Thomas Thiemann’s “Modern Canonical Quantum General Relativity”
Disclaimer: I’m told that many of the books that I love are considered dry, boring, long, and unexciting in the eyes of others (I certainly don’t think they are in the least!). So my opinions on this book as fascinating ought to be taken in this light. (End of disclaimer.)
I’m not sure how long ago Thomas Thiemann published his tome Modern Canonical Quantum General Relativity but I picked up a copy the other day when I was getting paper for my printer at the UC Davis bookstore. Please let me emphasize this is a technical monograph with the intended audience being mathematically savvy individuals, not an introductory text for the lazy layman (even the hyperactive layman may have some difficulty).
The forward is written by Christopher Isham, who gives a rather interesting personal background on his personal interest in quantum general relativity stemming from his encounter in 1969 with a researcher named Abdus Salam.
The approach taken was perturbative quantization, which Isham notes:
“The perturbative quantum field theory schemes foundered on interactable ultra-violet divergences and gave way to super-gravity — the super-symmetric extension of standard general relativity. In spute of initial optimism, this approach succumbed to the same disease and was eventually replaced by the far more ambitious superstring theories” (Forward, paragraph 2).
Isham continues to explain his interest in canonical quantum gravity and in particular the Wheeler-DeWitt equation. The Wheeler-DeWitt equation is the Hamiltonian constraint in general relativity, it is the constraint corresponding to the time re-parametrization invariance gauge. It should be unsurprising that it is nightmarishly complicated and extremely difficult to solve in its traditional form using the metric tensor as the canonical position variable.
Thomas Thiemann pioneered the Master Constraint programme and did a tremendous amount of work on the Hamiltonian constraint that deserves the utmost respect (see these technical papers for a “small” taste: “Quantum Spin Dynamics“,
“Quantum Spin Dynamics II. The Kernel of the Wheeler-DeWitt Constraint Operator“,
“QSD III : Quantum Constraint Algebra and Physical Scalar Product in Quantum General Relativity“,
“QSD IV : 2+1 Euclidean Quantum Gravity as a model to test 3+1 Lorentzian Quantum Gravity“,
“QSD V : Quantum Gravity as the Natural Regulator of Matter Quantum Field Theories“,
“QSD VI : Quantum Poincaré Algebra and a Quantum Positivity of Energy Theorem for Canonical Quantum Gravity“,
“Quantum Spin Dynamics VIII. The Master Constraint“,
“Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework“). He covers the quantum Wheeler-DeWitt equation in chapter 10 section 3 (”Derivation of the Hamiltonian Constraint Operator”) of his book.
The book is very well written (for a physics monograph). Even in the “Outline of the book”, Thiemann demonstrates a superior command of math. In this section of his book, he outlines and describes various other approaches to quantum gravity citing sources for the interested reader to read.
This is something I always appreciate in books.
Unfortunately, I am sick, so I have had plenty of time to read the book! I am however a slow reader because I’m also taking notes as I’m reading. It’s a force of habit I’ve had since I’ve started reading. This book is an excellent introduction to modern canonical quantum gravity, that is to say Loop Quantum Gravity.
It covers the steps taken to introduce the subject matter. He begins by introducing the (ADM) Hamiltonian formulation of general relativity and he does it with unparalleled clarity. Thiemann continues on this subject matter to deal with the gauge symmetries and their constraints and the geometric interpretation of these constraints. He found the Legendre transform and the gauge constraints, then gave a mathematical definition for functional differentiability (more precisely that “…a functional 
I thought “Uhoh, Thiemann’s decided to go purely mathematical with his presentation!” But this initial fear was unfounded, I am relieved to report. Thiemann provides definitions from pure math on various mathematical objects that are relevant to the topic of each chapter and future chapters. This is actually something I appreciate, it allows the book to be more self-contained.
The next topic Thiemann tackles are the philosophical issues like the problem with time, locality in a relational formalism of general relativity, and various interpretations of quantum theory. This concludes chapter 2.
So far I’m on chapter 4, but I’m fascinated with the book. It is of a high quality and I recommend it for anyone remotely interested in modern developments of canonical quantum gravity…provided the reader has knowledge of classical general relativity and quantum theory. I can’t stress how well written this book is, it certainly makes up for the lack of such high quality books on the subject (there are how many others? One? How many are there on String theory? A thousand or two?).
And now back to studying for my classes and preparing for finals.
