27,796 research outputs found
The microscopic dynamics of quantum space as a group field theory
We provide a rather extended introduction to the group field theory approach
to quantum gravity, and the main ideas behind it. We present in some detail the
GFT quantization of 3d Riemannian gravity, and discuss briefly the current
status of the 4-dimensional extensions of this construction. We also briefly
report on recent results obtained in this approach and related open issues,
concerning both the mathematical definition of GFT models, and possible avenues
towards extracting interesting physics from them.Comment: 60 pages. Extensively revised version of the contribution to
"Foundations of Space and Time: Reflections on Quantum Gravity", edited by G.
Ellis, J. Murugan, A. Weltman, published by Cambridge University Pres
Quantum statistics on graphs
Quantum graphs are commonly used as models of complex quantum systems, for
example molecules, networks of wires, and states of condensed matter. We
consider quantum statistics for indistinguishable spinless particles on a
graph, concentrating on the simplest case of abelian statistics for two
particles. In spite of the fact that graphs are locally one-dimensional, anyon
statistics emerge in a generalized form. A given graph may support a family of
independent anyon phases associated with topologically inequivalent exchange
processes. In addition, for sufficiently complex graphs, there appear new
discrete-valued phases. Our analysis is simplified by considering combinatorial
rather than metric graphs -- equivalently, a many-particle tight-binding model.
The results demonstrate that graphs provide an arena in which to study new
manifestations of quantum statistics. Possible applications include topological
quantum computing, topological insulators, the fractional quantum Hall effect,
superconductivity and molecular physics.Comment: 21 pages, 6 figure
Spin networks, quantum automata and link invariants
The spin network simulator model represents a bridge between (generalized)
circuit schemes for standard quantum computation and approaches based on
notions from Topological Quantum Field Theories (TQFT). More precisely, when
working with purely discrete unitary gates, the simulator is naturally modelled
as families of quantum automata which in turn represent discrete versions of
topological quantum computation models. Such a quantum combinatorial scheme,
which essentially encodes SU(2) Racah--Wigner algebra and its braided
counterpart, is particularly suitable to address problems in topology and group
theory and we discuss here a finite states--quantum automaton able to accept
the language of braid group in view of applications to the problem of
estimating link polynomials in Chern--Simons field theory.Comment: LateX,19 pages; to appear in the Proc. of "Constrained Dynamics and
Quantum Gravity (QG05), Cala Gonone (Italy) September 12-16 200
Three combinatorial models for affine sl(n) crystals, with applications to cylindric plane partitions
We define three combinatorial models for \hat{sl(n)} crystals, parametrized
by partitions, configurations of beads on an `abacus', and cylindric plane
partitions, respectively. These are reducible, but we can identify an
irreducible subcrystal corresponding to any dominant integral highest weight.
Cylindric plane partitions actually parametrize a basis for the tensor product
of an irreducible representation with the space spanned by all partitions. We
use this to calculate the partition function for a system of random cylindric
plane partitions. We also observe a form of rank level duality. Finally, we use
an explicit bijection to relate our work to the Kyoto path model.Comment: 29 pages, 14 Figures. v2: 5 new references. Minor corrections and
clarifications. v3: Section 4.2 correcte
Statistical Entropy of a BTZ Black Hole from Loop Quantum Gravity
We compute the statistical entropy of a BTZ black hole in the context of
three-dimensional Euclidean loop quantum gravity with a cosmological constant
. As in the four-dimensional case, a quantum state of the black hole
is characterized by a spin network state. Now however, the underlying colored
graph lives in a two-dimensional spacelike surface , and some
of its links cross the black hole horizon, which is viewed as a circular
boundary of . Each link crossing the horizon is colored by a
spin (at the kinematical level), and the length of the horizon is
given by the sum of the fundamental length contributions
carried by the spins of the links . We propose an
estimation for the number of the Euclidean BTZ
black hole microstates (defined on a fixed graph ) based on an analytic
continuation from the case to the case . In our model,
we show that reproduces the Bekenstein-Hawking
entropy in the classical limit. This asymptotic behavior is independent of the
choice of the graph provided that the condition
is satisfied, as it should be in three-dimensional quantum gravity.Comment: 14 pages. 1 figure. Paragraph added on page 7 to clarify the horizon
conditio
Quantum automata, braid group and link polynomials
The spin--network quantum simulator model, which essentially encodes the
(quantum deformed) SU(2) Racah--Wigner tensor algebra, is particularly suitable
to address problems arising in low dimensional topology and group theory. In
this combinatorial framework we implement families of finite--states and
discrete--time quantum automata capable of accepting the language generated by
the braid group, and whose transition amplitudes are colored Jones polynomials.
The automaton calculation of the polynomial of (the plat closure of) a link L
on 2N strands at any fixed root of unity is shown to be bounded from above by a
linear function of the number of crossings of the link, on the one hand, and
polynomially bounded in terms of the braid index 2N, on the other. The growth
rate of the time complexity function in terms of the integer k appearing in the
root of unity q can be estimated to be (polynomially) bounded by resorting to
the field theoretical background given by the Chern-Simons theory.Comment: Latex, 36 pages, 11 figure
The Feynman propagator for quantum gravity: spin foams, proper time, orientation, causality and timeless-ordering
We discuss the notion of causality in Quantum Gravity in the context of
sum-over-histories approaches, in the absence therefore of any background time
parameter. In the spin foam formulation of Quantum Gravity, we identify the
appropriate causal structure in the orientation of the spin foam 2-complex and
the data that characterize it; we construct a generalised version of spin foam
models introducing an extra variable with the interpretation of proper time and
show that different ranges of integration for this proper time give two
separate classes of spin foam models: one corresponds to the spin foam models
currently studied, that are independent of the underlying orientation/causal
structure and are therefore interpreted as a-causal transition amplitudes; the
second corresponds to a general definition of causal or orientation dependent
spin foam models, interpreted as causal transition amplitudes or as the Quantum
Gravity analogue of the Feynman propagator of field theory, implying a notion
of ''timeless ordering''.Comment: 8 pages; to appear in the Proceedings of the DICE 2004 Workshop "From
Decoherence and Emergent Classicality to Emergent Quantum Mechanics
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