Loop quantum gravity has emerged as a possible avenue towards the
quantization of general relativity. This book is an introduction to this
topic, which is at the level of a graduate student in Theoretical Physics.
The book covers: general relativity, hamiltonian mechanics, Yang-Mills
theories, quantum mechanics, quantum field theories and loop quantum
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
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.