2,547 research outputs found
2-vertex Lorentzian Spin Foam Amplitudes for Dipole Transitions
We compute transition amplitudes between two spin networks with dipole
graphs, using the Lorentzian EPRL model with up to two (non-simplicial)
vertices. We find power-law decreasing amplitudes in the large spin limit,
decreasing faster as the complexity of the foam increases. There are no
oscillations nor asymptotic Regge actions at the order considered, nonetheless
the amplitudes still induce non-trivial correlations. Spin correlations between
the two dipoles appear only when one internal face is present in the foam. We
compute them within a mini-superspace description, finding positive
correlations, decreasing in value with the Immirzi parameter. The paper also
provides an explicit guide to computing Lorentzian amplitudes using the
factorisation property of SL(2,C) Clebsch-Gordan coefficients in terms of SU(2)
ones. We discuss some of the difficulties of non-simplicial foams, and provide
a specific criterion to partially limit the proliferation of diagrams. We
systematically compare the results with the simplified EPRLs model, much faster
to evaluate, to learn evidence on when it provides reliable approximations of
the full amplitudes. Finally, we comment on implications of our results for the
physics of non-simplicial spin foams and their resummation.Comment: 27 pages + appendix, many figures. v2: one more numerical result,
plus minor amendment
Solving constraint-satisfaction problems with distributed neocortical-like neuronal networks
Finding actions that satisfy the constraints imposed by both external inputs
and internal representations is central to decision making. We demonstrate that
some important classes of constraint satisfaction problems (CSPs) can be solved
by networks composed of homogeneous cooperative-competitive modules that have
connectivity similar to motifs observed in the superficial layers of neocortex.
The winner-take-all modules are sparsely coupled by programming neurons that
embed the constraints onto the otherwise homogeneous modular computational
substrate. We show rules that embed any instance of the CSPs planar four-color
graph coloring, maximum independent set, and Sudoku on this substrate, and
provide mathematical proofs that guarantee these graph coloring problems will
convergence to a solution. The network is composed of non-saturating linear
threshold neurons. Their lack of right saturation allows the overall network to
explore the problem space driven through the unstable dynamics generated by
recurrent excitation. The direction of exploration is steered by the constraint
neurons. While many problems can be solved using only linear inhibitory
constraints, network performance on hard problems benefits significantly when
these negative constraints are implemented by non-linear multiplicative
inhibition. Overall, our results demonstrate the importance of instability
rather than stability in network computation, and also offer insight into the
computational role of dual inhibitory mechanisms in neural circuits.Comment: Accepted manuscript, in press, Neural Computation (2018
Gravity and Random Surfaces on the Lattice - A Review
We review recent work in the lattice approach to random surfaces and quantum
gravity. Our task is made somewhat easier by some very interesting results,
particularly in four dimensions, that have appeared recently and which are
reported elsewhere in these proceedings. Inevitably, given the scope of the
review and the limitations of space, the presentation will omit work of
importance and be telegraphic in discussing work that is included, for which
apologies are offered in advance. After the customary brief historical
introduction we work our way in dimensional order from one up to four
dimensions before closing with some remarks on the relation, if any, between
the various lattice models and ``real'' 4D gravity.Comment: 13 pages with 8 embedded eps figures, Latex+espcrc2.sty (included).
Plenary review talk at Lattice96 rendered, more or less, into written
English. Now with a minor typo in last section fixe
Spin Polaron Effective Magnetic Model for La_{0.5}Ca_{0.5}MnO_3
The conventional paradigm of charge order for La_{1-x}Ca_xMnO_3 for x=0.5 has
been challenged recently by a Zener polaron picture emerging from experiments
and theoretical calculations. The effective low energy Hamiltonian for the
magnetic degrees of freedom has been found to be a cubic Heisenberg model, with
ferromagnetic nearest neighbor and frustrating antiferromagnetic next nearest
neighbor interactions in the planes, and antiferromagnetic interaction between
planes. With linear spin wave theory and diagonalization of small clusters up
to 27 sites we find that the behavior of the model interpolates between the A
and CE-type magnetic structures when a frustrating intraplanar interaction is
tuned. The values of the interactions calculated by ab initio methods indicate
a possible non-bipartite picture of polaron ordering differing from the
conventional one.Comment: 21 pages and 8 figures (included), Late
Boundary Conformal Field Theory and Ribbon Graphs: a tool for open/closed string dualities
We construct and fully characterize a scalar boundary conformal field theory
on a triangulated Riemann surface. The results are analyzed from a string
theory perspective as tools to deal with open/closed string dualities.Comment: 40 pages, 7 figures; typos correcte
Fisher Metric, Geometric Entanglement and Spin Networks
Starting from recent results on the geometric formulation of quantum
mechanics, we propose a new information geometric characterization of
entanglement for spin network states in the context of quantum gravity. For the
simple case of a single-link fixed graph (Wilson line), we detail the
construction of a Riemannian Fisher metric tensor and a symplectic structure on
the graph Hilbert space, showing how these encode the whole information about
separability and entanglement. In particular, the Fisher metric defines an
entanglement monotone which provides a notion of distance among states in the
Hilbert space. In the maximally entangled gauge-invariant case, the
entanglement monotone is proportional to a power of the area of the surface
dual to the link thus supporting a connection between entanglement and the
(simplicial) geometric properties of spin network states. We further extend
such analysis to the study of non-local correlations between two non-adjacent
regions of a generic spin network graph characterized by the bipartite
unfolding of an Intertwiner state. Our analysis confirms the interpretation of
spin network bonds as a result of entanglement and to regard the same spin
network graph as an information graph, whose connectivity encodes, both at the
local and non-local level, the quantum correlations among its parts. This gives
a further connection between entanglement and geometry.Comment: 29 pages, 3 figures, revised version accepted for publicatio
A simply connected surface of general type with p_g=0 and K^2=2
In this paper we construct a simply connected, minimal, complex surface of
general type with p_g=0 and K^2=2 using a rational blow-down surgery and
Q-Gorenstein smoothing theory.Comment: 19 pages, 6 figures. To appear in Inventiones Mathematica
Minimizing Unsatisfaction in Colourful Neighbourhoods
Colouring sparse graphs under various restrictions is a theoretical problem
of significant practical relevance. Here we consider the problem of maximizing
the number of different colours available at the nodes and their
neighbourhoods, given a predetermined number of colours. In the analytical
framework of a tree approximation, carried out at both zero and finite
temperatures, solutions obtained by population dynamics give rise to estimates
of the threshold connectivity for the incomplete to complete transition, which
are consistent with those of existing algorithms. The nature of the transition
as well as the validity of the tree approximation are investigated.Comment: 28 pages, 12 figures, substantially revised with additional
explanatio
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