589 research outputs found
How the Jones Polynomial Gives Rise to Physical States of Quantum General Relativity
Solutions to both the diffeomorphism and the hamiltonian constraint of
quantum gravity have been found in the loop representation, which is based on
Ashtekar's new variables. While the diffeomorphism constraint is easily solved
by considering loop functionals which are knot invariants, there remains the
puzzle why several of the known knot invariants are also solutions to the
hamiltonian constraint. We show how the Jones polynomial gives rise to an
infinite set of solutions to all the constraints of quantum gravity thereby
illuminating the structure of the space of solutions and suggesting the
existance of a deep connection between quantum gravity and knot theory at a
dynamical level.Comment: 7p
Lattice knot theory and quantum gravity in the loop representation
We present an implementation of the loop representation of quantum gravity on
a square lattice. Instead of starting from a classical lattice theory,
quantizing and introducing loops, we proceed backwards, setting up constraints
in the lattice loop representation and showing that they have appropriate
(singular) continuum limits and algebras. The diffeomorphism constraint
reproduces the classical algebra in the continuum and has as solutions lattice
analogues of usual knot invariants. We discuss some of the invariants stemming
from Chern--Simons theory in the lattice context, including the issue of
framing. We also present a regularization of the Hamiltonian constraint. We
show that two knot invariants from Chern--Simons theory are annihilated by the
Hamiltonian constraint through the use of their skein relations, including
intersections. We also discuss the issue of intersections with kinks. This
paper is the first step towards setting up the loop representation in a
rigorous, computable setting.Comment: 23 pages, RevTeX, 14 figures included with psfi
Consistent canonical quantization of general relativity in the space of Vassiliev knot invariants
We present a quantization of the Hamiltonian and diffeomorphism constraint of
canonical quantum gravity in the spin network representation. The novelty
consists in considering a space of wavefunctions based on the Vassiliev knot
invariants. The constraints are finite, well defined, and reproduce at the
level of quantum commutators the Poisson algebra of constraints of the
classical theory. A similar construction can be carried out in 2+1 dimensions
leading to the correct quantum theory.Comment: 4 pages, RevTex, one figur
The Extended Loop Group: An Infinite Dimensional Manifold Associated with the Loop Space
A set of coordinates in the non parametric loop-space is introduced. We show
that these coordinates transform under infinite dimensional linear
representations of the diffeomorphism group. An extension of the group of loops
in terms of these objects is proposed. The enlarged group behaves locally as an
infinite dimensional Lie group. Ordinary loops form a subgroup of this group.
The algebraic properties of this new mathematical structure are analized in
detail. Applications of the formalism to field theory, quantum gravity and knot
theory are considered.Comment: The resubmited paper contains the title and abstract, that were
omitted in the previous version. 42 pages, report IFFI/93.0
Interacting Particles and Strings in Path and Surface Representations
Non-relativistic charged particles and strings coupled with abelian gauge
fields are quantized in a geometric representation that generalizes the Loop
Representation. We consider three models: the string in self-interaction
through a Kalb-Ramond field in four dimensions, the topological interaction of
two particles due to a BF term in 2+1 dimensions, and the string-particle
interaction mediated by a BF term in 3+1 dimensions. In the first case one
finds that a consistent "surface-representation" can be built provided that the
coupling constant is quantized. The geometrical setting that arises corresponds
to a generalized version of the Faraday's lines picture: quantum states are
labeled by the shape of the string, from which emanate "Faraday`s surfaces". In
the other models, the topological interaction can also be described by
geometrical means. It is shown that the open-path (or open-surface) dependence
carried by the wave functional in these models can be eliminated through an
unitary transformation, except by a remaining dependence on the boundary of the
path (or surface). These feature is closely related to the presence of
anomalous statistics in the 2+1 model, and to a generalized "anyonic behavior"
of the string in the other case.Comment: RevTeX 4, 28 page
Is the third coefficient of the Jones knot polynomial a quantum state of gravity?
Some time ago it was conjectured that the coefficients of an expansion of the
Jones polynomial in terms of the cosmological constant could provide an
infinite string of knot invariants that are solutions of the vacuum Hamiltonian
constraint of quantum gravity in the loop representation. Here we discuss the
status of this conjecture at third order in the cosmological constant. The
calculation is performed in the extended loop representation, a generalization
of the loop representation. It is shown that the the Hamiltonian does not
annihilate the third coefficient of the Jones polynomal () for general
extended loops. For ordinary loops the result acquires an interesting
geometrical meaning and new possibilities appear for to represent a
quantum state of gravity.Comment: 22 page
Finite, diffeomorphism invariant observables in quantum gravity
Two sets of spatially diffeomorphism invariant operators are constructed in
the loop representation formulation of quantum gravity. This is done by
coupling general relativity to an anti- symmetric tensor gauge field and using
that field to pick out sets of surfaces, with boundaries, in the spatial three
manifold. The two sets of observables then measure the areas of these surfaces
and the Wilson loops for the self-dual connection around their boundaries. The
operators that represent these observables are finite and background
independent when constructed through a proper regularization procedure.
Furthermore, the spectra of the area operators are discrete so that the
possible values that one can obtain by a measurement of the area of a physical
surface in quantum gravity are valued in a discrete set that includes integral
multiples of half the Planck area. These results make possible the construction
of a correspondence between any three geometry whose curvature is small in
Planck units and a diffeomorphism invariant state of the gravitational and
matter fields. This correspondence relies on the approximation of the classical
geometry by a piecewise flat Regge manifold, which is then put in
correspondence with a diffeomorphism invariant state of the gravity-matter
system in which the matter fields specify the faces of the triangulation and
the gravitational field is in an eigenstate of the operators that measure their
areas.Comment: Latex, no figures, 30 pages, SU-GP-93/1-
Vacuum stability conditions of the economical 3-3-1 model from copositivity
By applying copositivity criterion to the scalar potential of the economical
model, we derive necessary and sufficient bounded-from-below conditions
at tree level. Although these are a large number of intricate inequalities for
the dimensionless parameters of the scalar potential, we present general
enlightening relations in this work. Additionally, we use constraints coming
from the minimization of the scalar potential by means of the orbit space
method, the positivity of the squared masses of the extra scalars, the Higgs
boson mass, the gauge boson mass and its mixing angle with the SM
boson in order to further restrict the parameter space of this model.Comment: 22 pages, 7 figures, added text and references. Matches published
versio
Theoretical and numerical experiences on a test rig for active vibration control of mechanical systems with moving constraints
Active control of vibrations in mechanical systems has recently benefited of the remarkable development of robust control techniques. These control techniques are able to guarantee performances in spite of unavoidable modeling errors. They have been successfully codified and implemented for vibrating structures whose uncertain parameters could be assumed to be time-invariant. Unfortunately a wide class of mechanical systems, such as machine tools with carriage motion realized by a ball-screw, are characterized by time varying modal parameters. The focus of this paper is on modeling and controlling the vibrations of such systems. A test rig for active vibration control is presented. An analytical model of the test rig is synthesized starting by design data. Through experimental modal analysis, parametric identification and updating procedures, the model has been refined and a control system has been synthesized
The Extended Loop Representation of Quantum Gravity
A new representation of Quantum Gravity is developed. This formulation is
based on an extension of the group of loops. The enlarged group, that we call
the Extended Loop Group, behaves locally as an infinite dimensional Lie group.
Quantum Gravity can be realized on the state space of extended loop dependent
wavefunctions. The extended representation generalizes the loop representation
and contains this representation as a particular case. The resulting
diffeomorphism and hamiltonian constraints take a very simple form and allow to
apply functional methods and simplify the loop calculus. In particular we show
that the constraints are linear in the momenta. The nondegenerate solutions
known in the loop representation are also solutions of the constraints in the
new representation. The practical calculation advantages allows to find a new
solution to the Wheeler-DeWitt equation. Moreover, the extended representation
puts in a precise framework some of the regularization problems of the loop
representation. We show that the solutions are generalized knot invariants,
smooth in the extended variables, and any framing is unnecessary.Comment: 27 pages, report IFFC/94-1
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