Loop Quantum Gravity for Everyone

The following sections are included:

• Preface
• Acknowledgments
• Contents

Loop quantum gravity is one of the main contenders to unify Einstein’s general theory of relativity and quantum mechanics, therefore providing a quantum theory of gravity. If these words do not mean much to you, do not worry, we will define them in simple terms in the following chapters…

In this chapter we will talk about Einstein’s general theory of relativity. But to give some context, let us step back and talk about the other interactions in nature…

At the beginning of the 20th century physicists started studying matter at a microscopic scale. The idea of atoms goes back to the Greeks, but at that time it was just a speculation. In the 20th century the fact that matter was composed of atoms and that atoms were composed of a nucleus and electrons were experimentally verified. It was also noticed that at a microscopic scale things did not behave like objects in our everyday macroscopic life. Strange things happen…

We have explained what the last two terms mean, but what is this thing with loops? To discuss it, it is good to go back briefly to electromagnetism and one of its pioneers, Michael Faraday (Figure 4.1)…

We will introduce black holes and discuss their thermodynamics. This will require defining both terms.

Thermodynamics is the branch of physics that studies systems with incomplete information. A prototypical example of a thermodynamical system is a gas. Typically it is constituted by approximately 10²³ molecules. Obviously we cannot keep track of all of their positions and velocities. So we resort to “macro” variables to characterize its state, like the pressure and temperature…

This is the second application we wish to discuss, and it is called cosmology or the study of the universe as a whole. You may ask: how can they study the whole universe? The answer is: very coarsely. One ignores most details of it, and concentrates on a few of them. In the simplest example, the only degree of freedom considered is the scale of the universe. This is actually not a bad approximation: if one looks at things at a large scale, the universe is pretty homogeneous and isotropic (it looks the same in all places and in all directions; see Figure 6.1). That is not the case on a small scale: obviously the Solar System is different close to the Sun than close to Pluto, for example. But if one goes at the scales of galaxies or, even further, clusters of galaxies, things look pretty uniform…

We have discussed some of the difficulties encountered in handling Thiemann’s quantum version of the Einstein equations. This has led to the development of an alternative approach to the dynamics of the theory, known as spin foams. Many of the results in this area are highly technical and cannot be well covered in a book for the general public, here we will just give a general idea of what is being attempted. There is a technical book on the subject by Rovelli and Vidotto…

As we have mentioned, we do not know of any experiment that we can perform that requires quantum gravity to explain it, although we noted in Chapter 6 that some observational imprint could be left on the cosmic microwave background. The fundamental constants involved in quantum gravity are Newton’s constant G, the Planck constant of quantum mechanics ћ and the speed of light c. Out of the three of them one can construct a length, called the Planck length that is 10⁻³³ cm. That is 20 orders of magnitude smaller than the radius of a proton! That is the kind of scale where one expects quantum gravity to be relevant. Compared to it, macroscopic scales are enormous. The kind of granularity of space-time that is implied by the quantization of areas and volumes that we mentioned in loop quantum gravity has such a scale. That is why areas and volumes of macroscopic objects appear as smooth.

We have reviewed how the application of new techniques, based on the field lines that originated with Faraday long time ago and applied to gravity rewritten as Abhay Ashtekar taught us, has allowed to build a quantum theory of gravity. We still do not have complete control over it, but it has been applied to a couple of interesting situations and has yielded attractive results. There is much left to be done. But we hope the reader can appreciate that this mixture of the Faraday lines, the spin networks of Penrose, and the notion of diffeomorphism invariance have led to new insights on the problem of quantum gravity…

The following section is included:

• Index