1,154 research outputs found
Quantum gravity in three dimensions, Witten spinors and the quantisation of length
In this paper, I investigate the quantisation of length in euclidean quantum
gravity in three dimensions. The starting point is the classical hamiltonian
formalism in a cylinder of finite radius. At this finite boundary, a counter
term is introduced that couples the gravitational field in the interior to a
two-dimensional conformal field theory for an SU(2) boundary spinor, whose norm
determines the conformal factor between the fiducial boundary metric and the
physical metric in the bulk. The equations of motion for this boundary spinor
are derived from the boundary action and turn out to be the two-dimensional
analogue of the Witten equations appearing in Witten's proof of the positive
mass theorem. The paper concludes with some comments on the resulting quantum
theory. It is shown, in particular, that the length of a one-dimensional cross
section of the boundary turns into a number operator on the Fock space of the
theory. The spectrum of this operator is discrete and matches the results from
loop quantum gravity in the spin network representation.Comment: 22 pages, one figur
Complex Ashtekar variables, the Kodama state and spinfoam gravity
Starting from a Hamiltonian description of four dimensional general
relativity in presence of a cosmological constant we perform the program of
canonical quantisation. This is done using complex Ashtekar variables while
keeping the Barbero--Immirzi parameter real. Introducing the SL(2,C) Kodama
state formally solving all first class constraints we propose a spinfoam vertex
amplitude. We construct SL(2,C) boundary spinnetwork functions coloured by
finite dimensional representations of the group, and derive the skein relations
needed to calculate the amplitude. The space of boundary states is shown to
carry a representation of the holonomy flux algebra and can naturally be
equipped with a non-degenerate inner product. It fails to be positive definite,
but cylindrical consistency is perfectly satisfied.Comment: 30 pages, 3 figure
Discrete gravity as a topological field theory with light-like curvature defects
I present a model of discrete gravity, which is formulated in terms of a
topological gauge theory with defects. The theory has no local degrees of
freedom and the gravitational field is trivial everywhere except at a number of
colliding null surfaces, which represent a system of curvature defects
propagating at the speed of light. The underlying action is local and it is
studied in both its Lagrangian and Hamiltonian formulation. The canonically
conjugate variables on the null surfaces are a spinor and a spinor-valued
two-surface density, which are coupled to a topological field theory for the
Lorentz connection in the bulk. I discuss the relevance of the model for
non-perturbative approaches to quantum gravity, such as loop quantum gravity,
where similar variables have recently appeared as well.Comment: 52 pages, one figure, one table, v2: published version with extended
conclusio
- …