2,171 research outputs found
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
Critical behavior of dissipative two-dimensional spin lattices
We explore critical properties of two-dimensional lattices of spins
interacting via an anisotropic Heisenberg Hamiltonian and subject to incoherent
spin flips. We determine the steady-state solution of the master equation for
the density matrix via the corner-space renormalization method. We investigate
the finite-size scaling and critical exponent of the magnetic linear
susceptibility associated to a dissipative ferromagnetic transition. We show
that the Von Neumann entropy increases across the critical point, revealing a
strongly mixed character of the ferromagnetic phase. Entanglement is witnessed
by the quantum Fisher information which exhibits a critical behavior at the
transition point, showing that quantum correlations play a crucial role in the
transition even though the system is in a mixed state.Comment: Accepted for publication on Phys. Rev. B (6 pages, 5 figures
Consistent and mimetic discretizations in general relativity
A discretization of a continuum theory with constraints or conserved
quantities is called mimetic if it mirrors the conserved laws or constraints of
the continuum theory at the discrete level. Such discretizations have been
found useful in continuum mechanics and in electromagnetism. We have recently
introduced a new technique for discretizing constrained theories. The technique
yields discretizations that are consistent, in the sense that the constraints
and evolution equations can be solved simultaneously, but it cannot be
considered mimetic since it achieves consistency by determining the Lagrange
multipliers. In this paper we would like to show that when applied to general
relativity linearized around a Minkowski background the technique yields a
discretization that is mimetic in the traditional sense of the word. We show
this using the traditional metric variables and also the Ashtekar new
variables, but in the latter case we restrict ourselves to the Euclidean case.
We also argue that there appear to exist conceptual difficulties to the
construction of a mimetic formulation of the full Einstein equations, and
suggest that the new discretization scheme can provide an alternative that is
nevertheless close in spirit to the traditional mimetic formulations.Comment: 14 pages, Revtex, no figures, final version to appear in JM
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
Extended Loops: A New Arena for Nonperturbative Quantum Gravity
We propose a new representation for gauge theories and quantum gravity. It
can be viewed as a generalization of the loop representation. We make use of a
recently introduced extension of the group of loops into a Lie Group. This
extension allows the use of functional methods to solve the constraint
equations. It puts in a precise framework the regularization problems of the
loop representation. It has practical advantages in the search for quantum
states. We present new solutions to the Wheeler-DeWitt equation that reinforce
the conjecture that the Jones Polynomial is a state of nonperturbative quantum
gravity.Comment: 12pp, Revtex, no figures, CGPG-93/12-
Canonical quantum gravity in the Vassiliev invariants arena: II. Constraints, habitats and consistency of the constraint algebra
In a companion paper we introduced a kinematical arena for the discussion of
the constraints of canonical quantum gravity in the spin network representation
based on Vassiliev invariants. In this paper we introduce the Hamiltonian
constraint, extend the space of states to non-diffeomorphism invariant
``habitats'' and check that the off-shell quantum constraint commutator algebra
reproduces the classical Poisson algebra of constraints of general relativity
without anomalies. One can therefore consider the resulting set of constraints
and space of states as a consistent theory of canonical quantum gravity.Comment: 20 Pages, RevTex, many figures included with psfi
Loop Representations for 2+1 Gravity on a Torus
We study the loop representation of the quantum theory for 2+1 dimensional
general relativity on a manifold, , where
is the torus, and compare it with the connection representation
for this system. In particular, we look at the loop transform in the part of
the phase space where the holonomies are boosts and study its kernel. This
kernel is dense in the connection representation and the transform is not
continuous with respect to the natural topologies, even in its domain of
definition. Nonetheless, loop representations isomorphic to the connection
representation corresponding to this part of the phase space can still be
constructed if due care is taken. We present this construction but note that
certain ambiguities remain; in particular, functions of loops cannot be
uniquely associated with functions of connections.Comment: 24 journal or 52 preprint pages, revtex, SU-GP-93/3-
Non-Gaussianity in the Cosmic Microwave Background Anisotropies at Recombination in the Squeezed limit
We estimate analytically the second-order cosmic microwave background
temperature anisotropies at the recombination epoch in the squeezed limit and
we deduce the contamination to the primordial local non-Gaussianity. We find
that the level of contamination corresponds to f_NL^{con}=O(1) which is below
the sensitivity of present experiments and smaller than the value O(5) recently
claimed in the literature.Comment: LaTeX file; 15 pages. Slightly revised version. Main result unchange
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