# Gamma matrices

by countable
Tags: gamma, matrices
 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 non-trivial irreducible representation of the Cliford algebra, up to conjugacy" FYI The Clifford algebra is just the the relationship between gamma matrices: $$[\gamma_\mu,\gamma_\nu]=2\eta_{\nu\mu}$$ where [..] is the anticommutator rather than the commutator
 Sci Advisor P: 3,451 The γμ's are initially understood to be abstract objects satisfying the equation you've written. Representation means we assign a matrix to each γμ 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 γμ is a solution then the conjugate set γμ' = M γμ M-1 is also a solution. Irreducible means that the γμ'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.
P: 13
 Quote by Bill_K The γμ's are initially understood to be abstract objects satisfying the equation you've written. Representation means we assign a matrix to each γμ 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 γμ is a solution then the conjugate set γμ' = M γμ M-1 is also a solution. Irreducible means that the γμ'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.
thanks for the info Bill:)

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