12,521 research outputs found
Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction
A Planck-scale minimal observable length appears in many approaches to
quantum gravity. It is sometimes argued that this minimal length might conflict
with Lorentz invariance, because a boosted observer could see the minimal
length further Lorentz contracted. We show that this is not the case within
loop quantum gravity. In loop quantum gravity the minimal length (more
precisely, minimal area) does not appear as a fixed property of geometry, but
rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted
observer can see the same observable spectrum, with the same minimal area. What
changes continuously in the boost transformation is not the value of the
minimal length: it is the probability distribution of seeing one or the other
of the discrete eigenvalues of the area. We discuss several difficulties
associated with boosts and area measurement in quantum gravity. We compute the
transformation of the area operator under a local boost, propose an explicit
expression for the generator of local boosts and give the conditions under
which its action is unitary.Comment: 12 pages, 3 figure
The century of the incomplete revolution: searching for general relativistic quantum field theory
In fundamental physics, this has been the century of quantum mechanics and
general relativity. It has also been the century of the long search for a
conceptual framework capable of embracing the astonishing features of the world
that have been revealed by these two ``first pieces of a conceptual
revolution''. I discuss the general requirements on the mathematics and some
specific developments towards the construction of such a framework. Examples of
covariant constructions of (simple) generally relativistic quantum field
theories have been obtained as topological quantum field theories, in
nonperturbative zero-dimensional string theory and its higher dimensional
generalizations, and as spin foam models. A canonical construction of a general
relativistic quantum field theory is provided by loop quantum gravity.
Remarkably, all these diverse approaches have turn out to be related,
suggesting an intriguing general picture of general relativistic quantum
physics.Comment: To appear in the Journal of Mathematical Physics 2000 Special Issu
Spin Networks and Recoupling in Loop Quantum Gravity
I discuss the role played by the spin-network basis and recoupling theory (in
its graphical tangle-theoretic formulation) and their use for performing
explicit calculations in loop quantum gravity. In particular, I show that
recoupling theory allows the derivation of explicit expressions for the
eingenvalues of the quantum volume operator. An important side result of these
computations is the determination of a scalar product with respect to which
area and volume operators are symmetric, and the spin network states are
orthonormal.Comment: 8 pages, LaTeX3e, To appear in the Proceedings of the 2nd Conference
on Constrained Dynamics and Quantum Gravity, Santa Margherita, Italy, 17-21
September 199
A simple background-independent hamiltonian quantum model
We study formulation and probabilistic interpretation of a simple
general-relativistic hamiltonian quantum system. The system has no unitary
evolution in background time. The quantum theory yields transition
probabilities between measurable quantities (partial observables). These
converge to the classical predictions in the limit. Our main tool
is the kernel of the projector on the solutions of Wheeler-deWitt equation,
which we analyze in detail. It is a real quantity, which can be seen as a
propagator that propagates "forward" as well as "backward" in a local parameter
time. Individual quantum states, on the other hand, may contain only "forward
propagating" components. The analysis sheds some light on the interpretation of
background independent transition amplitudes in quantum gravity
Discreteness of area and volume in quantum gravity
We study the operator that corresponds to the measurement of volume, in
non-perturbative quantum gravity, and we compute its spectrum. The operator is
constructed in the loop representation, via a regularization procedure; it is
finite, background independent, and diffeomorphism-invariant, and therefore
well defined on the space of diffeomorphism invariant states (knot states). We
find that the spectrum of the volume of any physical region is discrete. A
family of eigenstates are in one to one correspondence with the spin networks,
which were introduced by Penrose in a different context. We compute the
corresponding component of the spectrum, and exhibit the eigenvalues
explicitly. The other eigenstates are related to a generalization of the spin
networks, and their eigenvalues can be computed by diagonalizing finite
dimensional matrices. Furthermore, we show that the eigenstates of the volume
diagonalize also the area operator. We argue that the spectra of volume and
area determined here can be considered as predictions of the
loop-representation formulation of quantum gravity on the outcomes of
(hypothetical) Planck-scale sensitive measurements of the geometry of space.Comment: 36 pages, latex, 13 figures uuencode
Graviton propagator from background-independent quantum gravity
We study the graviton propagator in euclidean loop quantum gravity, using the
spinfoam formalism. We use boundary-amplitude and group-field-theory
techniques, and compute one component of the propagator to first order, under a
number of approximations, obtaining the correct spacetime dependence. In the
large distance limit, the only term of the vertex amplitude that contributes is
the exponential of the Regge action: the other terms, that have raised doubts
on the physical viability of the model, are suppressed by the phase of the
vacuum state, which is determined by the extrinsic geometry of the boundary.Comment: 6 pages. Substantially revised second version. Improved boundary
state ansat
Unitary dynamics of spherical null gravitating shells
The dynamics of a thin spherically symmetric shell of zero-rest-mass matter
in its own gravitational field is studied. A form of action principle is used
that enables the reformulation of the dynamics as motion on a fixed background
manifold. A self-adjoint extension of the Hamiltonian is obtained via the group
quantization method. Operators of position and of direction of motion are
constructed. The shell is shown to avoid the singularity, to bounce and to
re-expand to that asymptotic region from which it contracted; the dynamics is,
therefore, truly unitary. If a wave packet is sufficiently narrow and/or
energetic then an essential part of it can be concentrated under its
Schwarzschild radius near the bounce point but no black hole forms. The quantum
Schwarzschild horizon is a linear combination of a black and white hole
apparent horizons rather than an event horizon.Comment: 26 pages, Latex, no figures; definitive version, to be published in
Nuclear Physics
A spin foam model without bubble divergences
We present a spin foam model in which the fundamental ``bubble amplitudes''
(the analog of the one-loop corrections in quantum field theory) are finite as
the cutoff is removed. The model is a natural variant of the field theoretical
formulation of the Barrett-Crane model. As the last, the model is a quantum BF
theory plus an implementation of the constraint that reduces BF theory to
general relativity. We prove that the fundamental bubble amplitudes are finite
by constructing an upper bound, using certain inequalities satisfied by the
Wigner (3n)j-symbols, which we derive in the paper. Finally, we present
arguments in support of the conjecture that the bubble diagrams of the model
are finite at all orders.Comment: 19 page
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