1,148 research outputs found
An Invitation to Higher Gauge Theory
In this easy introduction to higher gauge theory, we describe parallel
transport for particles and strings in terms of 2-connections on 2-bundles.
Just as ordinary gauge theory involves a gauge group, this generalization
involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie
group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes,
which play an important role in string theory and multisymplectic geometry.
Second, every group representation gives a Lie 2-group; the representation of
the Lorentz group on 4d Minkowski spacetime gives the Poincar\'e 2-group, which
leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint
representation of any Lie group on its own Lie algebra gives a 'tangent
2-group', which serves as a gauge 2-group in 4d BF theory, which has
topological gravity as a special case. Fourth, every Lie group has an 'inner
automorphism 2-group', which serves as the gauge group in 4d BF theory with
cosmological constant term. Fifth, every Lie group has an 'automorphism
2-group', which plays an important role in the theory of nonabelian gerbes. And
sixth, every compact simple Lie group gives a 'string 2-group'. We also touch
upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra
that governs 11-dimensional supergravity.Comment: 60 pages, based on lectures at the 2nd School and Workshop on Quantum
Gravity and Quantum Geometry at the 2009 Corfu Summer Institut
Spin Foam Models of Yang-Mills Theory Coupled to Gravity
We construct a spin foam model of Yang-Mills theory coupled to gravity by
using a discretized path integral of the BF theory with polynomial interactions
and the Barret-Crane ansatz. In the Euclidian gravity case we obtain a vertex
amplitude which is determined by a vertex operator acting on a simple spin
network function. The Euclidian gravity results can be straightforwardly
extended to the Lorentzian case, so that we propose a Lorentzian spin foam
model of Yang-Mills theory coupled to gravity.Comment: 10 page
Positivity of Spin Foam Amplitudes
The amplitude for a spin foam in the Barrett-Crane model of Riemannian
quantum gravity is given as a product over its vertices, edges and faces, with
one factor of the Riemannian 10j symbols appearing for each vertex, and simpler
factors for the edges and faces. We prove that these amplitudes are always
nonnegative for closed spin foams. As a corollary, all open spin foams going
between a fixed pair of spin networks have real amplitudes of the same sign.
This means one can use the Metropolis algorithm to compute expectation values
of observables in the Riemannian Barrett-Crane model, as in statistical
mechanics, even though this theory is based on a real-time (e^{iS}) rather than
imaginary-time (e^{-S}) path integral. Our proof uses the fact that when the
Riemannian 10j symbols are nonzero, their sign is positive or negative
depending on whether the sum of the ten spins is an integer or half-integer.
For the product of 10j symbols appearing in the amplitude for a closed spin
foam, these signs cancel. We conclude with some numerical evidence suggesting
that the Lorentzian 10j symbols are always nonnegative, which would imply
similar results for the Lorentzian Barrett-Crane model.Comment: 15 pages LaTeX. v3: Final version, with updated conclusions and other
minor changes. To appear in Classical and Quantum Gravity. v4: corrects # of
samples in Lorentzian tabl
Volume and Quantizations
The aim of this letter is to indicate the differences between the
Rovelli-Smolin quantum volume operator and other quantum volume operators
existing in the literature. The formulas for the operators are written in a
unifying notation of the graph projective framework. It is clarified whose
results apply to which operators and why.Comment: 8 page
Asymptotics of 10j symbols
The Riemannian 10j symbols are spin networks that assign an amplitude to each
4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This
amplitude is a function of the areas of the 10 faces of the 4-simplex, and
Barrett and Williams have shown that one contribution to its asymptotics comes
from the Regge action for all non-degenerate 4-simplices with the specified
face areas. However, we show numerically that the dominant contribution comes
from degenerate 4-simplices. As a consequence, one can compute the asymptotics
of the Riemannian 10j symbols by evaluating a `degenerate spin network', where
the rotation group SO(4) is replaced by the Euclidean group of isometries of
R^3. We conjecture formulas for the asymptotics of a large class of Riemannian
and Lorentzian spin networks in terms of these degenerate spin networks, and
check these formulas in some special cases. Among other things, this conjecture
implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the
Riemannian ones.Comment: 25 pages LaTeX with 8 encapsulated Postscript figures. v2 has various
clarifications and better page breaks. v3 is the final version, to appear in
Classical and Quantum Gravity, and has a few minor corrections and additional
reference
Quantum Theory of Gravity I: Area Operators
A new functional calculus, developed recently for a fully non-perturbative
treatment of quantum gravity, is used to begin a systematic construction of a
quantum theory of geometry. Regulated operators corresponding to areas of
2-surfaces are introduced and shown to be self-adjoint on the underlying
(kinematical) Hilbert space of states. It is shown that their spectra are {\it
purely} discrete indicating that the underlying quantum geometry is far from
what the continuum picture might suggest. Indeed, the fundamental excitations
of quantum geometry are 1-dimensional, rather like polymers, and the
3-dimensional continuum geometry emerges only on coarse graining. The full
Hilbert space admits an orthonormal decomposition into finite dimensional
sub-spaces which can be interpreted as the spaces of states of spin systems.
Using this property, the complete spectrum of the area operators is evaluated.
The general framework constructed here will be used in a subsequent paper to
discuss 3-dimensional geometric operators, e.g., the ones corresponding to
volumes of regions.Comment: 33 pages, ReVTeX, Section 4 Revised: New results on the effect of
topology of a surface on the eigenvalues and eigenfunctions of its area
operator included. The proof of the bound on the level spacing of eigenvalues
(for large areas) simplified and its ramification to the Bekenstein-Mukhanov
analysis of black-hole evaporation made more explicit. To appear in CQ
Self-referential Monte Carlo method for calculating the free energy of crystalline solids
A self-referential Monte Carlo method is described for calculating the free energy of crystalline solids. All Monte Carlo methods for the free energy of classical crystalline solids calculate the free-energy difference between a state whose free energy can be calculated relatively easily and the state of interest. Previously published methods employ either a simple model crystal, such as the Einstein crystal, or a fluid as the reference state. The self-referential method employs a radically different reference state; it is the crystalline solid of interest but with a different number of unit cells. So it calculates the free-energy difference between two crystals, differing only in their size. The aim of this work is to demonstrate this approach by application to some simple systems, namely, the face centered cubic hard sphere and Lennard-Jones crystals. However, it can potentially be applied to arbitrary crystals in both bulk and confined environments, and ultimately it could also be very efficient
Positivity in Lorentzian Barrett-Crane Models of Quantum Gravity
The Barrett-Crane models of Lorentzian quantum gravity are a family of spin
foam models based on the Lorentz group. We show that for various choices of
edge and face amplitudes, including the Perez-Rovelli normalization, the
amplitude for every triangulated closed 4-manifold is a non-negative real
number. Roughly speaking, this means that if one sums over triangulations,
there is no interference between the different triangulations. We prove
non-negativity by transforming the model into a ``dual variables'' formulation
in which the amplitude for a given triangulation is expressed as an integral
over three copies of hyperbolic space for each tetrahedron. Then we prove that,
expressed in this way, the integrand is non-negative. In addition to implying
that the amplitude is non-negative, the non-negativity of the integrand is
highly significant from the point of view of numerical computations, as it
allows statistical methods such as the Metropolis algorithm to be used for
efficient computation of expectation values of observables.Comment: 13 page
Cosmological Deformation of Lorentzian Spin Foam Models
We study the quantum deformation of the Barrett-Crane Lorentzian spin foam
model which is conjectured to be the discretization of Lorentzian Plebanski
model with positive cosmological constant and includes therefore as a
particular sector quantum gravity in de-Sitter space. This spin foam model is
constructed using harmonic analysis on the quantum Lorentz group. The
evaluation of simple spin networks are shown to be non commutative integrals
over the quantum hyperboloid defined as a pile of fuzzy spheres. We show that
the introduction of the cosmological constant removes all the infrared
divergences: for any fixed triangulation, the integration over the area
variables is finite for a large class of normalization of the amplitude of the
edges and of the faces.Comment: 37 pages, 7 figures include
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