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Gamma matrices 
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#1
Mar1911, 07:15 AM

P: 13

I'm doing a course which assumes knowledge of Group Theory  unfortunately I don't have very much.
Can someone please explain this statement to me (particularly the bits in bold): "there is only one nontrivial irreducible representation of the Cliford algebra, up to conjugacy" FYI The Clifford algebra is just the the relationship between gamma matrices: [tex][\gamma_\mu,\gamma_\nu]=2\eta_{\nu\mu}[/tex] where [..] is the anticommutator rather than the commutator 


#2
Mar1911, 09:49 AM

Sci Advisor
Thanks
P: 4,160

The γ_{μ}'s are initially understood to be abstract objects satisfying the equation you've written. Representation means we assign a matrix to each γ_{μ} and interpret the equation as a matrix equation, where the RHS contains the identity matrix I. Conjugacy refers to the fact that for any matrix M, if γ_{μ} is a solution then the conjugate set γ_{μ}' = M γ_{μ} M^{1} is also a solution. Irreducible means that the γ_{μ}'s are not simultaneously block diagonal, nor conjugate to a set that is block diagonal.
Putting that all together, it means that you can write down any set of γ matrices you can think of that solve the equation, and be assured that any other set γ' you might have chosen instead is related to your set by a conjugation. 


#3
Mar1911, 09:32 PM

P: 13




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