783 research outputs found
Geometric spin foams, Yang-Mills theory and background-independent models
We review the dual transformation from pure lattice gauge theory to spin foam
models with an emphasis on a geometric viewpoint. This allows us to give a
simple dual formulation of SU(N) Yang-Mills theory, where spin foam surfaces
are weighted with the exponentiated area. In the case of gravity, we introduce
a symmetry condition which demands that the amplitude of an individual spin
foam depends only on its geometric properties and not on the lattice on which
it is defined. For models that have this property, we define a new sum over
abstract spin foams that is independent of any choice of lattice or
triangulation. We show that a version of the Barrett-Crane model satisfies our
symmetry requirement.Comment: 28 pages, 27 diagrams, typos correcte
Generalized Schroedinger equation in Euclidean field theory
We investigate the idea of a "general boundary" formulation of quantum field
theory in the context of the Euclidean free scalar field. We propose a precise
definition for an evolution kernel that propagates the field through arbitrary
spacetime regions. We show that this kernel satisfies an evolution equation
which governs its dependence on deformations of the boundary surface and
generalizes the ordinary (Euclidean) Schroedinger equation. We also derive the
classical counterpart of this equation, which is a Hamilton-Jacobi equation for
general boundary surfaces.Comment: 25 pages, 11 figure
Quantum geometry from phase space reduction
In this work we give an explicit isomorphism between the usual spin network
basis and the direct quantization of the reduced phase space of tetrahedra. The
main outcome is a formula that describes the space of SU(2) invariant states by
an integral over coherent states satisfying the closure constraint exactly, or
equivalently, as an integral over the space of classical tetrahedra. This
provides an explicit realization of theorems by Guillemin--Sternberg and Hall
that describe the commutation of quantization and reduction. In the final part
of the paper, we use our result to express the FK spin foam model as an
integral over classical tetrahedra and the asymptotics of the vertex amplitude
is determined.Comment: 33 pages, 1 figur
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