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