• 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 , 22,741 comments
1,470 users with positive rep
818 active unimported users
More ...

  Division algebras and Spinors

+ 4 like - 0 dislike

I was reading the paper "Division Algebras and Supersymmetry I " By John Baez and John Huerta. 

In this paper he constructs representations of Spinors in space time with signature (d+1,1)(d=1,2,4,8) using the Reals, Complex, Quarterniions and Octernions respectively. 

In his paper(pdf) on page 6 he says, 

  • When K = R, S+ ∼ S− is the Majorana spinor representation of Spin(2, 1).
  •  When K = C, S+ ∼ S− is the Majorana spinor representation of Spin(3, 1).
  • When K = H, S+ and S− are the Weyl spinor representations of Spin(5, 1).
  • When K = O, S+ and S− are the Majorana–Weyl spinor representations of Spin(9, 1).

Please refer to the paper for further details about their construction. 

Is this construction Exhaustive? How does one construct the other possible representations for instance the Weyl representations in (3,1) using division algebras?

Also I request recommendations for further reading material on this subject, which is suitable for a student of physics. 

asked Sep 8, 2014 in Theoretical Physics by Prathyush (705 points) [ revision history ]
edited Sep 8, 2014 by Prathyush

1 Answer

+ 5 like - 0 dislike

The normed division algebras are sporadic and exist only in dimensions $d=1,2,4,8$. As a consequence, the spinor representations related to these are also only sporadic, for a few dimensions and forms.  One can be happy for every such sporadic correspondence found, but there is no general machinery that would guarantee, e.g., a  Weyl representations of Spin(3,1). One would have to try to construct it and succeed or fail. 

 There is another Baez paper that reviews much of sporadic niceties potentially relevant for physics:

J. C. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), 145–205. arXiv:math/0105155

answered Sep 8, 2014 by Arnold Neumaier (15,787 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