• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,345 answers , 22,719 comments
1,470 users with positive rep
818 active unimported users
More ...

  Dirac bracket and second class constraints in first-order gravity formalism

+ 6 like - 0 dislike

In the first order formulation of general relativity, the frame field $e_{\mu}^a$ and $\mathrm{SO}(3,1)$ spin connection $\omega_{\mu c}^b$ are independent variables. In the Hamiltonian formulation of this theory, one finds that there are second-class constraints.

According to Dirac, the way to deal with these second-class constraints when quantising is to first define the Dirac bracket, which is essentially a new Poisson bracket that 'respects the constraints', in the sense that the Dirac bracket of any two constraints is another constraint, and then proceed with the quantisation procedure.

After looking a little bit in the literature, I have been unable to find any paper that actually attempts to construct the Dirac bracket for the first-order formulation of general relativity. And indeed it seems people go to lengths to reformulate gravity so that it doesn't have any second class constraints from the get-go (e.g. using the Ashtekar variables). My question is, has the Dirac bracket for first-order gravity been constructed? If so, a reference would be great.

This post imported from StackExchange Physics at 2014-08-12 09:33 (UCT), posted by SE-user Steven
asked Aug 12, 2014 in Theoretical Physics by Steven (45 points) [ no revision ]
retagged Aug 12, 2014

1 Answer

+ 2 like - 0 dislike

In the particle-physics-oriented part of the theoretical physics community, it was becoming increasingly clear that the Dirac bracket is at most a complicated piece of formalism that isn't able to solve any real physical problems and make theories well-defined or finite or renormalizable etc.

So the people who are playing with such tools applied to quantized gravity are still close to the loop quantum gravity community. The Dirac bracket quantization for the Ashtekar-Barbero form of gravity has been attempted in papers by Sergei Alexandrov, e.g.


The paper has 77 respectable followups but I think that none of them really uses the results in any meaningful way.

This post imported from StackExchange Physics at 2014-08-12 09:33 (UCT), posted by SE-user Luboš Motl
answered Aug 12, 2014 by Luboš Motl (10,278 points) [ no revision ]

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:
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