1,443 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
Categorified Symplectic Geometry and the String Lie 2-Algebra
Multisymplectic geometry is a generalization of symplectic geometry suitable
for n-dimensional field theories, in which the nondegenerate 2-form of
symplectic geometry is replaced by a nondegenerate (n+1)-form. The case n = 2
is relevant to string theory: we call this 2-plectic geometry. Just as the
Poisson bracket makes the smooth functions on a symplectic manifold into a Lie
algebra, the observables associated to a 2-plectic manifold form a "Lie
2-algebra", which is a categorified version of a Lie algebra. Any compact
simple Lie group G has a canonical 2-plectic structure, so it is natural to
wonder what Lie 2-algebra this example yields. This Lie 2-algebra is
infinite-dimensional, but we show here that the sub-Lie-2-algebra of
left-invariant observables is finite-dimensional, and isomorphic to the already
known "string Lie 2-algebra" associated to G. So, categorified symplectic
geometry gives a geometric construction of the string Lie 2-algebra.Comment: 16 page
Description of the control system design for the SSF PMAD DC testbed
The Power Management and Distribution (PMAD) DC Testbed Control System for Space Station Freedom was developed using a top down approach based on classical control system and conventional terrestrial power utilities design techniques. The design methodology includes the development of a testbed operating concept. This operating concept describes the operation of the testbed under all possible scenarios. A unique set of operating states was identified and a description of each state, along with state transitions, was generated. Each state is represented by a unique set of attributes and constraints, and its description reflects the degree of system security within which the power system is operating. Using the testbed operating states description, a functional design for the control system was developed. This functional design consists of a functional outline, a text description, and a logical flowchart for all the major control system functions. Described here are the control system design techniques, various control system functions, and the status of the design and implementation
Power system monitoring and source control of the Space Station Freedom DC power system testbed
Unlike a terrestrial electric utility which can purchase power from a neighboring utility, the Space Station Freedom (SSF) has strictly limited energy resources; as a result, source control, system monitoring, system protection, and load management are essential to the safe and efficient operation of the SSF Electric Power System (EPS). These functions are being evaluated in the DC Power Management and Distribution (PMAD) Testbed which NASA LeRC has developed at the Power System Facility (PSF) located in Cleveland, Ohio. The testbed is an ideal platform to develop, integrate, and verify power system monitoring and control algorithms. State Estimation (SE) is a monitoring tool used extensively in terrestrial electric utilities to ensure safe power system operation. It uses redundant system information to calculate the actual state of the EPS, to isolate faulty sensors, to determine source operating points, to verify faults detected by subsidiary controllers, and to identify high impedance faults. Source control and monitoring safeguard the power generation and storage subsystems and ensure that the power system operates within safe limits while satisfying user demands with minimal interruptions. System monitoring functions, in coordination with hardware implemented schemes, provide for a complete fault protection system. The objective of this paper is to overview the development and integration of the state estimator and the source control algorithms
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
Dual variables and a connection picture for the Euclidean Barrett-Crane model
The partition function of the SO(4)- or Spin(4)-symmetric Euclidean
Barrett-Crane model can be understood as a sum over all quantized geometries of
a given triangulation of a four-manifold. In the original formulation, the
variables of the model are balanced representations of SO(4) which describe the
quantized areas of the triangles. We present an exact duality transformation
for the full quantum theory and reformulate the model in terms of new variables
which can be understood as variables conjugate to the quantized areas. The new
variables are pairs of S^3-values associated to the tetrahedra. These
S^3-variables parameterize the hyperplanes spanned by the tetrahedra (locally
embedded in R^4), and the fact that there is a pair of variables for each
tetrahedron can be viewed as a consequence of an SO(4)-valued parallel
transport along the edges dual to the tetrahedra. We reconstruct the parallel
transport of which only the action of SO(4) on S^3 is physically relevant and
rewrite the Barrett-Crane model as an SO(4) lattice BF-theory living on the
2-complex dual to the triangulation subject to suitable constraints whose form
we derive at the quantum level. Our reformulation of the Barrett-Crane model in
terms of continuous variables is suitable for the application of various
analytical and numerical techniques familiar from Statistical Mechanics.Comment: 33 pages, LaTeX, combined PiCTeX/postscript figures, v2: note added,
TeX error correcte
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
Gauge symmetries in spinfoam gravity: the case for "cellular quantization"
The spinfoam approach to quantum gravity rests on a "quantization" of BF
theory using 2-complexes and group representations. We explain why, in
dimension three and higher, this "spinfoam quantization" must be amended to be
made consistent with the gauge symmetries of discrete BF theory. We discuss a
suitable generalization, called "cellular quantization", which (1) is finite,
(2) produces a topological invariant, (3) matches with the properties of the
continuum BF theory, (4) corresponds to its loop quantization. These results
significantly clarify the foundations - and limitations - of the spinfoam
formalism, and open the path to understanding, in a discrete setting, the
symmetry-breaking which reduces BF theory to gravity.Comment: 6 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
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