3 research outputs found
The graviton vacuum as a distributional state in kinematic Loop Quantum Gravity
The quantum behaviour of weak gravitational fields admits an adequate, albeit
approximate, description by those graviton states in which the expectation
values and fluctuations of the linearised gravitational field are small. Such
states must approximate corresponding states in full quantum gravity. We
analyse the nature of this approximation for the graviton vacuum state in the
context of kinematical Loop Quantum Gravity (LQG) wherein the constraints are
ignored. We identify the graviton vacuum state with kinematically
non-normalizable, distributional states in LQG by demanding that relations
between linearised operator actions on the former are mirrored by those of
their non-linear counterparts on the latter. We define a semi- norm on the
space of kinematical distributions and show that the identification is
approximate upto distributions which are small in this semi-norm. We argue that
our candidate states are annihilated by the linearised constraints (expressed
as operators in the full theory) to leading order in the parameter
characterising the approximation. This suggests the possibility, in a scheme
such as ours, of solving the full constraints order by order in this parameter.
The main drawback of our considerations is that they depend on certain
auxilliary constructions which, though mathematically well defined, do not
arise from physical insight. Our work is an attempt to implement an earlier
proposal of Iwasaki and Rovelli.Comment: 44 pages, no figure
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
Gauge Field Theory Coherent States (GCS) : IV. Infinite Tensor Product and Thermodynamical Limit
In the canonical approach to Lorentzian Quantum General Relativity in four
spacetime dimensions an important step forward has been made by Ashtekar, Isham
and Lewandowski some eight years ago through the introduction of an appropriate
Hilbert space structure.
This Hilbert space, together with its generalization due to Baez and Sawin,
is appropriate for semi-classical quantum general relativity if the spacetime
is spatially compact. In the spatially non-compact case, however, an extension
of the Hilbert space is needed in order to approximate metrics that are
macroscopically nowhere degenerate.
For this purpose, in this paper we apply von Neumann's theory of the Infinite
Tensor Product (ITP) of Hilbert Spaces to Quantum General Relativity. The
cardinality of the number of tensor product factors can take the value of any
possible Cantor aleph as is needed for our problem, where a Hilbert space is
attached to each edge of an arbitrarily complicated, generally infinite graph.
The new framework opens a pandora's box full of techniques, appropriate to
pose fascinating physical questions such as quantum topology change,
semi-classical quantum gravity, effective low energy physics etc. from the
universal point of view of the ITP. In particular, the study of photons and
gravitons propagating on fluctuating quantum spacetimes is now in reach, the
topic of the next paper in this series.Comment: 60 pages, LATEX, no figure