• 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
5,064 questions , 2,215 unanswered
5,347 answers , 22,734 comments
1,470 users with positive rep
818 active unimported users
More ...

  Representation theory of SO(1, d-1) and O(1, d-1)

+ 1 like - 0 dislike

Under a Lorentz transformation, a spinor living in \(d\) dimensions transforms as

\(\psi (x) \rightarrow \psi'(x') = e^{\frac{1}{2} \lambda^{\mu \nu} \Sigma_{\mu \nu}} \psi (x)\)

up to some numerical factors in the exponential from convention. This is only true if we're dealing with the orthochronous proper Lorentz transforms \(SO^+ (1,3)\), because the projective spinor representations mean we can just deal with the algebra, and the \(\Sigma_{\mu \nu}\) are representations of \(so (1, d-1)\). How does this change when we want to think about \(P\) and \(T\)? That is, how does one derive the action of elements of \(O (1, d-1)\) on spinors?

The only progress I've made towards understanding this is the coordinate-dependent interpretation put forth in standard texts on QFT, where \(P\)  and \(T\) act with some product of gamma matrices. I was looking for a slightly more general definition.

A related question: is the number of of disconnected components of \(O (1, d-1)\) the same for even and odd dimension?

asked Sep 1, 2018 in Mathematics by Sam Makhoul (25 points) [ no revision ]
recategorized Sep 4, 2018 by Dilaton

For $d \geq 2$, $O(1,d-1)$ has always four connected components.

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