22 research outputs found
Group field theories generating polyhedral complexes
Group field theories are a generalization of matrix models which provide both
a second quantized reformulation of loop quantum gravity as well as generating
functions for spin foam models. While states in canonical loop quantum gravity,
in the traditional continuum setting, are based on graphs with vertices of
arbitrary valence, group field theories have been defined so far in a
simplicial setting such that states have support only on graphs of fixed
valency. This has led to the question whether group field theory can indeed
cover the whole state space of loop quantum gravity. In this contribution based
on [1] I present two new classes of group field theories which satisfy this
objective: i) a straightforward, but rather formal generalization to multiple
fields, one for each valency and ii) a simplicial group field theory which
effectively covers the larger state space through a dual weighting, a technique
common in matrix and tensor models. To this end I will further discuss in some
detail the combinatorial structure of the complexes generated by the group
field theory partition function. The new group field theories do not only
strengthen the links between the mentioned quantum gravity approaches but,
broadening the theory space of group field theories, they might also prove
useful in the investigation of renormalizability.Comment: accepted for publication in PoS, Frontiers of Fundamental Physics 14
(AMU Marseille
Spectral dimension of quantum geometries
The spectral dimension is an indicator of geometry and topology of spacetime
and a tool to compare the description of quantum geometry in various approaches
to quantum gravity. This is possible because it can be defined not only on
smooth geometries but also on discrete (e.g., simplicial) ones. In this paper,
we consider the spectral dimension of quantum states of spatial geometry
defined on combinatorial complexes endowed with additional algebraic data: the
kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the
effects of topology and discreteness of classical discrete geometries are
studied in a systematic manner. We look for states reproducing the spectral
dimension of a classical space in the appropriate regime. We also test the
hypothesis that in LQG, as in other approaches, there is a scale dependence of
the spectral dimension, which runs from the topological dimension at large
scales to a smaller one at short distances. While our results do not give any
strong support to this hypothesis, we can however pinpoint when the topological
dimension is reproduced by LQG quantum states. Overall, by exploring the
interplay of combinatorial, topological and geometrical effects, and by
considering various kinds of quantum states such as coherent states and their
superpositions, we find that the spectral dimension of discrete quantum
geometries is more sensitive to the underlying combinatorial structures than to
the details of the additional data associated with them.Comment: 39 pages, 18 multiple figures. v2: discussion improved, minor typos
correcte
Renormalization in combinatorially non-local field theories: the Hopf algebra of 2-graphs
It is well known that the mathematical structure underlying renormalization
in perturbative quantum field theory is based on a Hopf algebra of Feynman
diagrams. A precondition for this is locality of the field theory.
Consequently, one might suspect that non-local field theories such as matrix or
tensor field theories cannot benefit from a similar algebraic understanding.
Here I show that, on the contrary, the renormalization and perturbative
diagramatics of a broad class of such field theories is based in the same way
on a Hopf algebra. These theories are characterized by interaction vertices
with graphs as external structure leading to Feynman diagrams which can be
summed up under the concept of "2-graphs". From the renormalization
perspective, such graph-like interactions are as much local as point-like
interactions. They differ in combinatorial details as I exemplify with the
central identity for the perturbative series of combinatorial correlation
functions. This sets the stage for a systematic study of perturbative
renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger
equations, for a number of combinatorially non-local field theories with
possible applications to quantum gravity, statistical models and more.Comment: 22 pages, v2 minor adaptions for consistency with arXiv:2103.0113
Discrete quantum geometries and their effective dimension
In several approaches towards a quantum theory of gravity, such as group
field theory and loop quantum gravity, quantum states and histories of the
geometric degrees of freedom turn out to be based on discrete spacetime. The
most pressing issue is then how the smooth geometries of general relativity,
expressed in terms of suitable geometric observables, arise from such discrete
quantum geometries in some semiclassical and continuum limit. In this thesis I
tackle the question of suitable observables focusing on the effective dimension
of discrete quantum geometries. For this purpose I give a purely combinatorial
description of the discrete structures which these geometries have support on.
As a side topic, this allows to present an extension of group field theory to
cover the combinatorially larger kinematical state space of loop quantum
gravity. Moreover, I introduce a discrete calculus for fields on such
fundamentally discrete geometries with a particular focus on the Laplacian.
This permits to define the effective-dimension observables for quantum
geometries. Analysing various classes of quantum geometries, I find as a
general result that the spectral dimension is more sensitive to the underlying
combinatorial structure than to the details of the additional geometric data
thereon. Semiclassical states in loop quantum gravity approximate the classical
geometries they are peaking on rather well and there are no indications for
stronger quantum effects. On the other hand, in the context of a more general
model of states which are superposition over a large number of complexes, based
on analytic solutions, there is a flow of the spectral dimension from the
topological dimension on low energy scales to a real number on
high energy scales. In the particular case of these results allow to
understand the quantum geometry as effectively fractal.Comment: PhD thesis, Humboldt-Universit\"at zu Berlin;
urn:nbn:de:kobv:11-100232371;
http://edoc.hu-berlin.de/docviews/abstract.php?id=4204
N=4 Multi-Particle Mechanics, WDVV Equation and Roots
We review the relation of N=4 superconformal multi-particle models on the
real line to the WDVV equation and an associated linear equation for two
prepotentials, F and U. The superspace treatment gives another variant of the
integrability problem, which we also reformulate as a search for closed flat
Yang-Mills connections. Three- and four-particle solutions are presented. The
covector ansatz turns the WDVV equation into an algebraic condition, for which
we give a formulation in terms of partial isometries. Three ideas for
classifying WDVV solutions are developed: ortho-polytopes, hypergraphs, and
matroids. Various examples and counterexamples are displayed
Group field theories for all loop quantum gravity
Group field theories represent a 2nd quantized reformulation of the loop
quantum gravity state space and a completion of the spin foam formalism. States
of the canonical theory, in the traditional continuum setting, have support on
graphs of arbitrary valence. On the other hand, group field theories have
usually been defined in a simplicial context, thus dealing with a restricted
set of graphs. In this paper, we generalize the combinatorics of group field
theories to cover all the loop quantum gravity state space. As an explicit
example, we describe the GFT formulation of the KKL spin foam model, as well as
a particular modified version. We show that the use of tensor model tools
allows for the most effective construction. In order to clarify the
mathematical basis of our construction and of the formalisms with which we
deal, we also give an exhaustive description of the combinatorial structures
entering spin foam models and group field theories, both at the level of the
boundary states and of the quantum amplitudes.Comment: version published in New Journal of Physic
Dimensional flow in discrete quantum geometries
11 pags.; 6 figs.; PACS numbers: 04.60.-m, 04.60.Pp© 2015 American Physical Society. In various theories of quantum gravity, one observes a change in the spectral dimension from the topological spatial dimension d at large length scales to some smaller value at small, Planckian scales. While the origin of such a flow is well understood in continuum approaches, in theories built on discrete structures a firm control of the underlying mechanism is still missing. We shed some light on the issue by presenting a particular class of quantum geometries with a flow in the spectral dimension, given by superpositions of states defined on regular complexes. For particular superposition coefficients parametrized by a real number 0magic number> DS≃2 for the spectral dimension of spacetime, appearing so often in quantum gravity, is reproduced as well. These results apply, in particular, to special superpositions of spin-network states in loop quantum gravity, and they provide more solid indications of dimensional flow in this approach.Peer Reviewe
One-matrix differential reformulation of two-matrix models
Differential reformulations of field theories are often used for explicit
computations. We derive a one-matrix differential formulation of two-matrix
models, with the help of which it is possible to diagonalize the one- and
two-matrix models using a formula by Itzykson and Zuber that allows
diagonalizing differential operators with respect to matrix elements of
Hermitian matrices. We detail the equivalence between the expressions obtained
by diagonalizing the partition function in differential or integral
formulation, which is not manifest at first glance. For one-matrix models, this
requires transforming certain derivatives to variables. In the case of
two-matrix models, the same computation leads to a new determinant formulation
of the partition function, and we discuss potential applications to new
orthogonal polynomials methods.Comment: 25 pages, 2 figure