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 is functionally differentiable at…”).

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.

Interesting Paper on the Immirzi Parameter, and more…

In Loop Quantum Gravity, the Immirzi parameter is somewhat troubling…largely because it’s such an odd value.

It was once calculated to be some ugly value like:

Or something like that. Bizarre!

Well, a paper came out that argued that:

Microscopic state counting for a black hole in Loop Quantum Gravity yields a result proportional to horizon area, and inversely proportional to Newton’s constant and the Immirzi parameter. It is argued here that before this result can be compared to the Bekenstein-Hawking entropy of a macroscopic black hole, the scale dependence of both Newton’s constant and the area must be accounted for. The two entropies could then agree for any value of the Immirzi parameter, if a certain renormalization property holds.

(Emphasis added). Fascinating!

The paper is called “Renormalization and black hole entropy in Loop Quantum Gravity” by Ted Jacobson.

There was a second paper that crossed my eye because of something that stuck out from Carlip’s Class. This student with a British accent asked why not place the universe within a box? We do it for the Hydrogen atom, among other things, so why not do it for quantum gravity?

Well, the obvious answer is the philosophical problems with this in the context of classical general relativity. However, a paper has come out about this very subject! It’s very exciting purely from the nostalgic feeling the abstract conjures:

The curvature perturbation in a box by David H. Lyth

The stochastic properties of cosmological perturbations are best defined through the Fourier expansion in a finite box. I discuss the reasons for that with reference the curvature perturbation, and explore some issues arising from it.

Next on the reading List is a lengthy piece dealing with particle propagators in arbitrary backgrounds.

This piece fascinates me partially because it comforts my inner general relativist in dealing with background independency (or more spacifically, a sort of pseudo-background independency) for quantum theory.

Particle propagation in non-trivial backgrounds: a quantum field theory approach by Daniel Arteaga

The basic aim of the thesis is the study of the propagation of particles and quasiparticles in non-trivial backgrounds from the quantum field theory point of view. By “non-trivial background” we mean either a non-vacuum state in Minkowski spacetime or an arbitrary state in a curved spacetime. Starting with the case of a flat spacetime, the basic properties of the particle and quasiparticle propagation are analyzed using two different methods other than the conventional mean-field-based techniques: on the one hand, the quantum state corresponding to the quasiparticle excitation is explicitly constructed; on the other hand, the spectral representation of the two-point propagators is analyzed. Both methods lead to the same results: the energy and decay rate of the quasiparticles are determined by the real and imaginary parts of the retarded self-energy respectively. These general results are applied to two particular quantum systems: first, a scalar particle immersed in a thermal graviton bath; second, a simplified atomic model, seizing the opportunity to connect with other statistical and first-quantized approaches. In the second part of the thesis the results are extended to curved spacetime. Working with a quasilocal quasiparticle concept the flat-spacetime results are recovered. In cosmology, within the adiabatic approximation, it is possible to go beyond the flat spacetime results and find additional effects due to the universe expansion. The cosmologically-induced effects are analyzed, obtaining that there might be an additional contribution to the particle decay due to the universe expansion. In the de Sitter case, this additional contribution coincides with the decay rate in a thermal bath in a flat spacetime at the corresponding de Sitter temperature.

Fascinating papers, all of them well worth reading.