Canonical Quantum Gravity: Fundamentals And Recent Developments

Fundamentals and Recent Developments

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

• Publisher: WSPC
• Year: 2014
• eBook ISBN: 9789814556668
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

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.

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.

Chapter 1

Introduction to General Relativity

In this chapter we will analyze the affine and metric properties of a 4-dimensional manifold, stressing the tensor character of all fundamental geometric objects. Then we will introduce the concepts of geodesics, curvature, and geodesic deviation in order to characterize the motion of test-particles in a generic spacetime. By means of the fundamental General Relativity and Equivalence principles we develop the basic ideas at the ground of General Relativity theory, matching them with the previously constructed formalism. Hence, we write down Einstein’s equations describing the geometrodynamics of spacetime and we also outline the role played by matter fields.

Finally, we provide an important formalism, commonly called vierbein representation, which allows to determine the existence of a local gauge-like symmetry associated to the Lorentz group which can be identified with an internal symmetry. In fact, the action of the local Lorentz group leaves unchanged both the metric and the Riemann tensors. In Chapter 7 this will be the starting point in order to represent General Relativity in the fashion of a non-Abelian Yang-Mills theory.

1.1 Parametric manifold representation

Let us consider a differentiable 4-dimensional manifold

embedded in a Minkowskian 5-dimensional space M⁵, endowed with the natural scalar product.

It is possible to define a parametric representation of

via a 5-dimensional vector,

, of which each component is a function of 4 parameters u^μ. We can project

on an orthonormal basis {

}

where the · indicates the common scalar product using the Minkowski metric η_IJ = diag{−1, 1, 1, 1, 1}. The manifold is said to be infinitely differentiable if X^I (u^μ) ∈ C^∞, for I = 0, …, 4.

On the other hand, it is also possible to define an adapted basis on

, which is given at each point by the set of the vectors which are tangent to the manifold itself (see figure 1.1). The tangent vectors {

} read

Figure 1.1 A drawing of the embedded manifold

described by the 5-dimensional vector

, with the basis vectors

.

Then, it follows that any vector

can be projected onto the basis {

} as follows

where the V ^I are the components in the 5-dimensional Minkowskian space. In this way we defined the contravariant components of

as V ^μ. It is possible to define the covariant components via the scalar product with the basis vectors

Hence, we generally obtained that V_μ ≠ V ^μ and that the two different kinds of components are linked by the non-trivial relation

1.2 Tensor formalism

The set of parameters {u^μ} defines a coordinate system adapted to the tangent hypersurface at every point of

. Any geometrical quantity belonging to the tangent hypersurface can be expressed as a function of {u^μ}.

If we choose a different parametrization of the manifold, we obtain a coordinate transformation which we request to be invertible and differentiable, i.e. a diffeomorphism, on

hence, we obtain the transformation of the basis vectors of the tangent hypersurfaces,

Vectors on

transforming this way, i.e. via the derivatives of the old variables with respect to the new ones, are called covariant. It then follows

vectors transforming as V^ρ are called contravariant, i.e. via the derivatives of the new variables with respect to the old ones.

A scalar quantity is an object independent of the choice of coordinates. We write it as a contraction of covariant and contravariant quantities

where we used the common property of the derivative chain rule

We define a tensor...

Table of contents

• Cover Page
• Halftitle Page
• Title Page
• Copyright
• Contents
• Preface
• List of Figures
• List of Notations
• 1.  Introduction to General Relativity
• 2.  Theory requirements Elements of Cosmology
• 3.  Constrained Hamiltonian Systems
• 4.  Lagrangian Formulations
• 5.  Quantization Methods
• 6.  Quantum Geometrodynamics
• 7.  Gravity as a Gauge Theory
• 8.  Loop Quantum Gravity
• 9.  Quantum Cosmology
• Bibliography
• Index