Canonical Big Operators

The original publication is available at http://www.springerlink.com/content/16v67m7248714568/International audienceIn this paper, we present an approach to describe uniformly iterated “big” operations and to provide lemmas that encapsulate all the commonly used reasoning steps on these constructs. We show that these iterated operations can be handled generically using the syntactic notation and canonical structure facilities provided by the Coq system. We then show how these canonical big operations played a crucial enabling role in the study of various parts of linear algebra and multi-dimensional real analysis, as illustrated by the formal proofs of the properties of determinants, of the Cayley-Hamilton theorem and of Kantorovitch's theorem

Non-standard loop quantum cosmology

We present results concerning the nature of the cosmological big bounce(BB) transition within the loop geometry underlying loop quantum cosmology (LQC). Our canonical quantization method is an alternative to the standard LQC. An evolution parameter we use has clear interpretation both at classical and quantum levels. The physical volume operator has discrete spectrum which is bounded from below. The minimum gap in the spectrum defines a quantum of the volume. The spectra of operators are parametrized by a free parameter to be determined.Comment: 4 pages, prepared for the proceedings of the Grassmannian Conference in Fundamental Cosmology (Grasscosmofun'09), 14-19 September 2009, Szczecin, Polan

Quantum Cosmology

We give an introduction into quantum cosmology with emphasis on its conceptual parts. After a general motivation we review the formalism of canonical quantum gravity on which discussions of quantum cosmology are usually based. We then present the minisuperspace Wheeler--DeWitt equation and elaborate on the problem of time, the imposition of boundary conditions, the semiclassical approximation, the origin of irreversibility, and singularity avoidance. Restriction is made to quantum geometrodynamics; loop quantum gravity and string theory are discussed in other contributions to this volume.Comment: 29 pages, 9 figures, contribution to "Beyond the Big Bang", ed. by R. Vaas (Springer 2008); typos corrected, reference adde

Dirac Fields in Loop Quantum Gravity and Big Bang Nucleosynthesis

Big Bang nucleosynthesis requires a fine balance between equations of state for photons and relativistic fermions. Several corrections to equation of state parameters arise from classical and quantum physics, which are derived here from a canonical perspective. In particular, loop quantum gravity allows one to compute quantum gravity corrections for Maxwell and Dirac fields. Although the classical actions are very different, quantum corrections to the equation of state are remarkably similar. To lowest order, these corrections take the form of an overall expansion-dependent multiplicative factor in the total density. We use these results, along with the predictions of Big Bang nucleosynthesis, to place bounds on these corrections.Comment: 15 pages, 2 figures; v2: new discussion of relevance of quantum gravity corrections (Sec. II) and new estimates (Sec. V

Turning Big Bang into Big Bounce: I. Classical Dynamics

The big bounce (BB) transition within a flat Friedmann-Robertson-Walker model is analyzed in the setting of loop geometry underlying the loop cosmology. We solve the constraint of the theory at the classical level to identify physical phase space and find the Lie algebra of the Dirac observables. We express energy density of matter and geometrical functions in terms of the observables. It is the modification of classical theory by the loop geometry that is responsible for BB. The classical energy scale specific to BB depends on a parameter that should be fixed either by cosmological data or determined theoretically at quantum level, otherwise the energy scale stays unknown.Comment: 14 pages, 1 figure, version accepted for publication in Physical Review

Quantum D-modules and generalized mirror transformations

In the previous paper, the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c_1(M) of the tangent bundle is nef. In this paper, even when c_1(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem by Coates and Givental. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees.Comment: 54 pages; v2: corrected typos; v3: added reference; v4: corrected errors in the proof of the main theorem; v5: major revision. In the main theorem, we added a condition on the ambient toric variet

Gravity and the Quantum

The goal of this article is to present a broad perspective on quantum gravity for \emph{non-experts}. After a historical introduction, key physical problems of quantum gravity are illustrated. While there are a number of interesting and insightful approaches to address these issues, over the past two decades sustained progress has primarily occurred in two programs: string theory and loop quantum gravity. The first program is described in Horowitz's contribution while my article will focus on the second. The emphasis is on underlying ideas, conceptual issues and overall status of the program rather than mathematical details and associated technical subtleties.Comment: A general review of quantum gravity addresed non-experts. To appear in the special issue `Space-time Hundred Years Later' of NJP; J.Pullin and R. Price (editors). Typos and an attribution corrected; a clarification added in section 2.