- #1
maverick280857
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Hi,
While reading "Superspace: One Thousand and One Lessons in Supersymmetry" by Gates et al. I came across the following paragraph:
Maybe I haven't understood what exactly they're trying to say here, but
1. Why is the Lorentz Group SL(2, R) instead of SL(2, C)?
2. Why is the two-component spinor real? (Well I guess this follows from question 1).
Any response will be much appreciated!
PS - The book is available freely from http://arxiv.org/abs/hep-th/0108200.
Thanks!
EDIT: I understand that the proper orthochronous Lorentz group [itex]L_{+}^{\uparrow}[/itex] is homomorphic to SL(2, C), i.e. for any [itex]M \in [/itex] SL(2, C), there exists a Lorentz matrix
[tex]\Lambda = \lambda(M) \in L_{+}^{\uparrow}[/tex]
such that [itex]\Lambda(M_1) \Lambda(M_2) = \Lambda(M_1 M_2)[/itex]
and [itex]\Lambda^{-1}(M) = \Lambda(M^{-1})[/itex]
But how does going from 4 to 3 dimensions take us from SL(2, C) to SL(2,R)?
While reading "Superspace: One Thousand and One Lessons in Supersymmetry" by Gates et al. I came across the following paragraph:
Our three-dimensional notation is as follows: In three-dimensional spacetime (with signature -++) the Lorentz group is SL(2, R) (instead of SL(2, C)) and the corresponding fundamental representation acts on a real Majorana two-component spinor [itex]\psi^\alpha = (\psi^+, \psi^-)[/itex].
Maybe I haven't understood what exactly they're trying to say here, but
1. Why is the Lorentz Group SL(2, R) instead of SL(2, C)?
2. Why is the two-component spinor real? (Well I guess this follows from question 1).
Any response will be much appreciated!
PS - The book is available freely from http://arxiv.org/abs/hep-th/0108200.
Thanks!
EDIT: I understand that the proper orthochronous Lorentz group [itex]L_{+}^{\uparrow}[/itex] is homomorphic to SL(2, C), i.e. for any [itex]M \in [/itex] SL(2, C), there exists a Lorentz matrix
[tex]\Lambda = \lambda(M) \in L_{+}^{\uparrow}[/tex]
such that [itex]\Lambda(M_1) \Lambda(M_2) = \Lambda(M_1 M_2)[/itex]
and [itex]\Lambda^{-1}(M) = \Lambda(M^{-1})[/itex]
But how does going from 4 to 3 dimensions take us from SL(2, C) to SL(2,R)?
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