278 research outputs found
Note on Self-Duality and the Kodama State
An interesting interplay between self-duality, the Kodama (Chern-Simons)
state and knot invariants is shown to emerge in the quantum theory of an
Abelian gauge theory. More precisely, when a self-dual representation of the
CCR is chosen, the corresponding vacuum in the Schroedinger representation is
precisely given by the Kodama state. Several consequences of this construction
are explored.Comment: 4 pages, no figures. References and discussion added. Final version
to appear in PR
Loop quantum gravity and Planck-size black hole entropy
The Loop Quantum Gravity (LQG) program is briefly reviewed and one of its
main applications, namely the counting of black hole entropy within the
framework is considered. In particular, recent results for Planck size black
holes are reviewed. These results are consistent with an asymptotic linear
relation (that fixes uniquely a free parameter of the theory) and a logarithmic
correction with a coefficient equal to -1/2. The account is tailored as an
introduction to the subject for non-experts.Comment: 21 pages, 5 figures. Contribution to the Proceedings of the NEB XII
International Conferenc
Quantum Superposition Principle and Geometry
If one takes seriously the postulate of quantum mechanics in which physical
states are rays in the standard Hilbert space of the theory, one is naturally
lead to a geometric formulation of the theory. Within this formulation of
quantum mechanics, the resulting description is very elegant from the
geometrical viewpoint, since it allows to cast the main postulates of the
theory in terms of two geometric structures, namely a symplectic structure and
a Riemannian metric. However, the usual superposition principle of quantum
mechanics is not naturally incorporated, since the quantum state space is
non-linear. In this note we offer some steps to incorporate the superposition
principle within the geometric description. In this respect, we argue that it
is necessary to make the distinction between a 'projective superposition
principle' and a 'decomposition principle' that extend the standard
superposition principle. We illustrate our proposal with two very well known
examples, namely the spin 1/2 system and the two slit experiment, where the
distinction is clear from the physical perspective. We show that the two
principles have also a different mathematical origin within the geometrical
formulation of the theory.Comment: 10 pages, no figures. References added. V3 discussion expanded and
new results added, 14 pages. Dedicated to Michael P. Ryan on the occasion of
his sixtieth bithda
Quantum Structure of Geometry: Loopy and fuzzy?
In any attempt to build a quantum theory of gravity, a central issue is to
unravel the structure of space-time at the smallest scale. Of particular
relevance is the possible definition of coordinate functions within the theory
and the study of their algebraic properties, such as non-commutativity. Here we
approach this issue from the perspective of loop quantum gravity and the
picture of quantum geometry that the formalism offers. In particular, as we
argue here, this emerging picture has two main elements: i) The nature of the
quantum geometry at Planck scale is one-dimensional, polymeric with quantized
geometrical quantities and; ii) Appropriately defined operators corresponding
to coordinates by means of intrinsic, relational, constructions become
non-commuting. This particular feature of the operators, that operationally
localize points on space, gives rise to an emerging geometry that is also, in a
precise sense, fuzzy.Comment: 9 pages, no figure
Transcending Big Bang in Loop Quantum Cosmology: Recent Advances
We discuss the way non-perturbative quantization of cosmological spacetimes
in loop quantum cosmology provides insights on the physics of Planck scale and
the resolution of big bang singularity. In recent years, rigorous examination
of mathematical and physical aspects of the quantum theory has led to a
consistent quantization which is consistent and physically viable and some
early ideas have been ruled out. The latter include so called `physical
effects' originating from modifications to inverse scale factors in the flat
models. The singularity resolution is understood to originate from the
non-local nature of curvature in the quantum theory and the underlying polymer
representation. Using an exactly solvable model various insights have been
gained. The model predicts a generic occurrence of bounce for states in the
physical Hilbert space and a supremum for the spectrum of the energy density
operator. It also provides answers to the growth of fluctuations, showing that
semi-classicality is preserved to an amazing degree across the bounce.Comment: Invited plenary talk at the Sixth International Conference on
Gravitation and Cosmology, IUCAA (Pune). 13 pages, 3 figure
A Gaussian Weave for Kinematical Loop Quantum Gravity
Remarkable efforts in the study of the semi-classical regime of kinematical
loop quantum gravity are currently underway. In this note, we construct a
``quasi-coherent'' weave state using Gaussian factors. In a similar fashion to
some other proposals, this state is peaked in both the connection and the spin
network basis. However, the state constructed here has the novel feature that,
in the spin network basis, the main contribution for this state is given by the
fundamental representation, independently of the value of the parameter that
regulates the Gaussian width.Comment: 15 pages, 3 figures, Revtex file. Comments added and references
updated. Final version to appear in IJMP-
Black hole entropy quantization
Ever since the pioneer works of Bekenstein and Hawking, black hole entropy
has been known to have a quantum origin. Furthermore, it has long been argued
by Bekenstein that entropy should be quantized in discrete (equidistant) steps
given its identification with horizon area in (semi-)classical general
relativity and the properties of area as an adiabatic invariant. This lead to
the suggestion that black hole area should also be quantized in equidistant
steps to account for the discrete black hole entropy. Here we shall show that
loop quantum gravity, in which area is {\it not} quantized in equidistant steps
can nevertheless be consistent with Bekenstein's equidistant entropy proposal
in a subtle way. For that we perform a detailed analysis of the number of
microstates compatible with a given area and show consistency with the
Bekenstein framework when an oscillatory behavior in the entropy-area relation
is properly interpreted.Comment: 4 pages, 4 figures. Version to be published in PR
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