between two bodies make use of Green tensors. The expansion of this tensor in polarizations allows for an anatomic interpretation of the interaction. In planar systems with partial translation invariance, the Weyl representation of the Green tensor is often applied. Although it is transverse in this representation, we argue that the field it describes contains nonradiative parts as well. This can be seen by calculating the Green tensor in the momentum representation and observing certain cancellations among longitudinal and transverse contributions.

High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon

A formalism is developed for calculating high coefficients c r of the Weyl (high energy) expansion for the trace of the resolvent of the Laplace operator in a domain B with smooth boundary ∂ B The c r are used to test the following conjectures. ( a ) The sequence of c r diverges factorially, controlled by the shortest accessible real or complex periodic geodesic. ( b ) If this is a 2-bounce orbit, it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the Möbius strip which is the space of chords of B . ( c ) This orbit gives an exponential contribution to the remainder when the Weyl series, truncated at its least term, is subtracted from the resolvent; the exponential switches on smoothly (according to an error function) where it is smallest, that is across the negative energy axis (Stokes line). These conjectures are motivated by recent results in asymptotics. They survive tests for the circle billiard, and for a family of curves with 2 and 3 bulges, where the dominant orbit is not always the shortest and is sometimes complex. For some systems which are not smooth billiards (e. g. a particle on a ring, or in a billiard where ∂ B is a polygon), the Weyl series terminates and so no geodesics are accessible; for a particle on a compact surface of constant negative curvature, only the complex geodesics are accessible from the Weyl series.