Author | Francesco Cianfrani, Giovanni Montani, Matteo Lulli, and Orchidea Maria Lecian | |

Isbn | 9789814556644 | |

File size | 2.4 MB | |

Year | 2014 | |

Pages | 324 | |

Language | English | |

File format | ||

Category | physics |

CANONICAL QUANTUM
GRAVITY
Fundamentals and Recent Developments
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CANONICAL QUANTUM
GRAVITY
Fundamentals and Recent Developments
Francesco Cianfrani
University of Wrocław, Poland
Orchidea Maria Lecian
“Sapienza”, University of Rome, Italy,
Max Planck Institute for Gravitational Physics, Germany &
ICRA International Center for Relativistic Astrophysics, Italy
Matteo Lulli
“Sapienza”, University of Rome, Italy
Giovanni Montani
ENEA - C.R. Frascati, UTFUS-MAG & “Sapienza”, University of Rome, Italy
World Scientific
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CANONICAL QUANTUM GRAVITY
Fundamentals and Recent Developments
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 978-981-4556-64-4
Printed in Singapore
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Contents
Preface
xi
List of Figures
xix
List of Notations
xxi
1.
Introduction to General Relativity
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
Parametric manifold representation . . . . . . . . .
Tensor formalism . . . . . . . . . . . . . . . . . . .
Aﬃne properties of the manifold . . . . . . . . . .
1.3.1 Ordinary derivative . . . . . . . . . . . . .
1.3.2 Covariant derivative . . . . . . . . . . . . .
1.3.3 Properties of the aﬃne connections . . . .
Metric properties of the manifold . . . . . . . . . .
1.4.1 Metric tensor . . . . . . . . . . . . . . . . .
1.4.2 Christoﬀel symbols . . . . . . . . . . . . .
Geodesic equation and parallel transport . . . . . .
Levi-Civita tensor . . . . . . . . . . . . . . . . . .
Volume element and covariant divergence . . . . .
Gauss and Stokes theorems . . . . . . . . . . . . .
The Riemann tensor . . . . . . . . . . . . . . . . .
1.9.1 Levi-Civita construction . . . . . . . . . .
1.9.2 Algebraic properties and Bianchi identities
Geodesic deviation . . . . . . . . . . . . . . . . . .
Einstein’s equations . . . . . . . . . . . . . . . . .
1.11.1 Equivalence Principle . . . . . . . . . . . .
1.11.2 Theory requirements . . . . . . . . . . . .
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vi
1.12
2.
Elements of Cosmology
2.1
2.2
2.3
2.4
2.5
3.
The Robertson-Walker geometry . .
Kinematics of the Universe . . . . .
Isotropic Universe dynamics . . . . .
Universe thermal history . . . . . . .
2.4.1 Universe critical parameters
Inﬂationary paradigm . . . . . . . .
2.5.1 Standard Model paradoxes .
2.5.2 Inﬂation mechanism . . . . .
2.5.3 Re-heating phase . . . . . .
3.3
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Preliminaries . . . . . . . . . . . . . . . . .
Constrained systems . . . . . . . . . . . . .
3.2.1 Primary and secondary constraints
3.2.2 First- and second-class constraints .
Canonical transformations . . . . . . . . . .
3.3.1 Strongly canonical transformations
3.3.2 Weakly canonical transformations .
3.3.3 Gauged canonical transformations .
Electromagnetic ﬁeld . . . . . . . . . . . . .
3.4.1 Modiﬁed Lagrangian formulation .
3.4.2 Hamiltonian formulation . . . . . .
3.4.3 Gauge transformations . . . . . . .
3.4.4 Gauged canonicity . . . . . . . . . .
4.2
Metric representation . . . . . . . . . . . .
4.1.1 Einstein-Hilbert formulation . . . .
4.1.2 Stress-Energy tensor . . . . . . . . .
4.1.3 ΓΓ formulation . . . . . . . . . . . .
4.1.4 Dirac formulation . . . . . . . . . .
4.1.5 General f (R) Lagrangian densities
4.1.6 Palatini formulation . . . . . . . . .
ADM formalism . . . . . . . . . . . . . . .
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Lagrangian Formulations
4.1
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Constrained Hamiltonian Systems
3.1
3.2
4.
1.11.3 Field equations . . . . . . . . . . . . . . . . . . .
Vierbein representation . . . . . . . . . . . . . . . . . . .
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4.3
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Quantization Methods
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5.3
5.4
5.5
6.
4.2.1 Spacetime foliation and extrinsic curvature
4.2.2 Gauss-Codazzi equation . . . . . . . . . . .
4.2.3 ADM Lagrangian density . . . . . . . . . .
Boundary terms . . . . . . . . . . . . . . . . . . .
4.3.1 Gibbons-York-Hawking boundary term . .
4.3.2 Comparison among diﬀerent formulations .
vii
Classical and quantum dynamics . . . . . . . . . . . . . .
5.1.1 Dirac observables . . . . . . . . . . . . . . . . . .
5.1.2 Poisson brackets and commutators . . . . . . . . .
5.1.3 Schrödinger representation . . . . . . . . . . . . .
5.1.4 Heisenberg representation . . . . . . . . . . . . . .
5.1.5 The Schrödinger equation . . . . . . . . . . . . . .
5.1.6 Quantum to classical correspondence: HamiltonJacobi equation . . . . . . . . . . . . . . . . . . .
5.1.7 Semi-classical states . . . . . . . . . . . . . . . . .
Weyl quantization . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Weyl systems . . . . . . . . . . . . . . . . . . . .
5.2.2 The Stone-von Neumann uniqueness theorem . . .
GNS construction . . . . . . . . . . . . . . . . . . . . . . .
Polymer representation . . . . . . . . . . . . . . . . . . . .
5.4.1 Diﬀerence operators versus diﬀerential operators .
5.4.2 The polymer representation of Quantum Mechanics
5.4.3 Kinematics . . . . . . . . . . . . . . . . . . . . . .
5.4.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . .
5.4.5 Continuum limit . . . . . . . . . . . . . . . . . . .
Quantization of Hamiltonian constraints . . . . . . . . . .
5.5.1 Non-relativistic particle . . . . . . . . . . . . . . .
5.5.2 Relativistic particle . . . . . . . . . . . . . . . . .
5.5.3 Scalar ﬁeld . . . . . . . . . . . . . . . . . . . . . .
5.5.4 The group averaging technique . . . . . . . . . . .
Quantum Geometrodynamics
6.1
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The Hamiltonian structure of gravity . . . . . . . . . . . . 145
6.1.1 ADM Hamiltonian density . . . . . . . . . . . . . 145
6.1.2 Constraints in the 3+1 representation . . . . . . . 147
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viii
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6.2
6.3
6.4
7.
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Gravity as a Gauge Theory
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7.4
8.
The Hamilton-Jacobi equation for the gravitational ﬁeld . . . . . . . . . . . . . . . . . . . . . .
ADM reduction of the Hamiltonian dynamics . . . . . . .
Quantization of the gravitational ﬁeld . . . . . . . . . . .
6.3.1 Quantization of the primary constraints . . . . . .
6.3.2 Quantization of the supermomentum constraint .
6.3.3 The Wheeler-DeWitt equation . . . . . . . . . . .
Shortcomings of the Wheeler-DeWitt approach . . . . . .
6.4.1 The deﬁnition of the Hilbert space . . . . . . . . .
6.4.2 The functional nature of the theory . . . . . . . .
6.4.3 The frozen formalism: the problem of time . . . .
Gauge theories . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 The Yang-Mills formulation . . . . . . . . . . . .
7.1.2 Hamiltonian formulation . . . . . . . . . . . . . .
7.1.3 Lattice gauge theories . . . . . . . . . . . . . . . .
Gravity as a gauge theory of the Lorentz group? . . . . .
7.2.1 Spinors in curved spacetime . . . . . . . . . . . .
7.2.2 Comparison between gravity and Yang-Mills
theories . . . . . . . . . . . . . . . . . . . . . . . .
Poincaré gauge theory . . . . . . . . . . . . . . . . . . . .
Holst action . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Lagrangian formulation . . . . . . . . . . . . . . .
7.4.2 Hamiltonian formulation . . . . . . . . . . . . . .
7.4.3 Ashtekar variables . . . . . . . . . . . . . . . . . .
7.4.4 Removing the time-gauge condition . . . . . . . .
7.4.5 The Kodama functional as a classical solution of
the constraints . . . . . . . . . . . . . . . . . . . .
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Loop Quantum Gravity
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8.2
8.3
Smeared variables . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Why a reformulation in terms of holonomies? . . .
Hilbert space representation of the holonomy-ﬂux algebra
8.2.1 Holonomy-ﬂux algebra . . . . . . . . . . . . . . .
8.2.2 Kinematical Hilbert space . . . . . . . . . . . . .
Kinematical constraints . . . . . . . . . . . . . . . . . . .
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8.3.1
8.4
8.5
8.6
8.7
8.8
9.
Solution of the Gauss constraint: invariant spinnetworks . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 Three-diﬀeomorphisms invariance and s-knots . .
Geometrical operators: discrete spectra . . . . . . . . . .
8.4.1 Area operator . . . . . . . . . . . . . . . . . . . .
8.4.2 Volume operator . . . . . . . . . . . . . . . . . . .
The scalar constraint operator . . . . . . . . . . . . . . .
8.5.1 Spin foams and the Hamiltonian constraint . . . .
Open issues in Loop Quantum Gravity . . . . . . . . . . .
8.6.1 Algebra of the constraints . . . . . . . . . . . . .
8.6.2 Semiclassical limit . . . . . . . . . . . . . . . . . .
8.6.3 Physical scalar product . . . . . . . . . . . . . . .
8.6.4 On the physical meaning of the Immirzi parameter
Master Constraint and Algebraic Quantum Gravity . . . .
The picture of quantum spacetime . . . . . . . . . . . . .
Quantum Cosmology
9.1
9.2
9.3
9.4
9.5
9.6
The minisuperspace model . . . . . . . . . . . . . . . . .
General behavior of Bianchi models . . . . . . . . . . . .
9.2.1 Quantum Picture . . . . . . . . . . . . . . . . .
9.2.2 Matter contribution . . . . . . . . . . . . . . . .
Bianchi I model . . . . . . . . . . . . . . . . . . . . . . .
Bianchi IX model . . . . . . . . . . . . . . . . . . . . . .
9.4.1 Quantum dynamics . . . . . . . . . . . . . . . .
9.4.2 Semiclassical behavior . . . . . . . . . . . . . . .
9.4.3 The quantum behavior of the isotropic Universe
BKL conjecture . . . . . . . . . . . . . . . . . . . . . . .
Cosmology in LQG . . . . . . . . . . . . . . . . . . . . .
9.6.1 Loop Quantum Cosmology . . . . . . . . . . . .
9.6.2 Other approaches . . . . . . . . . . . . . . . . .
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270
Bibliography
275
Index
295
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Preface
The request of a coherent quantization for the gravitational ﬁeld dynamics emerges as a natural consequence of Einstein’s Equations: the energymomentum of a ﬁeld is source of the spacetime curvature and therefore
its microscopic quantum features must be reﬂected onto microgravity effects. The use of expectation values as sources is well-grounded as a ﬁrst
approximation only, and does not fulﬁll the requirements of a fundamental
theory.
However, as is well-known, the achievement of a consistent Quantum
Gravity theory remains a complete open task, due to a plethora of diﬀerent subtleties, which are here summarized in the two main categories: i)
General Relativity is a background-independent theory and therefore any
analogy with the non-Abelian gauge formalisms must deal with the concept of a dynamic metric ﬁeld; ii) the implementation in the gravitational
sector of standard prescriptions, associated with the quantum mechanics
paradigms, appears as a formal procedure, whose real physical content is
still elusive.
There exists a qualitative consensus on the idea that a convincing solution to the Quantum Gravity problem will not arise before General Relativity and Quantum Mechanics are both deeply revised in view of a converging
picture. Indeed, over the last ﬁfteen years, the three most promising approaches to Quantum Gravity (i.e. String Theories, Loop Quantum Gravity
and Non-commutative Geometries) revealed common features in deﬁning a
“lattice” nature for the microphysics of spacetime. This consideration makes
clear that, up to the best of our present understanding, the main eﬀort to
improve fundamental formalisms must be the introduction of a “cut-oﬀ”
physics, able to replace the notion of a spacetime continuum with a consistent discrete scenario. The correctness of such a statement will probably
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be regarded as the main success of the end of the last century reached
in Theoretical Physics. A proper task for the present century is now to
constrain the morphology of such a discrete microstructure of spacetime,
to get, at least, a phenomenological description for Quantum Gravity effects. From a theoretical point of view, an important aim would consist
of a uniﬁed picture containing common features of the present approaches,
but synthesized into a more powerful mathematical language. On the one
hand, it would be important to recognize non-commutative properties in the
loop representation of spacetime; on the other hand, transporting the background independence of the “spin networks” into the interaction framework
characterizing String Theories would constitute a relevant progress. The
viability of these two goals is an intuitive perspective, but it contrasts with
the rigidity of the corresponding formalisms, which conﬁrms the request for
more general investigation tools.
The simplest approach to the quantization of General Relativity relies on
the implementation of the canonical method on the phase-space structure
associated with the gravitational degrees of freedom. This attempt is the
one originally pursued by B. DeWitt in 1967 and it immediately revealed
all the pathologies contained in the canonical quantum geometrodynamics.
It was this clear inconsistence of the canonical quantum procedure (referred
to the second order formalism) that attracted the attention of a large number of researchers, active on the last four decades. This strong eﬀort of
re-analysis, which could give the feeling of an overestimation of the real
chances allowed by the canonical method, found its “merry ending” in the
developments in Loop Quantum Gravity and its applications. In fact, the
formulation of the Hamiltonian problem for General Relativity provided by
A. Ashtekar in 1986, allowed for a new paradigm for the canonical method,
in close analogy with an SU (2) gauge theory. The main success of Loop
Quantum Gravity is recognizing a discrete structure of spacetime, by starting from continuous variables in the phase space of the theory. The origin
of such a valuable result consists in the discrete nature acquired by the
spectra of areas and volumes, reﬂecting to some extent the compactness of
the SU (2) group emerging in this formulation. The merit of the loop quantization of gravity can be identiﬁed in the possibility to deal with non-local
geometric variables, like holonomies and ﬂuxes, instead of the simple metric
analysis faced by DeWitt. In fact, using the physical properties associated
with the connection and the so-called electric ﬁelds, the background independence of the theory naturally emerges, and the quantum structure of
spacetime comes out, as far as the Hilbert space is characterized via the
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xiii
spin-network basis.
Quantum Cosmology is a privileged arena in which diﬀerent Quantum
Gravity approaches can be implemented and it is expected to provide a
consistent picture of the Universe birth. In this respect, the standard minisuperspace implementations of both the Wheeler-DeWitt formulation and
of Loop Quantum Gravity (Loop Quantum Cosmology) are predictive to
some extent.
In particular the Wheeler-DeWitt equation, as restricted to the homogeneous cosmological spaces, presents some pleasant features, such that the
Universe dynamics resembles that of a scalar ﬁeld interacting with a nontrivial and non-perturbative potential. In this formulation, the cosmological
singularity is not removed, but a consistent Universe evolution throughout
the Planck era can be clearly traced.
For what concerns Loop Quantum Cosmology, a Big-Bounce emerges
due to the discreteness of the space volume spectrum, and it oﬀers a viable
semiclassical picture of the singularity removal by a phenomenological repulsion of the high curvature gravity. Nonetheless, a critical revision of the
standard loop Universe quantization can be given, based on observing how
the homogeneity restriction implies a gauge ﬁxing on the SU (2) internal
symmetry of the full theory. A recent promising alternative reformulation of the Loop Quantum Gravity dynamics for the primordial Universe
is presented together with its basic tools (“in primis” the presence of an
intertwiner structure), which makes it closer to the full theory than the
Loop Quantum Cosmology paradigm.
The analysis of Quantum Cosmology here presented is probably one of
the most original contributions oﬀered in this volume to the comprehension of the impact that quantum physics can have on the dynamics of the
gravitational ﬁeld. It can be regarded as the prize the reader gets out of
the study of the preceding chapters concerning the successes and the limits
of the full Quantum Gravity theory.
The aim of this book is to provide a detailed account of the physical
content emerging from the canonical approach to Quantum Gravity. All
the crucial steps in our presentation have a rather pedagogical character,
providing the reader with the necessary tools to become involved in the
ﬁeld. Such a pedagogical aspect is then balanced and completed by subtle
discussions on speciﬁc topics which we regard as relevant for the physical
insight they outline on the treated questions. Our analysis is not aimed
at convincing the reader about a pre-constituted point of view instead, our
principal goal is to review the picture of Canonical Quantum Gravity on
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CANONICAL QUANTUM GRAVITY
the basis of the concrete facts at the ground of its clear successes, and also
of its striking shortcomings.
In order to focus our attention on the physical questions aﬀecting the
consistence of a canonical quantum model, when extended to the gravitational sector, we provide a critical discussion of all the fundamental concepts
of quantum physics, by stressing the peculiarity of the gravitational case in
their respect. All the key features of Loop Quantum Gravity are described
in some details, presenting the speciﬁc technicalities, but privileging their
physical interpretation. Finally, all the open questions concerning the faced
topics are clearly outlined by a precise criticism on the weak points present
in the quantum construction.
The detailed structure of the book is
• the main goal of the ﬁrst chapter is to provide the basic formalisms
and concepts at the ground of Einsteinian theory of gravity. We
start by constructing the tensor representation of a manifold differential geometry in terms of the so-called embedding procedure,
i.e. we consider a parametric representation for a generic hypersurface. The concepts of spacetime curvature and geodesic deviation
are then fully addressed. We discuss the fundamental principles of
General Relativity and show how they ﬁx the form of Einstein’s
equations. As a last step we provide the vierbein formalism for
further applications throughout the book.
• in the second chapter we provide a brief presentation of the Universe evolution based on the isotropic Robertson-Walker geometry.
We ﬁrst discuss the morphology of a homogeneous and isotropic
Universe and its kinematic implications, able to account for observations such as the photons redshift and the Hubble law. Then, we
address the dynamic features of the isotropic Universe as described
by Einstein’s equations. The structure of the dynamic equations
is outlined by characterizing the matter source via a perfect ﬂuid
endowed with a proper equation of state. This allows one to develop a brief sketch of the Universe thermal history, describing the
diﬀerent stages of its evolution. Finally, we analyze the inﬂationary
paradigm, oﬀering a compact review of the main shortcomings of
the standard cosmological model, as well as the basic ideas at the
ground of the inﬂationary scenario.
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• in the third chapter, we introduce the main concepts of the standard Hamiltonian formulation for constrained systems, namely the
distinction between primary and secondary constraints as well as
between ﬁrst-class and second-class constraints. Moreover the definition of canonical transformations is reviewed and broadened introducing the concepts of weak- and gauged-canonicity. In order
to give some practical insight, we use an example based on the
free electromagnetic ﬁeld where all the concepts previously faced
are sketched in a rather pedagogical way. In this respect we will
present the L. Castellani algorithm for constructing the generators
of gauge transformations.
• in the fourth chapter, the Lagrangian formulation of General Relativity is critically discussed giving diﬀerent but equivalent (under
certain hypothesis) Lagrangian densities for the gravitational ﬁeld.
We start by introducing the commonly used Einstein-Hilbert Lagrangian density followed by the study of the action principle when
matter is included. We discuss the non-covariant Lagrangian density approach due to Einstein and Dirac. The study of an extended
theory of gravity in the form of an f (R) Lagrangian density is developed followed by description of the Palatini formulation. The ADM
formulation is presented using the embedding technique developed
in the ﬁrst chapter and a critical discussion on the boundary terms
of some of the formulations is given allowing one to perform the
transformation linking some of the presented approaches, namely
the ADM to Einstein’s and Dirac’s ones.
• in the ﬁfth chapter, we recall the basic features of Quantum Mechanics. In particular, we review the deﬁnition of quantum states as
elements of a Hilbert space and of observables as linear operators.
The classical to quantum correspondence between Poisson brackets
and commutators is discussed as a tool for the quantization of a
generic mechanical system. Then, we present the Schrödinger and
Heisenberg representations. The Hamilton-Jacobi equation and the
development of semiclassical coherent states are analyzed in order
to bridge the quantum formulation with the classical world. Furthermore, we introduce Weyl quantization and outline the main
steps of the GNS construction. The polymer representation is then
presented as an application. Finally, the quantization of some
physically-relevant constrained systems and its shortcomings are
discussed.
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CANONICAL QUANTUM GRAVITY
• in the sixth chapter, the standard Hamiltonian formulation of General Relativity is reviewed. Then, the two possible scenarios for the
canonical quantization of the gravitation ﬁeld are illustrated. On
one hand, we present the reduction to the canonical form and its
main issues are brieﬂy outlined. On the other hand, we focus on
the Wheeler-De Witt approach. We emphasize how the implementation of the constraints via the Dirac prescription leads to the
Wheeler-DeWitt equation. The main technical and interpretative
shortcomings of this formulation (which prevents it from being a
well-grounded quantum description for the gravitational ﬁeld) are
discussed: the absence of a real Hilbert space structure, the need for
a consistent regularization scheme and the problem of time. Some
proposals to identify a time variable are presented: the BrownKuchař model, the multitime approach and the Vilenkin proposal.
• in the seventh chapter, Yang-Mills gauge theories are presented.
The classical Lagrangian and Hamiltonian formulations are analyzed. Then, we introduce some tools which are useful for lattice
quantization (spin networks and spin foams). Furthermore, a formal comparison between Yang-Mills models and General Relativity
is performed by outlining the similarities but also the unavoidable
diﬀerences. Poincaré gauge theory is then discussed as an alternative formulation in which gravity behaves as a Yang-Mills interaction. The Holst formulation for gravity is then revised and the
emergence of a SU(2) gauge symmetry is emphasized. The analysis
of the associated constrained system in vacuum and in the presence of matter ﬁelds is carefully discussed in a generic local Lorenz
frame. Finally, the Kodama state is presented.
• the eighth chapter is devoted to Loop Quantum Gravity. The
role of holonomies and ﬂuxes in quantum theories is brieﬂy discussed, while the quantization of the corresponding algebra in Loop
Quantum Gravity is presented. The main achievements of this
framework are emphasized: the deﬁnition of a kinematical Hilbert
space, the success in the quantization of kinematical constraints,
the achievement of discrete spectra for the area and volume operators, the implementation of the scalar constraint on a quantum
level. A careful explanation of the main technical and conceptual
issues (the study of the dynamics, the semiclassical limit, the constraint algebra, the role of the Immirzi parameter) is given, while
the Master Constraint program and the Algebraic Quantum Grav-
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ity setup are brieﬂy discussed as attempts towards the solution of
some shortcomings. Finally, a comparison between the picture of
the quantum spacetime coming out from the Loop Quantum Gravity and the Wheeler-DeWitt frameworks is performed.
• in the ninth chapter, Quantum Cosmology is presented as an arena
in which the predictions of Quantum Gravity can be investigated.
The minisuperspace approximation is discussed: we point out the
simpliﬁcations that the restriction to homogeneous models provides
for the quantization issue, such that a proper quantum description
of the Bianchi type I and IX models in the Wheeler-DeWitt approach can be given, even though this is not generically enough to
obtain a consistent quantum model for the Universe (the presence
of singularity and the chaos in Bianchi IX). Then, the paradigm
of Loop Quantum Cosmology is introduced and the bouncing solution replacing the Big-Bang singularity is explicitly shown to arise.
Finally, a critical discussion on the foundation of Loop Quantum
Cosmology is given and a new research line for the cosmological
sector of Loop Quantum Gravity, namely Quantum Reduced Loop
Gravity, is presented.
We would like to thank Abhay Ashtekar for interesting discussions on
Loop Quantum Gravity during the First Stueckelberg Workshop (Pescara,
June 25-July 1 2006), and which inspired the writing of this book.
We would also like to thank Richard Arnowitt, Stanley Deser and
Charles W. Misner for their comments and discussion on the Hamiltonian
formulation of General Relativity and for having called our attention on
the speciﬁc relevance of reference [17].
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CANONICAL QUANTUM GRAVITY
The work of FC was supported by funds provided by the National Science Center under the agreement DEC- 2011/02/A/ST2/00294.
The work of OML was partially supported by the research grant ’Reﬂections on the Hyperbolic Plane’ from the Albert Einstein Institute for
Gravitational Physics - MPI, Potsdam-Golm, and partially by the research
grant ’Classical and Quantum Physics of the Primordial Universe’ from
Sapienza University of Rome- Physics Department, Rome.
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List of Figures
1.1
1.2
Manifold embedded in a Minkowskian space . . . . . . . . . . .
Riemann Levi-Civita construction . . . . . . . . . . . . . . . .
4.1
ADM foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.1
8.2
8.3
SU (2) invariant spin network . . . . . . . . . . . . . . . . . . . 210
Tetrahedron adapted to a graph . . . . . . . . . . . . . . . . . 218
Quantum space in LQG . . . . . . . . . . . . . . . . . . . . . . 228
9.1
9.2
9.3
Bianchi IX potential . . . . . . . . . . . . . . . . . . . . . . . . 246
Potential for closed isotropic Universe . . . . . . . . . . . . . . 254
Bounce in LQC . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
xix
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Author Francesco Cianfrani, Giovanni Montani, Matteo Lulli, and Orchidea Maria Lecian Isbn 9789814556644 File size 2.4 MB Year 2014 Pages 324 Language English File format PDF Category Physics Book Description: FacebookTwitterGoogle+TumblrDiggMySpaceShare This book aims to present a pedagogical and self-consistent treatment of the canonical approach to Quantum Gravity, starting from its original formulation to the most recent developments in the field. We start with an innovative and enlightening introduction to the formalism and concepts on which General Relativity has been built, giving all the information necessary in the later analysis. A brief sketch of the Standard Cosmological Model describing the Universe evolution is also given alongside the analysis of the inflationary mechanism. After deepening the fundamental properties of constrained dynamic systems, the Lagrangian approach to the Einsteinian Theory is presented in some detail, underlining the parallelism with non-Abelian gauge theories. Then, the basic concepts of the canonical approach to Quantum Mechanics are provided, focusing on all those formulations which are relevant for the Canonical Quantum Gravity problem. The Hamiltonian formulation of General Relativity and its constrained structure is then analyzed by comparing different formulations. The resulting quantum dynamics, described by the WheelerDeWitt equation, is fully discussed in order to outline its merits and limits. Afterwards, the reformulation of Canonical Quantum Gravity in terms of the AshtekarBarberoImmirzi variables is faced by a detailed discussion of the resulting Loop Quantum Gravity Theory. Finally, we provide a consistent picture of canonical Quantum Cosmology by facing the main features of the WheelerDeWitt equation for the homogeneous Bianchi models and then by a detailed treatment of Loop Quantum Cosmology, including very recent developments. Contents: Introduction to General Relativity Elements of Cosmology Constrained Hamiltonian Systems Lagrangian Formulations Quantization Methods Quantum Geometrodynamics Gravity as a Gauge Theory Loop Quantum Gravity Quantum Cosmology Readership: Researchers in theoretical physics, quantum physics, general relativity and astrophysics. Download (2.4 MB) Quantum Gravity Recent Developments in Gravitation: Cargèse 1978 Mathematical Physics: Proceedings of the 12th Regional Conference A First Course In Loop Quantum Gravity Advanced Concepts In Particle And Field Theory Load more posts