162 research outputs found
The Tensor Theory Space
The tensor track is a background-independent discretization of quantum
gravity which includes a sum over all topologies. We discuss how to define a
functional renormalization group flow and the Wetterich equation in the
corresponding theory space. This space is different from the Einsteinian theory
space of asymptotic safety. It includes all fixed-rank tensor-invariant
interactions, hence generalizes matrix models and the (Moyal) non-commutative
field theory space.Comment: This short note is intended as a complement to arXiv:1311.1461, to
appear in the Proceedings of the Workshop on Noncommutative Field Theory and
Gravity in Corfu September 2013, Fortshritt. Phys. 201
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
Random Tensors and Quantum Gravity
We provide an informal introduction to tensor field theories and to their
associated renormalization group. We focus more on the general motivations
coming from quantum gravity than on the technical details. In particular we
discuss how asymptotic freedom of such tensor field theories gives a concrete
example of a natural "quantum relativity" postulate: physics in the deep
ultraviolet regime becomes asymptotically more and more independent of any
particular choice of Hilbert basis in the space of states of the universe.Comment: Section 6 is essentially reproduced from author's arXiv:1507.04190
for self-contained purpose of the revie
Why are tensor field theories asymptotically free?
In this pedagogic letter we explain the combinatorics underlying the generic
asymptotic freedom of tensor field theories. We focus on simple combinatorial
models with a propagator and quartic interactions and on the comparison
between the intermediate field representations of the vector, matrix and tensor
cases. The transition from asymptotic freedom (tensor case) to asymptotic
safety (matrix case) is related to the crossing symmetry of the matrix vertex
whereas in the vector case, the lack of asymptotic freedom ("Landau ghost"), as
in the ordinary scalar case, is simply due to the absence of any wave function
renormalization at one loop.Comment: 8 pages, 3 figure
Constructive Tensor Field Theory
We provide an up-to-date review of the recent constructive program for field
theories of the vector, matrix and tensor type, focusing not on the models
themselves but on the mathematical tools used.Comment: arXiv admin note: text overlap with arXiv:1401.500
Loop Vertex Expansion for Higher Order Interactions
This note provides an extension of the constructive loop vertex expansion to
stable interactions of arbitrarily high order, opening the way to many
applications. We treat in detail the example of the field
theory in zero dimension. We find that the important feature to extend the loop
vertex expansion is not to use an intermediate field representation, but rather
to force integration of exactly one particular field per vertex of the initial
action.Comment: 16 pages, 2 figures, V2, minor modification
Spheres are rare
We prove that triangulations of homology spheres in any dimension grow much
slower than general triangulations. Our bound states in particular that the
number of triangulations of homology spheres in 3 dimensions grows at most like
the power 1/3 of the number of general triangulations.Comment: 14 pages, 1 figur
The two dimensional Hubbard Model at half-filling: I. Convergent Contributions
We prove analyticity theorems in the coupling constant for the Hubbard model
at half-filling. The model in a single renormalization group slice of index
is proved to be analytic in for for some constant
, and the skeleton part of the model at temperature (the sum of all
graphs without two point insertions) is proved to be analytic in for
. These theorems are necessary steps towards
proving that the Hubbard model at half-filling is {\it not} a Fermi liquid (in
the mathematically precise sense of Salmhofer).Comment: 31 pages, 2 figure
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