13 research outputs found
Spin Foams Without Spins
We formulate the spin foam representation of discrete SU(2) gauge theory as a
product of vertex amplitudes each of which is the spin network generating
function of the boundary graph dual to the vertex. In doing so the sums over
spins have been carried out. The boundary data of each n-valent node is
explicitly reduced with respect to the local gauge invariance and has a
manifest geometrical interpretation as a framed polyhedron of fixed total area.
Ultimately, sums over spins are traded for contour integrals over simple poles
and recoupling theory is avoided using generating functions.Comment: 21 pages, 2 figure
Generating Functionals for Spin Foam Amplitudes
We construct a generating functional for the exact evalutation of a coherent
representation of spin network amplitudes. This generating functional is
defined for arbitrary graphs and depends only on a pair of spinors for each
edge. The generating functional is a meromorphic polynomial in the spinor
invariants which is determined by the cycle structure of the graph.
The expansion of the spin network generating function is given in terms of a
newly recognized basis of SU(2) intertwiners consisting of the monomials of the
holomorphic spinor invariants. This basis is labelled by the degrees of the
monomials and is thus discrete. It is also overcomplete, but contains the
precise amount of data to specify points in the classical space of closed
polyhedra, and is in this sense coherent. We call this new basis the
discrete-coherent basis.
We focus our study on the 4-valent basis, which is the first non-trivial
dimension, and is also the case of interest for Quantum Gravity. We find simple
relations between the new basis, the orthonormal basis, and the coherent basis.
Finally we discuss the process of coarse graining moves at the level of the
generating functionals and give a general prescription for arbitrary graphs. A
direct relation between the polynomial of cycles in the spin network generating
functional and the high temperature loop expansion of the 2d Ising model is
found.Comment: PhD Thesis, 128 page
Lieb-Robinson bounds with dependence on interaction strengths
We propose new Lieb-Robinson bounds (bounds on the speed of propagation of
information in quantum systems) with an explicit dependence on the interaction
strengths of the Hamiltonian. For systems with more than two interactions it is
found that the Lieb-Robinson speed is not always algebraic in the interaction
strengths. We consider Hamiltonians with any finite number of bounded operators
and also a certain class of unbounded operators. We obtain bounds and
propagation speeds for quantum systems on lattices and also general graphs
possessing a kind of homogeneity and isotropy. One area for which this
formalism could be useful is the study of quantum phase transitions which occur
when interactions strengths are varied.Comment: 19 pages, 1 figure, minor modification
On the exact evaluation of spin networks
We introduce a fully coherent spin network amplitude whose expansion
generates all SU(2) spin networks associated with a given graph. We then give
an explicit evaluation of this amplitude for an arbitrary graph. We show how
this coherent amplitude can be obtained from the specialization of a generating
functional obtained by the contraction of parametrized intertwiners a la
Schwinger. We finally give the explicit evaluation of this generating
functional for arbitrary graphs
Pachner moves in a 4d Riemannian holomorphic Spin Foam model
In this work we study a Spin Foam model for 4d Riemannian gravity, and
propose a new way of imposing the simplicity constraints that uses the recently
developed holomorphic representation. Using the power of the holomorphic
integration techniques, and with the introduction of two new tools: the
homogeneity map and the loop identity, for the first time we give the analytic
expressions for the behaviour of the Spin Foam amplitudes under 4-dimensional
Pachner moves. It turns out that this behaviour is controlled by an insertion
of nonlocal mixing operators. In the case of the 5-1 move, the expression
governing the change of the amplitude can be interpreted as a vertex
renormalisation equation. We find a natural truncation scheme that allows us to
get an invariance up to an overall factor for the 4-2 and 5-1 moves, but not
for the 3-3 move. The study of the divergences shows that there is a range of
parameter space for which the 4-2 move is finite while the 5-1 move diverges.
This opens up the possibility to recover diffeomorphism invariance in the
continuum limit of Spin Foam models for 4D Quantum Gravity.Comment: 48 pages, 30 figure
A spin foam model for general Lorentzian 4-geometries
We derive simplicity constraints for the quantization of general Lorentzian
4-geometries. Our method is based on the correspondence between coherent states
and classical bivectors and the minimization of associated uncertainties. For
spacelike geometries, this scheme agrees with the master constraint method of
the model by Engle, Pereira, Rovelli and Livine (EPRL). When it is applied to
general Lorentzian geometries, we obtain new constraints that include the EPRL
constraints as a special case. They imply a discrete area spectrum for both
spacelike and timelike surfaces. We use these constraints to define a spin foam
model for general Lorentzian 4-geometries.Comment: 27 pages, 1 figure; v4: published versio
Coherent states for continuous spectrum operators with non-normalizable fiducial states
The problem of building coherent states from non-normalizable fiducial states
is considered. We propose a way of constructing such coherent states by
regularizing the divergence of the fiducial state norm. Then, we successfully
apply the formalism to particular cases involving systems with a continuous
spectrum: coherent states for the free particle and for the inverted oscillator
are explicitly provided. Similar ideas can be used for other
systems having non-normalizable fiducial states.Comment: 17 pages, typos corrected, references adde