Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,786 comments
1,470 users with positive rep
820 active unimported users
More ...

  ADM Hamiltonian formalism and Quantum gravity

+ 2 like - 0 dislike
1172 views

is there a Hamiltonian reformultion of gravity ?=? if so if we use the usual Quantization scheme we can not we quantizy the gravity ??

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

This post imported from StackExchange Physics at 2014-04-11 15:51 (UCT), posted by SE-user Jose Javier Garcia
asked Aug 17, 2012 in Theoretical Physics by Jose Javier Garcia (70 points) [ no revision ]

1 Answer

+ 4 like - 0 dislike

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 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, 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).

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.).

This post imported from StackExchange Physics at 2014-04-11 15:51 (UCT), posted by SE-user Alex Nelson
answered Aug 17, 2012 by Alex Nelson (100 points) [ no revision ]
Disclaimer: I am leaving on a trip in 15 minutes, and won't be able to answer any follow ups for a few days (or weeks) depending on when I'll get internet access again!

This post imported from StackExchange Physics at 2014-04-11 15:51 (UCT), posted by SE-user Alex Nelson
i gave realativity in college we had RAY D'INVERNO 'introduction to einstein relativity ' but perhaps this is not enough.. thanks

This post imported from StackExchange Physics at 2014-04-11 15:51 (UCT), posted by SE-user Jose Javier Garcia

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysi$\varnothing$sOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...