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 ($J_3$) for general extended loops. For ordinary loops the result acquires an interesting geometrical meaning and new possibilities appear for $J_3$ 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 $3-3-1$ 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 $Z'$ gauge boson mass and its mixing angle with the SM $Z$ 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