651 research outputs found
Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models
This is the third paper in our series of five in which we test the Master
Constraint Programme for solving the Hamiltonian constraint in Loop Quantum
Gravity. In this work we analyze models which, despite the fact that the phase
space is finite dimensional, are much more complicated than in the second
paper: These are systems with an SL(2,\Rl) gauge symmetry and the
complications arise because non -- compact semisimple Lie groups are not
amenable (have no finite translation invariant measure). This leads to severe
obstacles in the refined algebraic quantization programme (group averaging) and
we see a trace of that in the fact that the spectrum of the Master Constraint
does not contain the point zero. However, the minimum of the spectrum is of
order which can be interpreted as a normal ordering constant arising
from first class constraints (while second class systems lead to normal
ordering constants). The physical Hilbert space can then be be obtained after
subtracting this normal ordering correction.Comment: 33 pages, no figure
Quantum resolution of black hole singularities
We study the classical and quantum theory of spherically symmetric spacetimes
with scalar field coupling in general relativity. We utilise the canonical
formalism of geometrodynamics adapted to the Painleve-Gullstrand coordinates,
and present a new quantisation of the resulting field theory. We give an
explicit construction of operators that capture curvature properties of the
spacetime and use these to show that the black hole curvature singularity is
avoided in the quantum theory.Comment: 5 pages, version to appear in CQ
On (Cosmological) Singularity Avoidance in Loop Quantum Gravity
Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological
sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of
the phase space of classical General Relativity to spatially homogeneous
situations which is then quantized by the methods of LQG. Thus, LQC is a
quantum mechanical toy model (finite number of degrees of freedom) for LQG(a
genuine QFT with an infinite number of degrees of freedom) which provides
important consistency checks. However, it is a non trivial question whether the
predictions of LQC are robust after switching on the inhomogeneous fluctuations
present in full LQG. Two of the most spectacular findings of LQC are that 1.
the inverse scale factor is bounded from above on zero volume eigenstates which
hints at the avoidance of the local curvature singularity and 2. that the
Quantum Einstein Equations are non -- singular which hints at the avoidance of
the global initial singularity. We display the result of a calculation for LQG
which proves that the (analogon of the) inverse scale factor, while densely
defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in
full LQG, if curvature singularity avoidance is realized, then not in this
simple way. In fact, it turns out that the boundedness of the inverse scale
factor is neither necessary nor sufficient for curvature singularity avoidance
and that non -- singular evolution equations are neither necessary nor
sufficient for initial singularity avoidance because none of these criteria are
formulated in terms of observable quantities.After outlining what would be
required, we present the results of a calculation for LQG which could be a
first indication that our criteria at least for curvature singularity avoidance
are satisfied in LQG.Comment: 34 pages, 16 figure
The Phoenix Project: Master Constraint Programme for Loop Quantum Gravity
The Hamiltonian constraint remains the major unsolved problem in Loop Quantum
Gravity (LQG). Seven years ago a mathematically consistent candidate
Hamiltonian constraint has been proposed but there are still several unsettled
questions which concern the algebra of commutators among smeared Hamiltonian
constraints which must be faced in order to make progress. In this paper we
propose a solution to this set of problems based on the so-called {\bf Master
Constraint} which combines the smeared Hamiltonian constraints for all smearing
functions into a single constraint. If certain mathematical conditions, which
still have to be proved, hold, then not only the problems with the commutator
algebra could disappear, also chances are good that one can control the
solution space and the (quantum) Dirac observables of LQG. Even a decision on
whether the theory has the correct classical limit and a connection with the
path integral (or spin foam) formulation could be in reach. While these are
exciting possibilities, we should warn the reader from the outset that, since
the proposal is, to the best of our knowledge, completely new and has been
barely tested in solvable models, there might be caveats which we are presently
unaware of and render the whole {\bf Master Constraint Programme} obsolete.
Thus, this paper should really be viewed as a proposal only, rather than a
presentation of hard results, which however we intend to supply in future
submissions.Comment: LATEX, uses AMSTE
Thiemann transform for gravity with matter fields
The generalised Wick transform discovered by Thiemann provides a
well-established relation between the Euclidean and Lorentzian theories of
general relativity. We extend this Thiemann transform to the Ashtekar
formulation for gravity coupled with spin-1/2 fermions, a non-Abelian
Yang-Mills field, and a scalar field. It is proved that, on functions of the
gravitational and matter phase space variables, the Thiemann transform is
equivalent to the composition of an inverse Wick rotation and a constant
complex scale transformation of all fields. This result holds as well for
functions that depend on the shift vector, the lapse function, and the Lagrange
multipliers of the Yang-Mills and gravitational Gauss constraints, provided
that the Wick rotation is implemented by means of an analytic continuation of
the lapse. In this way, the Thiemann transform is furnished with a geometric
interpretation. Finally, we confirm the expectation that the generator of the
Thiemann transform can be determined just from the spin of the fields and give
a simple explanation for this fact.Comment: LaTeX 2.09, 14 pages, no figure
Semiclassical quantisation of space-times with apparent horizons
Coherent or semiclassical states in canonical quantum gravity describe the
classical Schwarzschild space-time. By tracing over the coherent state
wavefunction inside the horizon, a density matrix is derived.
Bekenstein-Hawking entropy is obtained from the density matrix, modulo the
Immirzi parameter. The expectation value of the area and curvature operator is
evaluated in these states. The behaviour near the singularity of the curvature
operator shows that the singularity is resolved. We then generalise the results
to space-times with spherically symmetric apparent horizons.Comment: 52 pages, 4 figure
Absence of the Kasner singularity in the effective dynamics from loop quantum cosmology
In classical general relativity, the generic approach to the initial
singularity is usually understood in terms of the BKL scenario. In this
scenario, along with the Bianchi IX model, the exact, singular, Kasner solution
of vacuum Bianchi I model also plays a pivotal role. Using an effective
classical Hamiltonian obtained from loop quantization of vacuum Bianchi I
model, exact solution is obtained which is non-singular due to a discreteness
parameter. The solution is parameterized in exactly the same manner as the
usual Kasner solution and reduces to the Kasner solution as discreteness
parameter is taken to zero. At the effective Hamiltonian level, the avoidance
of Kasner singularity uses a mechanism distinct from the `inverse volume'
modifications characteristic of loop quantum cosmology.Comment: 4 pages, revtex4, no figure
Type I singularities and the Phantom Menace
We consider the future dynamics of a transient phantom dominated phase of the
universe in LQC and in the RS braneworld, which both have a non-standard
Friedmann equation. We find that for a certain class of potentials, the Hubble
parameter oscillates with simple harmonic motion in the LQC case and therefore
avoids any future singularity. For more general potentials we find that damping
effects eventually lead to the Hubble parameter becoming constant. On the other
hand in the braneworld case we find that although the type I singularity can be
avoided, the scale factor still diverges at late times.Comment: More references added. Final PRD versio
The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space
We combine I. background independent Loop Quantum Gravity (LQG) quantization
techniques, II. the mathematically rigorous framework of Algebraic Quantum
Field Theory (AQFT) and III. the theory of integrable systems resulting in the
invariant Pohlmeyer Charges in order to set up the general representation
theory (superselection theory) for the closed bosonic quantum string on flat
target space. While we do not solve the, expectedly, rich representation theory
completely, we present a, to the best of our knowledge new, non -- trivial
solution to the representation problem. This solution exists 1. for any target
space dimension, 2. for Minkowski signature of the target space, 3. without
tachyons, 4. manifestly ghost -- free (no negative norm states), 5. without
fixing a worldsheet or target space gauge, 6. without (Virasoro) anomalies
(zero central charge), 7. while preserving manifest target space Poincar\'e
invariance and 8. without picking up UV divergences. The existence of this
stable solution is exciting because it raises the hope that among all the
solutions to the representation problem (including fermionic degrees of
freedom) we find stable, phenomenologically acceptable ones in lower
dimensional target spaces, possibly without supersymmetry, that are much
simpler than the solutions that arise via compactification of the standard Fock
representation of the string. Moreover, these new representations could solve
some of the major puzzles of string theory such as the cosmological constant
problem. The solution presented in this paper exploits the flatness of the
target space in several important ways. In a companion paper we treat the more
complicated case of curved target spaces.Comment: 46 p., LaTex2e, no figure
Three dimensional loop quantum gravity: physical scalar product and spin foam models
In this paper, we address the problem of the dynamics in three dimensional
loop quantum gravity with zero cosmological constant. We construct a rigorous
definition of Rovelli's generalized projection operator from the kinematical
Hilbert space--corresponding to the quantization of the infinite dimensional
kinematical configuration space of the theory--to the physical Hilbert space.
In particular, we provide the definition of the physical scalar product which
can be represented in terms of a sum over (finite) spin-foam amplitudes.
Therefore, we establish a clear-cut connection between the canonical
quantization of three dimensional gravity and spin-foam models. We emphasize
two main properties of the result: first that no cut-off in the kinematical
degrees of freedom of the theory is introduced (in contrast to standard
`lattice' methods), and second that no ill-defined sum over spins (`bubble'
divergences) are present in the spin foam representation.Comment: Typos corrected, version appearing in Class. Quant. Gra
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