709 research outputs found
On the constraint algebra of quantum gravity in the loop representation
Although an important issue in canonical quantization, the problem of
representing the constraint algebra in the loop representation of quantum
gravity has received little attention. The only explicit computation was
performed by Gambini, Garat and Pullin for a formal point-splitting
regularization of the diffeomorphism and Hamiltonian constraints. It is shown
that the calculation of the algebra simplifies considerably when the
constraints are expressed not in terms of generic area derivatives but rather
as the specific shift operators that reflect the geometric meaning of the
constraints.Comment: 17 pages, LaTeX, 2 figures included with eps
An analytical expression for the third coefficient of the Jones Polynomial
An analytical expression for the third coefficient of the Jones Polynomial
P_J[\gamma,\, {\em e}^q] in the variable is reported. Applications of the
result in Quantum Gravity are considered.Comment: 9 page
More on the exponential bound of four dimensional simplicial quantum gravity
A crucial requirement for the standard interpretation of Monte Carlo
simulations of simplicial quantum gravity is the existence of an exponential
bound that makes the partition function well-defined. We present numerical data
favoring the existence of an exponential bound, and we argue that the more
limited data sets on which recently opposing claims were based are also
consistent with the existence of an exponential bound.Comment: 10 pages, latex, 2 figure
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
Absence of barriers in dynamical triangulation
Due to the unrecognizability of certain manifolds there must exist pairs of
triangulations of these manifolds that can only be reached from each other by
going through an intermediate state that is very large. This might reduce the
reliability of dynamical triangulation, because there will be states that will
not be reached in practice. We investigate this problem numerically for the
manifold , which is known to be unrecognizable, but see no sign of these
unreachable states.Comment: 8 pages, LaTeX2e source with postscript resul
Quantum Einstein-Maxwell Fields: A Unified Viewpoint from the Loop Representation
We propose a naive unification of Electromagnetism and General Relativity
based on enlarging the gauge group of Ashtekar's new variables. We construct
the connection and loop representations and analyze the space of states. In the
loop representation, the wavefunctions depend on two loops, each of them
carrying information about both gravitation and electromagnetism. We find that
the Chern-Simons form and the Jones Polynomial play a role in the model.Comment: 13pp. no figures, Revtex, UU-HEP-92/9, IFFI 92-1
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
NON-PERTURBATIVE SOLUTIONS FOR LATTICE QUANTUM GRAVITY
We propose a new, discretized model for the study of 3+1-dimensional
canonical quantum gravity, based on the classical SL(2,\C)-connection
formulation. The discretization takes place on a topological - lattice
with periodic boundary conditions. All operators and wave functions are
constructed from one-dimensional link variables, which are regarded as the
fundamental building blocks of the theory. The kinematical Hilbert space is
spanned by polynomials of certain Wilson loops on the lattice and is manifestly
gauge- and diffeomorphism- invariant. The discretized quantum Hamiltonian maps this space into itself. We find a large sector of solutions to the
discretized Wheeler-DeWitt equation , which are labelled by
single and multiple Polyakov loops. These states have a finite norm with
respect to a natural scalar product on the space of holomorphic
SL(2,\C)-Wilson loops. We also investigate the existence of further solutions
for the case of the -lattice. - Our results provide for the first time a
rigorous, regularized framework for studying non-perturbative quantum gravity.Comment: 26 pages, 2 figures (postscript, compressed and uuencoded), TeX, Jan
9
Does loop quantum gravity imply Lambda=0?
We suggest that in a recently proposed framework for quantum gravity, where
Vassiliev invariants span the the space of states, the latter is dramatically
reduced if one has a non-vanishing cosmological constant. This naturally
suggests that the initial state of the universe should have been one with
.Comment: 5 pages, one figure included with psfi
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-
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