Canonical Quantum Gravity Fundamentals and Recent Developments 1st Edition by Francesco Cianfrani, Orchidea Maria Lecian, Matteo Lulli, Giovanni Montani 9814556645 9789814556644

Canonical Quantum Gravity Fundamentals and Recent Developments 1st Edition by Francesco Cianfrani, Orchidea Maria Lecian, Matteo Lulli, Giovanni Montani – Ebook PDF Instant Download/Delivery: 9814556645, 9789814556644

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ISBN 10: 9814556645 

ISBN 13: 9789814556644

Author: Francesco Cianfrani, Orchidea Maria Lecian, Matteo Lulli, Giovanni Montani

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 Wheeler–DeWitt equation, is fully discussed in order to outline its merits and limits. Afterwards, the reformulation of Canonical Quantum Gravity in terms of the Ashtekar–Barbero–Immirzi 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 Wheeler–DeWitt equation for the homogeneous Bianchi models and then by a detailed treatment of Loop Quantum Cosmology, including very recent developments.

Table of contents:

1. Introduction to General Relativity

1.1 Parametric manifold representation

1.2 Tensor formalism

1.3 Affine properties of the manifold

1.3.1 Ordinary derivative

1.3.2 Covariant derivative

1.3.3 Properties of the affine connections

1.4 Metric properties of the manifold

1.4.1 Metric tensor

1.4.2 Christoffel symbols

1.5 Geodesic equation and parallel transport

1.6 Levi-Civita tensor

1.7 Volume element and covariant divergence

1.8 Gauss and Stokes theorems

1.9 The Riemann tensor

1.9.1 Levi-Civita construction

1.9.2 Algebraic properties and Bianchi identities .

1.10 Geodesic deviation

1.11 Einstein’s equations

1.11.1 Equivalence Principle

2. Theory requirements Elements of Cosmology

2.1 The Robertson-Walker geometry

2.2 Kinematics of the Universe

2.3 Isotropic Universe dynamics

2.4 Universe thermal history

2.4.1 Universe critical parameters

2.5 Inflationary paradigm

2.5.1 Standard Model paradoxes

2.5.2 Inflation mechanism

2.5.3 Re-heating phase

3. Constrained Hamiltonian Systems

3.1 Preliminaries

3.2 Constrained systems

3.2.1 Primary and secondary constraints

3.2.2 First- and second-class constraints

3.3 Canonical transformations

3.3.1 Strongly canonical transformations

3.3.2 Weakly canonical transformations

3.3.3 Gauged canonical transformations

3.4 Electromagnetic field

3.4.1 Modified Lagrangian formulation

3.4.2 Hamiltonian formulation

3.4.3 Gauge transformations

3.4.4 Gauged canonicity

4. Lagrangian Formulations

4.1 Metric representation

4.1.1 Einstein-Hilbert formulation

4.1.2 Stress-Energy tensor

4.1.3 rr formulation

4.1.4 Dirac formulation

4.1.5 General f (R) Lagrangian densities

4.1.6 Palatini formulation

4.2 ADM formalism

4.2.1 Spacetime foliation and extrinsic curvature.

4.2.2 Gauss-Codazzi equation

4.2.3 ADM Lagrangian density

4.3 Boundary terms

4.3.1 Gibbons-York-Hawking boundary term

4.3.2 Comparison among different formulations

5. Quantization Methods

5.1 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: Hamilton- Jacobi equation

5.1.7 Semi-classical states

5.2 Weyl quantization

5.2.1 Weyl systems

5.2.2 The Stone-von Neumann uniqueness theorem

5.3 GNS construction

5.4 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

5.5 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

6. Quantum Geometrodynamics

6.1 The Hamiltonian structure of gravity

6.1.1 ADM Hamiltonian density

6.1.2 Constraints in the 3+1 representation

6.1.3 The Hamilton-Jacobi equation for the gravitational field

6.2 ADM reduction of the Hamiltonian dynamics

6.3 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

6.4 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.

7. Gravity as a Gauge Theory

7.1 Gauge theories

7.1.1 The Yang-Mills formulation

7.1.2 Hamiltonian formulation

7.1.3 Lattice gauge theories

7.2 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

7.3 Poincaré gauge theory

7.4 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

8. Loop Quantum Gravity

8.1 Smeared variables

8.1.1 Why a reformulation in terms of holonomies?

8.2 Hilbert space representation of the holonomy-flux algebra

8.2.1 Holonomy-flux algebra

8.2.2 Kinematical Hilbert space

8.3 Kinematical constraints

8.3.1 Solution of the Gauss constraint: invariant spinnetworks

8.3.2 Three-diffeomorphisms invariance and s-knots

8.4 Geometrical operators: discrete spectra

8.4.1 Area operator

8.4.2 Volume operator

8.5 The scalar constraint operator

8.5.1 Spin foams and the Hamiltonian constraint.

8.6 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

8.7 Master Constraint and Algebraic Quantum Gravity.

8.8 The picture of quantum spacetime

9. Quantum Cosmology

9.1 The minisuperspace model

9.2 General behavior of Bianchi models

9.2.1 Quantum Picture

9.2.2 Matter contribution

9.3 Bianchi I model

9.4 Bianchi IX model

9.4.1 Quantum dynamics

9.4.2 Semiclassical behavior

9.4.3 The quantum behavior of the isotropic Universe .

9.5 BKL conjecture

9.6 Cosmology in LQG

9.6.1 Loop Quantum Cosmology

9.6.2 Other approaches

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Tags: Francesco Cianfrani, Orchidea Maria Lecian, Matteo Lulli, Canonical