ADM Hamiltonian formalism and Quantum gravity

Question

Is there a Hamiltonian reformulation of gravity ? If so using the usual Quantization scheme we cannot quantize gravity ??

In terms of a Gauge Theory with the potential $A_{\mu}^{i}$ how can we get the Schroedinger equation ?

Answer 1

1. Short Answer. You're interested, probably, in the ADM formalism and its quantization. This "has happened" (in the sense we can write down the equations) but "cannot be solved" (in the sense we don't know how to solve them!).

The fields taken are the ("spatial") metric tensor and its conjugate momentum is, more or less, its time derivative. This might seem strange at first, but what happens in the canonical formalism is we take a foliation of spacetime: i.e., we split spacetime into space+time.

There is an extensive literature dedicated to this subject. There are several different Hamiltonian formulations of classical GR, but usually people use the ADM formalism (named after its founders Arnowitt, Deser, and Misner). There is another approach, Loop Quantum Gravity, which more closely resembles other common quantum field theories.

2. Know Classical GR First! If you don't know classical General Relativity, you need to learn that first. You should study it first classically, then quantum mechanically. It's hard enough at the classical level!

You should be able to read through Misner, Thorne, and Wheeler's Gravitation or Wald's General Relativity. If you haven't started learning relativity, I highly recommend Bernard Schultz's A First Course in General Relativity first, then Poisson's A Toolkit for Relativity (or his lecture notes linked below), and then read through either Misner, Thorne, and Wheeler or Wald (or both!).

3. What Quantization Scheme to Use? Well, in the 80 years people have pursued quantum gravity, they have tried every quantization scheme you could think of!

To get a sense of how complicated quantum gravity is (and an overview of the different approached), I recommend reading Rovelli's "Notes for a brief history of quantum gravity" arXiv:gr-qc/0006061, which is an easy read.

Just a few remarks about quantization schemes: quantization is always problematical on its own, in the sense that presumably nature is already quantum and formulating a procedure to go from classical to quantum is nonsensical. This is discussed in many articles, I'll give a few free good references, e.g., S Twareque Ali and Miroslav Engliš' "Quantization Methods: A Guide for Physicists and Analysts" (arXiv:math-ph/0405065) and MJ Gotay's " Obstructions to Quantization" (arXiv:math-ph/9809011).

4. Reading List on Canonical Quantum Gravity. I'll just give a few references on canonical gravity, both classical and quantum. These I have found useful:

1. Claus Kiefer's Quantum Gravity. Just published in its third edition, and a great overview of the field.
2. Arnowitt, Deser, and Misner's "The Dynamics of General Relativity" arXiv:gr-qc/0405109 first describes the (classical) Hamiltonian formalism.
3. Eric Poisson's "Advanced General Relativity" lecture notes (the basis for his book) describes the mathematics and physical intuition underpinning the foliation of spacetime into space + time. Great preparation for more advanced relativity!
4. Yvon Choquet-Bruhat's General Relativity and the Einstein Equations (2009) discusses the conformal treatment of canonical GR quite beautifully, which is not discussed in any other textbooks I've found.
5. Steve Carlip's Quantum Gravity in 2+1 Dimensions (2003), beautifully discusses many aspects of quantum gravity which are exemplified in 2+1 dimensional models. (Don't dismiss 2+1d gravity because it's "trivial", it actually allows us to gaze at the problems underpinning 3+1d quantum gravity.)
6. Martin Bojowald's Canonical Gravity and Applications: Cosmology, Black Holes, and Quantum Gravity (2011). If you get only one book, get this one! It discusses exactly what you are looking for at the advanced undergraduate level.

5. Quantum Field Theory. You also need to know how to quantize fields, specifically constrained systems. This is a tricky subject, and there is no single book I'd recommend because each book discusses one aspect or one approach really well.

The usual text on quantizing constrained systems is Henneaux and Teiteilboim's Quantization of Gauge Systems (1994). It's not a good text, but it's the only text on the subject. (I had to work through the first few chapters by re-writing it in a mathematician's style, in the sense of "Each main idea gets a numbered item on a list, and elaborations on ideas get sub-items. Some items are theorems, others are definitions. A proof is a collection of steps.") I have heard of another book, which might serve well as preparation for Henneaux and Teiteilboim (but not a replacement!): Lev V Prokhorov and Sergei V Shabanov's Hamiltonian Mechanics of Gauge Systems (2011).

If you don't know quantum field theory, I recommend working your way through Brian Hatfield's Quantum Field Theory of Point Particles and Strings (1992) since it's the only textbook which actually discusses the functional Schrodinger equation, gives an "honest enough" account of the path integral, and shows all the calculations/steps in derivations. Caveat: the first edition of Hatfield, though cheaper secondhand, is full of typos...I took this as an exercise to double check every calculation in at least three different ways. The second edition has fixed the typos, or so I hear. (Usually Peskin and Schroeder's Introduction to Quantum Field Theory (1995) is recommended. This is a good text, but I found it slow reading.)

Personally, I preferred reading Ticciati's Quantum Field Theory for Mathematicians (2008) followed up by Edson de Faria and Welington de Melo's Mathematical Aspects of Quantum Field Theory (2010).

(Addendum: you might want to eventually work your way through the first volume of Weinberg's The Quantum Theory of Fields, since it uses the same metric signature as used in the ADM decomposition.)

One last book I'd recommend reading is NMJ Woodhouse's Geometric Quantization (1997), because it discusses an approach to quantization that isn't discussed in other texts.

6. Canonical Quantum Gravity Today? The canonical approach has led to Loop Quantum Gravity, which is actively researched today. There are a number of good books on this subject (Rovelli wrote one, Thiemann wrote another, etc.).