As it is said in the preface of the book, Loop quantum gravity has emerged

as a possible avenue towards the quantization of general relativity. This is

one of the main streams to attack this outstanding problem in mathematical physics,

although one has to recognize that it has had less impact than string and superstring

theories.

Before starting to read the book, I was delighted to have in my hands what

it seemed to be a leisurely introduction to a topic of such novelty, in

which my background as differential geometer and (mathematical) gauge

theorist could be well suited. The first chapters give overall

introductions in differential geometry, general relativity, semi-riemannian

geometry, always keeping an informal style. So informal that at some points

in touches the incorrectness. However, when reading these chapters, some of

the times even skipping parts (so basic for a mathematician), this was

to be forgiven. Around the second third of the book, physical

(and more interesting) ideas enter the discussion (Yang-Mills theories, quantum mechanics,

quantum field theory), and then the book swaps to the usual "physics

jargon", so extraneous to a mathematician. No much

intention not to lose people not already familiar with the theory

is shown in the text, accompanied by a lack of motivation when introducing physical concepts.

Moreover, not even for the experts the material is

going to be of much use, since details are skipped once and again. The third

part of the book (loop general relativity, loop quantum cosmology) is basically

a review of very technical material. This is where the book touches the topic

announced in the title, but by then all arguments follow a physics line of

reasoning: no rigorous proofs, renormalizations, divergences, etc.

Again, I have gone through a text in mathematical

physics on which my expectations have not been fulfilled. Maybe next time.