Quantum gravity booksreviewed by T. Nelson
Reviewed by T. Nelson
Loop quantum gravity is an up-and-coming competitor to string theory. It is a way of merging general relativity with quantum field theory in order to understand how gravity relates to the other three fundamental forces. Rather than starting from particle physics and working toward relativity, as string theory does, loop quantum gravity, or LQG, starts at relativity, that is to say with a geometrical interpretation, and works toward particle physics.
There are certain advantages of doing it this way. You don't end up with an embarrassingly large number of dimensions that have to be curled up and hidden someplace where nobody will notice. And you don't need an additional set of as-yet-unobserved particles as in supersymmetric string theory.
On the other hand, unlike string theory, LQG has not yet made any experimental predictions that are not already known in the standard model, known as QED. Partly this may be because LQG is so much younger than string theory. One prediction it does make is that instead of a Big Bang there might have been a Big Bounce.
Divergence, i.e. infinities in the theory and non-renormalizability historically were problems for LQG and which is where, say the authors, the field was stuck for a while, though recent work on quantized Hamiltonian constraints has reduced that problem. General relativity is an example of a non-renormalizable theory, which means an infinite number of terms might be needed to cancel the infinities if they tried to quantize it.
Despite these challenges, some physicists are starting to think the two theories can be merged. A timely article at Quanta Magazine says that many physicists now believe string theory and LQG may turn out not to be incompatible after all.
In this book it is assumed the reader is familiar with how to make classical Hamiltonians and how to do Legendre transforms. For background on this, chapters 19 and 20 of Penrose's The Road To Reality or Morin's textbook on classical mechanics are a good match. These concepts are used as analogies to explain basic quantum mechanics. For instance, the authors explain that the Poisson bracket, which plays an important role in the LQG theory, is a classical version of the commutator in QM.
On the back cover it says this book is suitable for readers with only a minimal knowledge of college level physics. This is most certainly not the case. However, the authors do provide the background the reader will need to make progress, although readers unfamiliar with the topics might find themselves making extensive use of the Internet in order to achieve it.
For example, on page 80 they introduce the Feynman propagator, which is an important concept in quantum field theory. All they say is that it has something to do with a time ordered product of two fields. Then they provide the formula for it and mention that it's the Fourier transform of the Green's function of the Klein-Gordon equation with the chosen boundary conditions.
Now, this is a perfectly good explanation, but a reader with no knowledge of QFT may start gasping for air at this point. A book called Student Guide to Quantum Field Theory by R.D. Klauber devotes eight pages and 31 equations to this one topic. And yet, somehow, Gambini and Pullin not only blast through it in a single paragraph, covering Wick's theorem and ultraviolet divergence in the next, their didactic skill is such that, if the reader is indeed still breathing at the end of it, he or she will actually understand some of the general mathematical and conceptual outlines of QFT.
So ... moving right along ... they reformulate general relativity in terms of Ashtekar's new variables, which were an early attempt at solving certain problems, and then they dive right into loop representation. At this point things start to get a bit technical.
What kind of loops are we talking about? Unlike the closed and open strings in string theory, LQG envisions intertwined curves called spin networks, which resemble several closed strings stuck together. The amount by which the value of a vector changes after traversing in a parallel manner around the closed loop is called a holonomy. It is a matrix that is invariant under gauge transformations, and thus a good candidate, say the authors, for an observable quantity. By tweaking this matrix, we can create spin networks. The dynamics of these spin networks, which are similar to Penrose's old spin networks from the 1950s, produce what's called a spin foam. In LQG, space is a quantized geometrical construct, and spacetime is a superposition of spin foams.
This is an exciting theory that suggests how one might create an utterly compelling picture in which matter, energy, and spacetime could be different manifestations of the same geometric objects. But all theories of quantum gravity still have problems. How, for example, the authors ask, can the idea of a fundamental length be reconciled with the Lorentz-Fitzgerald contraction?
The authors' goal was to provide students and non-professional physicists with enough information to grasp the basic concepts before deciding whether to pursue the topic in depth. So the formulas are given instead of derived, and complicated details are omitted to keep the book as concise as possible.
Concise does not necessarily mean easy, of course, and I'd be lying if I said I understood everything in this book. But confusion is the beginning of understanding. This book provides both confusion and understanding in abundance, and that may be just what we need to get us excited about quantum gravity.
The authors recommend Thiemann's more advanced book titled Modern Canonical Quantum General Relativity. Thiemann recommends Rovelli and Vidotto's book Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (reviewed at right). Rovelli and Vidotto recommend this book. These physicists just can't stop themselves from creating loops.
jan 18, 2016
Reviewed by T. Nelson
It seems to be a rule: the more difficult a book, the more likely it is to have the word ‘elementary’ in its title. This book continues in that grand tradition. In places it's very clear, and in others it's quite opaque.
My only consolation is that the more confused you are, the more you're learning. Confusion is just your brain trying to realign itself to a new worldview. The more confusion you can tolerate, the better your ability to learn. As a wise man once said, confusion is the beginning of understanding.
I have to say I learned a lot from this book. Rovelli and Vidotto are knowledgeable and impart a degree of enthusiasm, but their didactic skill isn't enough to overcome their subject matter if you don't have a strong background in the subject already. Before starting, I suggest getting a grounding in spinors (good tutorial in Penrose and another one here), tetrads in GR, Lie algebra, and of course QM and tensors. You're also expected to know exterior calculus. Fair enough, but would it have killed them to give us a definition? As I was trying to figure out what the heck a ‘∧’ was, all I could think was: throw me a bone, my kingdom for a bone.
Well, after a bit of Binging I found the bone, or at least a bone: α∧β(u,v) := α(u)β(v) − α(v)β(u).
This wedge operator is not to be confused with the cosmological constant Λ. The authors say a positive value of Λ is required by loop quantum gravity (LQG), but a negative one is required by string theory. (At the moment the accelerating universe suggests that Λ is >0; it's been estimated to be about +2.036×10−35 s−2 or about 10−122 in Planck units.)
What's the difference between covariant LQG and ordinary LQG? The authors say that unlike with metric space theories, discretizations of general covariant theories eliminate the need for a scaling parameter, which is good.
The section on Regge calculus in Chapters 4 and 5 is very good. And that's lucky, because the gist of quantum gravity is that geometry and space are quantized, and Regge calculus nicely describes the discretization of a 4-dimensional manifold. It approximates each quantum of spacetime as a sort of tetrahedron (actually the 4-dimensional equivalent, called a 4-simplex) which is very slightly lopsided so that traveling around them in a cyclic path leaves you with a small displacement. Of course space isn't really made of tetrahedra, since at this scale everything becomes very fuzzy and Heisenbergy. But Regge calculus does approximate the Riemannian curvature in an actually very nice way, though as far as I can see LQG is still a model without a mechanism: it does away with gravitons and it's hard (at least for me) to see how ordinary particles will produce gravity without them.
So a spinfoam is a collection of geometric objects glued together edge to edge. The spinfoam represents the evolution of a network of quantum states on a three-dimensional hypersurface. The advantage of quantizing spacetime is that it eliminates what Rovelli calls ultraviolet and infrared gravitational divergences: the IR ones are cut off by a positive Λ, and the UV ones by the Plank-scale quantization of geometry.
Of course there's no actual evidence that space is configured this way; it's only a mathematical model, and a complicated one at that, and LQG is very much a work in progress. But even though it may not be as exciting and exotic as all those 11-dimensional superstrings and hyperdimensional branes smashing into each other, maybe it's a bit more realistic.
nov 27, 2016