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

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

143 submissions , 120 unreviewed
3,899 questions , 1,377 unanswered
4,837 answers , 20,502 comments
1,470 users with positive rep
495 active unimported users
More ...

  The dimension of the subspace of flat spin connections

+ 4 like - 0 dislike
28 views

I am interested in the the flat spin connections in a Riemann spacetime of dimension 4. They appear in the context of the frame formalism of metric gravity theories. I believe that they form a subspace, homologous to the algebra of the Lorentz group, within the affine space of spin connections. The dimension of the subspace would be then 6. However, I cannot prove this idea myself. Does anybody can help (or correct) me with this? Sorry if the wording of the question is poor, but I am not a specialist in differential geometry.

This post imported from StackExchange MathOverflow at 2017-08-11 12:44 (UTC), posted by SE-user asierzm
asked Jul 11 in Theoretical Physics by asierzm (20 points) [ no revision ]
retagged Aug 11
In what sense do you use the word "homologous" here? Also, I don't see why the space of flat spin connections should be finite dimensional. Given any such flat connection, a local frame rotation will give you another connection, yet still flat. This already gives you an infinite dimensional space. Or I just don't understand what you are asking.

This post imported from StackExchange MathOverflow at 2017-08-11 12:44 (UTC), posted by SE-user Igor Khavkine
I am here referring to spin connections of pseudo Riemann spacetimes. The affine space of spin connections has dimension 24 (for a Riemann spacetime of dimension 4). The flat spin connections form a subspace. You are right that a local frame rotation or boost (for the Lorentz group) gives another flat spin connection. But, for affine spaces of spin connections there exist also translations of the original flat spin connection. The translations are done by tensors generated by the algebra of the Lorentz group (dimension 6). The subspace of the affine translations is what I am interested about.

This post imported from StackExchange MathOverflow at 2017-08-11 12:44 (UTC), posted by SE-user asierzm
The space of connections is infinite dimensional. If you do not impose any further restrictions on topology of your manifold, the space of flat connections is also infinite dimensional, even modulo gauge group.

This post imported from StackExchange MathOverflow at 2017-08-11 12:44 (UTC), posted by SE-user Misha
The spin connections is defined from the affine connection in a spinor bundle on a pseudo-Riemann manifold (signature 2). In tensor notation, a general spin connection on a dimension 4 Riemann manifold (spacetime) locally has 24 independent components. The dimension I refer to is defined locally like the dimension of the affine space containing it. But how much of these components are necessary for a general flat spin connection?

This post imported from StackExchange MathOverflow at 2017-08-11 12:44 (UTC), posted by SE-user asierzm

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$\hbar\varnothing$sicsOverflow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...