91 research outputs found
Causal Space-Times on a Null Lattice
I investigate a discrete model of quantum gravity on a causal null-lattice
with \SLC structure group. The description is geometric and foliates in a
causal and physically transparent manner. The general observables of this model
are constructed from local Lorentz symmetry considerations only. For smooth
configurations, the local lattice actions reduce to the Hilbert-Palatini
action, a cosmological term and the three topological terms of dimension four
of Pontyagin, Euler and Nieh-Yan. Consistency conditions for a topologically
hypercubic complex with null 4-simplexes are derived and a topological lattice
theory that enforces these non-local constraints is constructed. The lattice
integration measure is derived from an \SLC-invariant integration measure by
localization of the non-local structure group. This measure is unique up to a
density that depends on the local 4-volume. It can be expressed in terms of
manifestly coordinate invariant geometrical quantities. The density provides an
invariant regularization of the lattice integration measure that suppresses
configurations with small local 4-volumes. Amplitudes conditioned on geodesic
distances between local observables have a physical interpretation and may have
a smooth ultraviolet limit. Numerical studies on small lattices in the
unphysical strong coupling regime of large imaginary cosmological constant
suggest that this model of triangulated causal manifolds is finite. Two
topologically different triangulations of space-time are discussed: a single,
causally connected universe and a duoverse with two causally disjoint connected
components. In the duoverse, two hypercubic sublattices are causally disjoint
but the local curvature depends on fields of both sublattices. This may
simulate effects of dark matter in the continuum limit.Comment: Greatly improved version, new numerics, appendices, etc.. 42 pages,
14 figure
Irreducible Many-Body Casimir Energies of Intersecting Objects
The vacuum energy of a bosonic field interacting locally with objects is
decomposed into irreducible -body parts. The irreducible -body
contribution to the vacuum energy is finite if the common intersection of all objects is empty. I prove that the
perturbative expansion of the corresponding irreducible -body spectral
function \tphi^{(N)}(\beta) for vanishes to all orders even if
some of the objects intersect. These irreducible spectral functions and their
associated Casimir energies in principle can be computed numerically or
approximated semiclassically without regularization or implicit knowledge of
the spectrum. They are analytic in the parameters describing the relative
orientation and position of the individual objects and remain finite when some,
but not all, of the objects overlap. The Feynman-Kac theorem is used to
compute Casimir energies of a massless scalar field with potential scattering
and the finiteness of -body Casimir energies is shown explicitly in this
case. The irreducible -body contributions to the vacuum energy of a massless
scalar field with potential interactions is shown to be negative for an even-
and positive for an odd- number of objects. Some simple examples are used to
illustrate the analyticity of the -body Casimir energy and its sign. A
multiple scattering representation of the irreducible three-body Casimir energy
is given. It remains finite when any two of the three objects overlap.Comment: 6 pages, 2 figure
Local Measure of Convex Surfaces induced by the Wiener Measure of Paths
The Wiener measure induces a measure of closed, convex, (d-1)-dimensional,
Euclidean (hyper-)surfaces that are the convex hulls of closed d-dimensional
Brownian bridges. I present arguments and numerical evidence that this measure,
for odd d, is generated by a local classical action of length dimension two
that depends on geometric invariants of the (d-1)-dimensional surface only.Comment: Talk presented at QFEXT09 in Norman, Oklahoma (6 pages, 2 figs.
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