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Division algebras and Spinors

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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 (695 points) [ revision history ]
edited Sep 8, 2014 by Prathyush

1 Answer

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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 (12,570 points) [ no revision ]

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